Pre-Algebra, Student Edition

Authors Malloy • Molix-Bailey Price • Willard (bkdg)Created by Michael Trott with Mathematica. From Graphica 1, Copyrig...

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Authors Malloy • Molix-Bailey Price • Willard

(bkdg)Created by Michael Trott with Mathematica. From Graphica 1, Copyright ©1999 Wolfram Media, Inc., (b)Richard Cummins/SuperStock

About the Cover The Space Needle, designed as a symbol of the 1962 World’s Fair, is now the most popular tourist destination in Seattle, Washington. The Needle is 605 feet tall, which made it the tallest building west of the Mississippi River when it was built. The Needle can withstand a wind velocity of 200 miles per hour. It has also withstood several earthquakes, including one in 2001 that measured 6.8 on the Richter scale. You’ll learn more about how mathematics and architecture are related in Chapters 10 and 11. About the Graphics Created with Mathematica. A rectangular array of circles is progressively enlarged and rotated randomly. For more information, and for programs to construct such graphics, see: www.wolfram.com.

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Microsoft ® Excel® is a registered trademark of Microsoft Corporation in the United States and other countries. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-873818-0 MHID: 0-07-873818-0 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 043/071 16 15 14 13 12 11 10 09 08 07

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Start Smart Unit 1 Algebra and Integers 1

The Tools of Algebra

2

Integers

3

Equations

Unit 2 Algebra and Rational Numbers 4

Factors and Fractions

5

Rational Numbers

Unit 3 Linear Equations, Inequalities, and Functions 6

Ratio, Proportion, and Percent

7

Functions and Graphing

8

Equations and Inequalities

Unit 4 Applying Algebra to Geometry 9

Real Numbers and Right Triangles

10

Two-Dimensional Figures

11

Three-Dimensional Figures

Unit 5 Extending Algebra to Statistics and Polynomials 12

More Statistics and Probability

13

Polynomials and Nonlinear Functions

iii

Authors

Carol E. Malloy, Ph.D. Associate Professor University of North Carolina at Chapel Hill Chapel Hill, North Carolina

Rhonda J. Molix-Bailey Mathematics Consultant Mathematics by Design DeSoto, Texas

Jack Price, Ed.D. Professor Emeritus California State Polytechnic University Pomona, California

Teri Willard, Ed.D. Assistant Professor Department of Mathematics Central Washington University Ellensburg, Washington

Contributing Author Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas

iv Aaron Haupt

Meet the Authors at pre-alg.com

Consultants Glencoe/McGraw-Hill wishes to thank the following professionals for their feedback. They were instrumental in providing valuable input toward the development of this program.

Differentiated Instruction

Gifted and Talented

Mathematical Fluency

Nancy Frey, Ph.D. Associate Professor of Literacy San Diego State University San Diego, California

Ed Zaccaro Author Mathematics and science books for gifted children Bellevue, Iowa

Jason Mutford Mathematics Instructor Coxsackie-Athens Central School District Coxsackie, New York

English Language Learners Mary Avalos, Ph.D. Assistant Chair, Teaching and Learning Assistant Research Professor University of Miami, School of Education Coral Gables, Florida Jana Echevarria, Ph.D. Professor, College of Education California State University, Long Beach Long Beach, California Josefina V. Tinajero, Ph.D. Dean, College of Education The University of Texas at El Paso El Paso, Texas

Graphing Calculator Ruth M. Casey Mathematics Teacher Department Chair Anderson County High School Lawrenceburg, Kentucky Jerry Cummins Former President National Council of Supervisors of Mathematics Western Springs, Illinois

Learning Disabilities Kate Garnett, Ph.D. Chairperson, Coordinator Learning Disabilities School of Education Department of Special Education Hunter College, CUNY New York, New York

Pre-AP Dixie Ross AP Calculus Teacher Pflugerville High School Pflugerville, Texas

Reading and Vocabulary Douglas Fisher, Ph.D. Director of Professional Development and Professor City Heights Educational Collaborative San Diego State University San Diego, California Lynn T. Havens Director of Project CRISS Kalispell School District Kalispell, Montana

v

Teacher Reviewers

Each Teacher Reviewer reviewed at least two chapters of the Student Edition, giving feedback and suggestions for improving the effectiveness of the mathematics instruction. Chrissy Aldridge

Peter K. Christensen

Matt Gowdy

Teacher Charlotte Latin School Charlotte, North Carolina

Mathematics/AP Teacher Central High School Macon, Georgia

Mathematics Teacher Grimsley High School Greensboro, North Carolina

Harriette Neely Baker

Rebecca Claiborne

Wendy Hancuff

Mathematics Teacher South Mecklenburg High School Charlotte, North Carolina

Mathematics Department Chairperson George Washington Carver High School Columbus, Georgia

Teacher Jack Britt High School Fayetteville, North Carolina

Danny L. Barnes, NBCT

Laura Crook

Ernest A. Hoke Jr.

Mathematics Teacher Speight Middle School Stantonsburg, North Carolina

Mathematics Department Chair Middle Creek High School Apex, North Carolina

Mathematics Teacher E.B. Aycock Middle School Greenville, North Carolina

Aimee Barrette

Dayl F. Cutts

Carol B. Huss

Special Education Teacher Sedgefield Middle School Charlotte, North Carolina

Teacher Northwest Guilford High School Greensboro, North Carolina

Mathematics Teacher Independence High School Charlotte, North Carolina

Karen J. Blackert

Angela S. Davis

Deborah Ivy

Mathematics Teacher Myers Park High School Charlotte, North Carolina

Mathematics Teacher Bishop Spaugh Community Academy Charlotte, North Carolina

Mathematics Teacher Marie G. Davis Middle School Charlotte, North Carolina

Patricia R. Blackwell

Sheri Dunn-Ulm

Lynda B. (Lucy) Kay

Mathematics Department Chair East Mecklenburg High School Charlotte, North Carolina

Teacher Bainbridge High School Bainbridge, Georgia

Mathematics Department Chair Martin Middle School Raleigh, North Carolina

Rebecca B. Caison

Susan M. Fritsch

Julia Kolb

Mathematics Teacher Walter M. Williams High School Burlington, North Carolina

Mathematics Teacher, NBCT David W. Butler High School Matthews, North Carolina

Mathematics Teacher/Department Chair Leesville Road High School Raleigh, North Carolina

Myra Cannon

Dr. Jesse R. Gassaway

M. Kathleen Kroh

Mathematics Department Chair East Davidson High School Thomasville, North Carolina

Teacher Northwest Guilford Middle School Greensboro, North Carolina

Mathematics Teacher Z.B. Vance High School Charlotte, North Carolina

vi

Tosha S. Lamar

Susan M. Peeples

Mathematics Instructor Phoenix High School Lawrenceville, Georgia

Retired 8th Grade Mathematics Teacher Richland School District Two Columbia, South Carolina

Mathematics Teacher Clarke Central High School Athens, Georgia

Kay S. Laster

Carolyn G. Randolph

Elizabeth Webb

8th Grade Pre-Algebra/Algebra Teacher Rockingham County Middle School Reidsville, North Carolina

Mathematics Department Chair Kendrick High School Columbus, Georgia

Mathematics Department Chair Myers Park High School Charlotte, North Carolina

Joyce M. Lee

Tracey Shaw

Jack Whittemore

Lead Mathematics Teacher National Teachers Teaching with Technology Instructor George Washington Carver High School Columbus, Georgia

Mathematics Teacher Chatham Central High School Bear Creek, North Carolina

C & I Resource Teacher Charlotte-Mecklenburg Schools Charlotte, North Carolina

Marjorie Smith

Angela Whittington

Mathematics Teacher Eastern Randolph High School Ramseur, North Carolina

Mathematics Teacher North Forsyth High School Winston-Salem, North Carolina

McCoy Smith, III

Kentucky Consultants

Mathematics Department Chair Sedgefield Middle School Charlotte, North Carolina

Amy Adams Cash

Susan Marshall Mathematics Chairperson Kernodle Middle School Greensboro, North Carolina

Alice D. McLean Mathematics Coach West Charlotte High School Charlotte, North Carolina

Portia Mouton Mathematics Teacher Westside High School Macon, Georgia

Elaine Pappas Mathematics Department Chair Cedar Shoals High School Athens, Georgia

Bridget Sullivan 8th Grade Mathematics Teacher Northeast Middle School Charlotte, North Carolina

Marilyn R. Thompson Geometry/Mathematics Vertical Team Consultant Charlotte-Mecklenburg Schools Charlotte, North Carolina

Gwen Turner

Mathematics Educator/Department Chair Bowling Green High School Bowling Green, Kentucky

Susan Hack, NBCT Mathematics Teacher Oldham County High School Buckner, Kentucky

Kimberly L. Henderson Hockney Mathematics Educator Larry A. Ryle High School Union, Kentucky

vii

Unit 1

The Tools of Algebra 1-1 Using a Problem-Solving Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Reading Math: Translating Expressions into Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1-2 Numbers and Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1-3 Variables and Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Extend 1-3

Spreadsheet Lab: Expressions and Spreadsheets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Student Toolbox Prerequisite Skills • Get Ready for Chapter 1 25 • Get Ready for the Next Lesson 30, 36, 41, 47, 53, 59

Reading and Writing Mathematics • • • • •

Reading in the Content Area 37 Reading Math 31 Reading Math Tips 32, 33, 49 Vocabulary Link 37, 43, 44 Writing in Math 30, 36, 41, 47, 53, 58, 65

1-4 Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1-5 Variables and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1-6 Ordered Pairs and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Explore 1-7 Algebra Lab: Scatter Plots. . . . . . . . . . . . . . . . . . . . . . . . 60

1-7 Scatter Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Extend 1-7

Graphing Calculator Lab: Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Standardized Test Practice • Griddable 36 • Multiple Choice 30, 36, 41, 47, 51, 53, 59, 66 • Worked Out Example 50

H.O.T. Problems Higher Order Thinking • • • •

Challenge 30, 36, 41, 47, 53, 58, 65 Find the Error 35, 47 Number Sense 65 Open Ended 30, 35, 41, 47, 53, 58, 65 • Reasoning 36 • Select a Technique 53 • Which One Doesn’t Belong? 41

viii Kim Taylor/DK Limited/CORBIS

Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2

Integers

2-1 Integers and Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78 Explore 2-2 Algebra Lab: Adding Integers . . . . . . . . . . . . . . . . . . . . 84

2-2 Adding Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Reading Math: Learning Mathematics Vocabulary . . . . . . . . . . 91 Explore 2-3 Algebra Lab: Subtracting

Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2-3 Subtracting Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Explore 2-4 Algebra Lab: Multiplying Integers . . . . . . . . . . . . . . . . 99

2-4 Multiplying Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Explore 2-5 Algebra Lab: Dividing Integers . . . . . . . . . . . . . . . . . . 105

2-5 Dividing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 ASSESSMENT

Prerequisite Skills

Table of Contents

2-6 The Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Student Toolbox • Get Ready for Chapter 2 77 • Get Ready for the Next Lesson 83, 90, 97, 104, 110

Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Reading and Writing Mathematics

Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

• Reading in the Content Area 86 • Reading Math 91 • Reading Math Tips 78, 86, 100, 106, 113, 114 • Writing in Math 83, 90, 97, 104, 110, 115

Standardized Test Practice • Griddable 83, 104 • Multiple Choice 83, 90, 97, 102, 104, 110, 115 • Worked Out Example 101

H.O.T. Problems Higher Order Thinking • • • • • • •

Challenge 83, 90, 96, 110, 115 Find the Error 96, 115 Number Sense 82, 115 Open Ended 82, 90, 96, 110, 115 Select a Tool 104 Select a Technique 97 Which One Doesn’t Belong? 82

ix Kim Taylor/DK Limited/CORBIS

3

Equations 3-1 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3-2 Simplifying Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . 129 Explore 3-3 Algebra Lab: Solving Equations

Using Algebra Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3-3 Solving Equations by Adding or Subtracting . . . . . . . . . . . . . . . 136 3-4 Solving Equations by Multiplying or Dividing . . . . . . . . . . . . . . 141 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3-5 Solving Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Reading Math: Translating Verbal Problems into Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3-6 Writing Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Student Toolbox

3-7 Sequences and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Prerequisite Skills

3-8 Using Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

• Get Ready for Chapter 3 123 • Get Ready for the Next Lesson 128, 133, 140, 145, 151, 157, 161


Reading in the Content Area 136 Reading Math 152 Reading Math Tips 143, 162 Vocabulary Link 124, 129 Writing in Math 128, 133, 140, 145, 150, 157, 161, 167

Standardized Test Practice • Griddable 140, 151 • Multiple Choice 128, 133, 139, 140, 145, 151, 157, 161, 167 • Worked Out Example 138

H.O.T. Problems Higher Order Thinking • Challenge 128, 133, 140, 145, 150, 156, 161, 166 • Find the Error 128, 132, 156 • Number Sense 145, 157 • Open Ended 128, 132, 140, 145, 150, 156, 161, 166 • Reasoning 167 • Select a Technique 140 • Which One Doesn’t Belong? 132

x John Cancalosi/Stock Boston

Extend 3-8

Spreadsheet Lab: Perimeter and Area . . . . . . . . . . . . 168

ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Unit Factors and Fractions

4

4-1 Powers and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Extend 4-1

Algebra Lab: Base 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4-2 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4-3 Greatest Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4-4 Simplifying Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Reading Math: Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Student Toolbox

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Prerequisite Skills

4-5 Multiplying and Dividing Monomials . . . . . . . . . . . . . . . . . . . . . 203 Extend 4-5

Algebra Lab: A Half-Life Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

• Get Ready for Chapter 4 179 • Get Ready for the Next Lesson 184, 190, 195, 200, 207, 213

4-6 Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209


4-7 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

• • • • •

ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Reading in the Content Area 187 Reading Math 201 Reading Math Tips 180, 181, 205 Vocabulary Link 186 Writing in Math 184, 190, 195, 200, 207, 212, 218

Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Standardized Test Practice • Griddable 184, 218 • Multiple Choice 184, 190, 195, 198, 200, 207, 213, 218 • Worked Out Example 197–198

H.O.T. Problems Higher Order Thinking • Challenge 184, 189, 195, 200, 206, 212, 218 • Find the Error 189, 195 • Number Sense 189, 212, 218 • Open Ended 184, 189, 195, 200, 206, 212, 218 • Reasoning 207, 212 • Select a Tool 184 • Which One Doesn’t Belong? 200

xi John Cancalosi/Stock Boston

5

Rational Numbers 5-1 Writing Fractions as Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5-2 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5-3 Multiplying Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 239 5-4 Dividing Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5-5 Adding and Subtracting Like Fractions . . . . . . . . . . . . . . . . . . . . 250 Reading Math: Factors and Multiples . . . . . . . . . . . . . . . . . . . 255 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5-6 Least Common Multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Extend 5-6

Algebra Lab: Juniper Green . . . . . . . . . . . . . . . . . . . . 262

5-7 Adding and Subtracting Unlike Fractions . . . . . . . . . . . . . . . . . . 263

Student Toolbox Prerequisite Skills • Get Ready for Chapter 5 227 • Get Ready for the Next Lesson 233, 237, 244, 249, 254, 261, 272

5-8 Solving Equations with Rational Numbers . . . . . . . . . . . . . . . . . 268 Explore 5-9 Algebra Lab: Analyzing Data . . . . . . . . . . . . . . . . . . . 273

5-9 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Extend 5-9

Graphing Calculator Lab:

Mean and Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Reading and Writing Mathematics • • • • •

Reading in the Content Area 236 Reading Math 255 Reading Math Tips 234, 235, 245 Vocabulary Link 228 Writing in Math 233, 237, 244, 249, 254, 261, 267, 272, 278

Standardized Test Practice • Griddable 249, 272 • Multiple Choice 233, 237, 244, 249, 254, 261, 272, 277 • Worked Out Example 276–277

H.O.T. Problems Higher Order Thinking • Challenge 233, 237, 244, 249, 253, 261, 266, 272, 278 • Find the Error 243, 253, 267, 271 • Number Sense 233 • Open Ended 232, 237, 243, 249, 253, 261, 266, 278 • Reasoning 237 • Select a Technique 233 • Which One Doesn’t Belong? 272

xii Craig Tuttle/CORBIS


Unit 3 6


6-1 Ratios and Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 6-2 Proportional and Nonproportional Relationships . . . . . . . . . . . 297 Reading Math: Making Comparisons . . . . . . . . . . . . . . . . . . . 301 6-3 Using Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Extend 6-3

Algebra Lab: Capture-Recapture . . . . . . . . . . . . . . . . 307

6-4 Scale Drawings and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 6-5 Fractions, Decimals, and Percents . . . . . . . . . . . . . . . . . . . . . . . 313 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Explore 6-6 Algebra Lab: Using a Percent Model . . . . . . . . . . . . . 320

6-6 Using the Percent Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Student Toolbox Prerequisite Skills • Get Ready for Chapter 6 291 • Get Ready for the Next Lesson 296, 300, 306, 312, 318, 326, 331, 336, 342

6-7 Finding Percents Mentally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6-8 Using Percent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Extend 6-8

Spreadsheet Lab: Compound Interest . . . . . . . . . . . . 337

6-9 Percent of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 6-10 Using Sampling to Predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Reading and Writing Mathematics • • • •

Reading in the Content Area 297 Reading Math 301, 313 Reading Math Tips 298, 303, 334 Writing in Math 296, 300, 306, 312, 317, 326, 331, 336, 342, 347


Standardized Test Practice • Griddable 326 • Multiple Choice 296, 300, 306, 312, 318, 326, 331, 336, 340, 342, 347 • Worked Out Example 339–340

H.O.T. Problems Higher Order Thinking • Challenge 300, 306, 312, 317, 326, 331, 336, 341, 347 • Find the Error 312, 341 • Number Sense 317 • Open Ended 296, 300, 306, 312, 317, 326, 331, 336, 341, 347 • Reasoning 331 • Select a Technique 296 • Which One Doesn’t Belong? 317, 341

xiii Craig Tuttle/CORBIS

7

Functions and Graphing Explore 7-1 Algebra Lab: Input and Output . . . . . . . . . . . . . . . . . 358

7-1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Extend 7-1

Graphing Calculator Lab: Function Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

7-2 Representing Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Reading Math: Language of Functions . . . . . . . . . . . . . . . . . . . 370 7-3 Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 7-4 Constant Rate of Change and Direct Variation . . . . . . . . . . . . . . 376 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Explore 7-5 Algebra Lab: It’s All Downhill . . . . . . . . . . . . . . . . . . . 383

7-5 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384



Reading in the Content Area 359 Reading Math 370 Reading Math Tips 365, 366, 391 Vocabulary Link 360 Writing in Math 363, 369, 375, 381, 389, 394, 402, 406

Standardized Test Practice • Multiple Choice 363, 369, 375, 381, 387, 389, 394, 402, 406 • Worked Out Example 386

H.O.T. Problems Higher Order Thinking • Challenge 363, 369, 375, 388, 394, 402, 406 • Find the Error 388, 394 • Number Sense 369 • Open Ended 362, 369, 375, 381, 388, 394, 401, 406 • Reasoning 362, 381, 389 • Select a Tool 401

xiv Bill Ross/CORBIS

Extend 7-5

Graphing Calculator Lab: Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

7-6 Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Extend 7-6

Graphing Calculator Lab: The Family of Linear Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

7-7 Writing Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 7-8 Prediction Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

8


Explore 8-1 Algebra Lab: Equations with

Variables on Each Side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 8-1 Solving Equations with Variables on Each Side . . . . . . . . . . . . . 420 8-2 Solving Equations with Grouping Symbols . . . . . . . . . . . . . . . . . 424 Reading Math: Meanings of At Most and At Least . . . . . . . . . 429 8-3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 8-4 Solving Inequalities by Adding or Subtracting . . . . . . . . . . . . . . 435 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 8-5 Solving Inequalities by Multiplying or Dividing . . . . . . . . . . . . . 441 8-6 Solving Multi-Step Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . 446 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Student Toolbox Prerequisite Skills • Get Ready for Chapter 8 417 • Get Ready for the Next Lesson 423, 428, 434, 439, 445


Reading in the Content Area 430 Reading Math 429 Reading Math Tips 431 Writing in Math 423, 428, 434, 439, 445, 449

Standardized Test Practice • Griddable 445 • Multiple Choice 423, 428, 434, 439, 444, 445, 450 • Worked Out Example 442

H.O.T. Problems Higher Order Thinking • Challenge 423, 427, 434, 439, 444, 449 • Find the Error 439, 445, 449 • Number Sense 422, 428, 434 • Open Ended 423, 427, 434, 439, 444, 449 • Select a Tool 428 • Select a Technique 428

xv Bill Ross/CORBIS

Unit 4 9

Real Numbers and Right Triangles Explore 9-1 Algebra Lab: Squares and Square Roots . . . . . . . . . . 462

9-1 Squares and Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 9-2 The Real Number System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Reading Math: Learning Geometry Vocabulary. . . . . . . . . . . . 475


Reading and Writing Mathematics • Reading in the Content Area 464 • Reading Math 475, 497 • Reading Math Tips 464, 465, 476, 478, 485, 493 • Writing in Math 468, 474, 480, 490, 496, 502

9-3 Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Explore 9-4 Algebra Lab: The Pythagorean

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 9-4 The Pythagorean Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Extend 9-4

Algebra Lab: Graphing Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

9-5 The Distance Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 9-6 Similar Figures and Indirect Measurement . . . . . . . . . . . . . . . . 497 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Standardized Test Practice

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

• Griddable 488, 490 • Multiple Choice 468, 474, 481, 490, 496, 502 • Worked Out Example 486–487

Standardized Test Practice. . .

H.O.T. Problems Higher Order Thinking • Challenge 468, 473, 480, 490, 495, 502 • Find the Error 490, 502 • Number Sense 468 • Open Ended 468, 473, 480, 490, 495, 502 • Reasoning 468 • Select a Tool 480 • Select a Technique 495 • Which One Doesn’t Belong? 474

xvi CORBIS

. . . . . . . . . . . . . . . . . . . 508

10 Two-Dimensional Figures 10-1 Line and Angle Relationships. . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 10-2 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 10-3 Transformations on the Coordinate Plane . . . . . . . . . . . . . . . . . 524 Extend 10-3 Geometry Lab: Rotations . . . . . . . . . . . . . . . . . . . . . . 531

10-4 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Reading Math: Learning Mathematics Prefixes . . . . . . . . . . . . 538 10-5 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Extend 10-5 Geometry Lab: Tessellations. . . . . . . . . . . . . . . . . . . . 544

10-6 Area: Parallelograms, Triangles, and Trapezoids . . . . . . . . . . . 545 10-7 Circles: Circumference and Area . . . . . . . . . . . . . . . . . . . . . . . . 551 Extend 10-7 Spreadsheet Lab: Circle Graphs

and Spreadsheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 10-8 Area: Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 Extend 10-8 Spreadsheet Lab: Dilations and

Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570



Reading in the Content Area 512 Reading Math 538 Reading Math Tips 513, 515, 525 Vocabulary Link 518 Writing in Math 517, 523, 529, 536, 543, 550, 556, 562

Standardized Test Practice • Griddable 536, 562 • Multiple Choice 517, 523, 527, 530, 536, 543, 550, 556, 562 • Worked Out Example 525

H.O.T. Problems Higher Order Thinking • Challenge 517, 523, 529, 535, 543, 550, 556, 562 • Find the Error 523, 556 • Number Sense 556 • Open Ended 517, 529, 536, 543, 550, 556, 562 • Select a Tool 543 • Which One Doesn’t Belong? 529

xvii CORBIS

11 Three-Dimensional Figures Explore 11-1 Geometry Lab: Building Three-

Dimensional Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 11-1 Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Explore 11-2 Geometry Lab: Volume . . . . . . . . . . . . . . . . . . . . . . . 582

11-2 Volume: Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11-3 Volume: Pyramids, Cones, and Spheres . . . . . . . . . . . . . . . . . . . 589 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Explore 11-4 Geometry Lab: Exploring Lateral

Area and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 11-4 Surface Area: Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . 597


11-5 Surface Area: Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . 602 Explore 11-6 Geometry Lab: Similar Solids . . . . . . . . . . . . . . . . . . . 607

11-6 Similar Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Reading Math: Precision and Accuracy . . . . . . . . . . . . . . . . . . . 614 ASSESSMENT


Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

• • • •

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

Reading in the Content Area 583 Reading Math 614 Vocabulary Link 576, 597 Writing in Math 580, 588, 594, 601, 606, 613

Standardized Test Practice • Griddable 588 • Multiple Choice 581, 586, 588, 594, 601, 606, 613 • Worked Out Example 584–585

H.O.T. Problems Higher Order Thinking • Challenge 580, 588, 594, 601, 606, 613 • Find the Error 587, 612 • Number Sense 612 • Open Ended 580, 587, 593, 601, 605, 612 • Reasoning 580 • Select a Technique 593

xviii Antonio M. Rosario/Getty Images

Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Unit 5 Statistics and 12 More Probability 12-1 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 Extend 12-1 Graphing Calculator Lab: Stem-and-

Leaf Plots and Line Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 12-2 Measures of Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 12-3 Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Extend 12-3 Graphing Calculator Lab: Box-and-

Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 12-4 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 Extend 12-4 Graphing Calculator Lab: Histograms. . . . . . . . . . . . 650

12-5 Selecting an Appropriate Display . . . . . . . . . . . . . . . . . . . . . . . . 651 Extend 12-5 Spreadsheet Lab: Bar Graphs and

Student Toolbox Prerequisite Skills • Get Ready for Chapter 12 625 • Get Ready for the Next Lesson 631, 637, 642, 648, 656, 663, 669, 674, 680

Line Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 12-6 Misleading Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 Reading Math: Dealing with Bias . . . . . . . . . . . . . . . . . . . . . . . 664 12-7 Simple Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 12-8 Counting Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 Extend 12-8 Algebra Lab: Probability and

Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 12-9 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 676 Explore 12-10 Graphing Calculator Lab: Probability

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 12-10 Probability of Composite Events. . . . . . . . . . . . . . . . . . . . . . . . . 682 Extend 12-10 Algebra Lab: Simulations . . . . . . . . . . . . . . . . . . . . . . 688


Reading and Writing Mathematics • Reading in the Content Area 633 • Reading Math 664 • Reading Math Tips 634, 639, 645, 659, 660, 666, 676, 678, 682 • Writing in Math 631, 637, 642, 648, 655, 663, 669, 674, 680, 686

Standardized Test Practice • Griddable 642 • Multiple Choice 631, 637, 642, 648, 654, 656, 663, 669, 674, 680, 687 • Worked Out Example 653

H.O.T. Problems Higher Order Thinking • Challenge 630, 637, 642, 648, 655, 663, 669, 674, 680, 686 • Find the Error 630, 679, 686 • Number Sense 642 • Open Ended 636, 655, 663, 669, 674, 680, 686 • Reasoning 648, 674 • Select a Tool 655

xix

13 Polynomials and Nonlinear Functions Reading Math: Prefixes and Polynomials . . . . . . . . . . . . . . . . . 700 13-1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 Extend 13-1 Algebra Lab: Modeling Polynomials

with Algebra Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 13-2 Adding Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 13-3 Subtracting Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Explore 13-4 Algebra Lab: Modeling

Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 13-4 Multiplying a Polynomial by a Monomial . . . . . . . . . . . . . . . . . . 716 13-5 Linear and Nonlinear Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 720 13-6 Graphing Quadratic and Cubic Equations. . . . . . . . . . . . . . . . . . 726



Reading in the Content Area 701 Reading Math 700 Reading Math Tips 722 Writing in Math 704, 709, 713, 719, 724, 729

Standardized Test Practice • Multiple Choice 704, 709, 713, 718, 719, 725, 730 • Worked Out Example 717

H.O.T. Problems Higher Order Thinking • Challenge 704, 709, 713, 719, 724, 729 • Find the Error 704, 709 • Number Sense 729 • Open Ended 709, 713, 719, 724, 729 • Reasoning 719, 729 • Select a Tool 713 • Which One Doesn’t Belong? 724

xx Age fotostock/SuperStock

Extend 13-6 Graphing Calculator Lab: The Family of

Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

Student Handbook Built-In Workbooks Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761 Mixed Problem Solving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 Extend 10-6: Graphing Geometric Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 807 Preparing for Standardized Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Reference English-Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R1 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R24 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R58 Photo Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R77

xxi Age fotostock/SuperStock

Start Smart

tart mart Be a Better Problem Solver As you gear up to study mathematics, you are probably wondering, “What will I learn this year?” There are three focal points this year:

•

Use basic principles of algebra to analyze and represent proportional and nonproportional linear relationships,

• •

Apply operations with rational numbers, and Use probability and statistics to make predictions.

Along the way, you’ll learn more about problem solving, how to use the tools and language of mathematics, and how to THINK mathematically.

Start Smart 1 Digital Vision/Getty Images

Problem-Solving Strategy: Look for a Pattern R e a l - Wo r l d Problem Solving Tuning musical instruments is a science that also involves mathematics. All music is caused by vibrations. The number of vibrations per second is called the frequency. There is a pattern in the frequencies: the frequency of any note is 1.059 times the frequency of the note that is one-half step lower. There are many problem-solving strategies in mathematics. One of the most common is to look for a pattern. To use this strategy, analyze the first few numbers in a pattern and identify a rule that is used to go from the first number in the pattern to the second, and then to the third, and so on. Then use the rule to extend the pattern and find a solution.

E-MAIL Ramon got an E-mail from his friend Angela. After 10 minutes, he forwarded it to 2 of his friends. After 10 more minutes, those 2 friends forwarded it to 2 more friends. If the message was forwarded like this every 10 minutes, how many people received Angela’s E-mail message after 40 minutes?

1

EXPLORE What are you trying to find? Restate the problem in your own words. Use as few words as possible. You need to find the total number of people who received the E-mail. What other information do you need? Do you think you’ll need any additional information such as a formula or a measurement conversion? You do not need any additional information.

2 Start Smart David Young-Wolff/PhotoEdit

3

PLAN

SOLVE

Organize the data in a table. Look for a pattern in the data. Then extend the pattern. 1,

2, ×2

,

4, ×2

×2

, ×2

To continue the pattern, multiply each term by 2. 4×2=8

Time (min)

People Receiving Message

0

1

10

2

20

4

30

40

Start Smart

2

8 × 2 = 16

So, 1 + 2 + 4 + 8 + 16 or 31 people got the message.

4

CHECK

In the 40th minute, 16 people received the E-mail. Half as many received it each time before that. So, it is reasonable that the total will be less than 16 × 2 or 32.

Practice Solve each problem by looking for a pattern. 1. List the first five common multiples of 3, 4, and 6. Write an expression to

describe all common multiples of 3, 4, and 6. 2. Two workers can make two chairs in two days. How many chairs can

8 workers working at the same rate make in 20 days? 3. Courtney travels south on her bicycle riding

8 miles per hour. One hour later, her friend Horacio starts riding his bicycle from the same location. If he travels south at 10 miles per hour, how long will it take him to catch Courtney? 4. What is the perimeter of the twelfth figure?

Figure 1 Perimeter ⫽ 6

Figure 2 Perimeter ⫽8

Figure 3 Perimeter ⫽10

5. A ball bounces back 0.6 of its height on

every bounce. If a ball is dropped from 200 feet, how high does it bounce on the fifth bounce? Round to the nearest tenth. 6. A forest fire spread to 41 acres in 10 hours.

Each hour the fire spread to four more acres than the previous hour. How many acres were consumed during each hour of the fire?

Problem-Solving Strategy: Look for a Pattern John Evans

3

Problem-Solving Strategy: Make a Table or List One strategy for solving problems is to make a table or list. A table allows you to organize information in an understandable way. When you make a list, use an organized approach so you do not leave out important items.

A fruit machine accepts dollars, and each piece of fruit costs 65 cents. If the machine gives only nickels, dimes, and quarters, what combinations of those coins are possible as change for a dollar? The machine will give back $1.00 - $0.65 or 35 cents in change in a combination of nickels, dimes, and quarters. Make a table showing different combinations of nickels, dimes, and quarters that total 35 cents. Organize the table by starting with the combinations that include the most quarters. quarters

dimes

nickels

1

1

0

1

0

2

0

3

1

0

2

3

0

1

5

0

0

7

The total for each combination of the coins is 35 cents. There are 6 combinations possible.

Practice Solve each problem by making a table or list. 1. How many ways can you make change for a half-dollar using only nickels,

dimes, and quarters? 2. A number cube has faces numbered 1 to 6. If a red and a blue cube are

tossed and the faces landing up are added, how many ways can you roll a sum less than 8? 3. A penny, a nickel, a dime, and a quarter are in a purse. How many

amounts of money are possible if you grab two coins at random? 4. The three counters at the right are used for

a board game. If the counters are tossed, how many ways can at least one counter with Side A turn up? 5. How many ways can you receive change for a

Counters

Side 1 Side 2

Counter 1

A

B

Counter 2

A

C

Counter 3

B

C

quarter if at least one coin is a dime? 6. Jorge had 55 football cards. He traded 8 cards for 5 from Elise. He traded 6

more for 4 from Leon and 5 for 3 from Bret. Finally, he traded 12 cards for 9 from Ginger. How many cards does Jorge have now? 4 Start Smart

Start Smart

Problem-Solving Strategy: Work Backward In most problems, a set of conditions or facts is given and an end result must be found. However, some problems start with the result and ask for something that happened earlier. The strategy of working backward can be used to solve problems like this. To use this strategy, start with the end result and undo each step.

Kendrick spent half of the money he had this morning on lunch. After lunch, he loaned his friend a dollar. Now he has $1.50. How much money did Kendrick start with? Start with the end result, $1.50, and work backward to find the amount Kendrick started with. Kendrick now has $1.50. $1.50 Add $1.00 to undo Undo the $1 he loaned to his friend. +1.00 giving his friend $1.00. $2.50 Multiply by 2 to Undo the half he spent for lunch. × 2 undo spending half $5.00 the original amount. The amount Kendrick started with was $5.00. CHECK

Kendrick started with $5.00. If he spent half of that, or $2.50, on lunch and loaned his friend $1.00, he would have $1.50 left. This matches the amount stated in the problem, so the solution is correct.

Practice Solve each problem by working backward. 1. Tia used half of her allowance to buy a ticket to the class play. Then she

spent $1.75 for an ice cream cone. Now she has $2.25 left. How much is her allowance? 2. Lawanda put $15 of her paycheck in savings. Then she spent one-half of

what was left on clothes. She paid $24 for a concert ticket and later spent one-half of what was then left on a book. When she got home, she had $14 left. What was the amount of Lawanda’s paycheck? 3. A certain number is multiplied by 3, and then 5 is added to the result. The

final answer is 41. What is the number? 4. Mr. and Mrs. Delgado each own an equal number of shares of a stock.

Mr. Delgado sells one-third of his shares for $2700. What was the total value of Mr. and Mrs. Delgado’s stock before the sale? 5. A certain bacteria doubles its population every 12 hours. After 3 full days,

there are 1600 bacteria in a culture. How many bacteria were there at the beginning of the first day? 6. To catch a 7:30 A.M. bus, Don needs 30 minutes to get dressed, 30 minutes

for breakfast, and 15 minutes to walk to the bus stop. What time should he wake up? Problem-Solving Strategy: Work Backward PhotoLink/Getty Images

5

Problem-Solving Strategy: Guess and Check To solve some problems, you can make a reasonable guess and then check it in the problem. You can then use the results to improve your guess until you find the solution. This strategy is called guess and check.

The product of two consecutive even integers is 1088. What are the integers? The product is close to 1000. Make a guess. Let’s try 24 and 26.

24 × 26 = 624

This product is too low.

Adjust the guess upward. Try 30 and 32.

30 × 32 = 960

This product is still too low.

Adjust the guess upward again. Try 34 and 36.

34 × 36 = 1224

This product is too high.

Try between 30 and 34. Try 32 and 34.

32 × 34 = 1088

This is the correct product.

The integers are 32 and 34.

Practice Use the guess-and-check strategy to solve each problem. 1. The product of two consecutive odd integers is 783. What are the integers? 2. Brianne is three times as old as Camila. Four years from now she will be

just two times as old as Camila. How old are Brianne and Camila now? 3. Rafael is burning a CD for Selma. The CD will hold 35 minutes of music.

Which songs should he select from the list to record the maximum time on the CD without going over? Song

A

B

C

D

E

F

G

H

I

J

Time

5 min 4s

9 min 10 s

4 min 12 s

3 min 9s

3 min 44 s

4 min 30 s

5 min 0s

7 min 21 s

4 min 33 s

5 min 58 s

4. Each hand in the human body has 27 bones. There are 6 more bones in the

fingers than in the wrist. There are 3 fewer bones in the palm than in the wrist. How many bones are in each part of the hand? 5. The Science Club sold candy bars and soft pretzels to raise money for an

animal shelter. They raised a total of $62.75. They made 25¢ profit on each candy bar and 30¢ profit on each pretzel sold. How many of each did they sell? 6. Odell has the same number of quarters, dimes, and nickels. In all he has $4

in change. How many of each coin does he have? 7. Anita sold tickets to the school musical. She had 12 bills worth $175 for the

tickets she sold. If all the money was in $5 bills, $10 bills, and $20 bills, how many of each bill did she have? 6 Start Smart

Start Smart

Problem-Solving Strategy: Solve a Simpler Problem One of the strategies you can use to solve a problem is to solve a simpler problem. To use this strategy, first solve a simpler or more familiar case of the problem. Then use the same concepts and relationships to solve the original problem.

Find the sum of the numbers 1 through 500. Consider a simpler problem. Find the sum of the numbers 1 through 10. Notice that you can group the addends into partial sums as shown below. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 11 The number of sums is 5, or half the number of addends. 11 11 11 Each partial sum is 11, the sum 11 of the first and last numbers. The sum is 5 × 11 or 55. Use the same concepts to find the sum of the numbers 1 through 500. 1 + 2 + 3 + … + 499 + 500 = 250 × 501 Multiply half the number of addends, 250, by the sum of = 125,250 the first and last numbers, 501.

Practice Solve each problem by first solving a simpler problem. 1. Find the sum of the numbers 1 through 1000. 2. Find the number of squares of any size in the

game board shown at the right. 3. How many links are needed to join 30 pieces of

chain into one long chain? 4. Three people can pick six baskets of apples in one

hour. How many baskets of apples can 12 people pick in one-half hour? 5. A shirt shop has 112 orders for T-shirt designs.

Three designers can make 12 shirts in 2 hours. How many designers are needed to complete the orders in 8 hours? 6. Stamps for postcards cost $0.24, and stamps for first-class letters cost $0.39.

Diego wants to send postcards and letters to 10 friends. If he has $3.15 for stamps, how many postcards and how many letters can he send? Problem-Solving Strategy: Solve a Simpler Problem

7

Problem-Solving Strategy: Draw a Diagram Another strategy for solving problems is to draw a diagram. There will be times when a sketch or diagram will give you a better picture of how to tackle a mathematics problem. Adding details like units, labels, and numbers to the drawing or sketch can help you make decisions on how to solve the problem.

Imani is trying to determine the number of 9-inch tiles needed to cover her patio. The rectangular patio measures 8-feet by 10-feet. What is the minimum number of 9-inch tiles Imani should purchase? First, draw a diagram of the situation. Express the measurement of the patio in inches. If each tile is 9 inches square, the minimum number of tiles for the width of the patio is 96 ÷ 9 ≈ 10.7 or 11 tiles. The minimum number of tiles for the length of the patio is 120 ÷ 9 ≈ 13.3 or 14 tiles.

0ATIO

FT IN

So, the minimum number of tiles Imani needs to cover the patio is 11 × 14 or 154 tiles.

Practice Solve each problem by drawing a diagram. 1. The area of a rectangular flower bed is 24 square feet. If the sides are

whole number dimensions, how many combinations of lengths and widths are possible for the flower bed? List them. 2. Kevin was hired to paint a mural on a wall that measures 15 feet by

20 feet. Starting from the center of the wall, he will paint a square that measures 3 feet. The dimensions of the next square will be 1.5 times greater than and centered on the previous square. How many squares can Kevin paint on the wall? 3. A 500-gallon bathtub is being filled with water. Eighty gallons of water are

in the bathtub after 4 minutes. How long will it take to fill the bathtub? 4. It takes 42 minutes to cut a 2-inch by 4-inch piece of wood into 7 equally

sized pieces. How long will it take to cut a similar 2-inch by 4-inch into 4 equally sized pieces? 5. Find the number of line segments that can be drawn between any two

vertices of an octagon. 6. How many different teams of 3 players can be chosen from 8 players?

8 Start Smart

FT IN

Start Smart

Problem-Solving Strategy: Act It Out There are problem situations where acting the problem out will provide you with a visual image and provide direction for solving the problem. To act it out, you could use real people, models of the real objects, or manipulatives like tiles and cubes, which will help you visualize the problem and make decisions about how to proceed with solving the problem.

A quiz has 5 multiple-choice items each with choices A, B, C, and D. Is spinning a spinner with 4 equal sections to decide the answers a good strategy for taking the quiz? First, make a spinner with 4 equal sections and use it to act out taking the quiz. Each section represents one of the choices A, B, C, and D for each question. Suppose the correct answers to the quiz are D, D, B, C, and A. Do 5 trials. Answers

D

D

B

C

A

Number Correct

Trial 1

D

A

D

B

C

1

Trial 2

C

B

C

A

B

0

Trial 3

B

C

D

A

A

1

Trial 4

B

D

A

C

B

2

Trial 5

D

A

B

A

D

2

The experiment only produces 1 or 2 correct answers. So, spinning a spinner is not a good strategy to use to answer a multiple-choice quiz with 4 choices.

Practice Solve each problem by acting the problem out. 1. A test has 10 true-false questions. Is tossing a coin to decide the answers a

good strategy for taking the test? Explain. 2. A pizza parlor has thin crust and deep dish, 2 different cheeses, and

4 toppings. How many different one-cheese and one-topping pizzas can be ordered? 3. A field hockey conference has four teams. In how many ways can first,

second, and third place be awarded at the end of the season? 4. Waban makes 2 out of every 3 free throws he attempts. How many would

you expect him to make in his next 25 attempts? 5. A gumball machine has an equal number of red, yellow, and orange

gumballs. If it costs $0.25 per gumball, how much would you have to spend to get at least one gumball of each color? Problem-Solving Strategy: Act It Out

9

The Graphing Calculator This year, you may use an exciting tool to help you solve problems—a graphing calculator. Graphing Calculator Labs have been included in your textbook so you can use technology to explore concepts. These labs use the TI-83 Plus or TI-84 Plus calculator. A graphing calculator does more than just graph. You can also use it to calculate.

is used to enter equations.

Press 2nd to access the additional functions listed above each key.

The key is used to find the second power of a number or expression.

Press ON to turn on your calculator. Press 2nd [OFF] to turn off your calculator.

( ) is used to

indicate a negative or opposite value.

10 Start Smart Matt Meadows

Press 2nd [TABLE] to display a table of values for equations entered key. using the

Press CLEAR once to clear an entry. Press CLEAR twice to clear the screen. Use the operation keys to add, subtract, multiply or divide. Multiplication is shown as * on the screen and division is displayed as /.

The ENTER key acts like an equals button to evaluate an expression. It is also used to select menu items.

Start Smart

EXAMPLE

ntering and Evaluating Expressions

Evaluate each expression. a. -2[4(11 - 6)] + (-23) KEYSTROKES:

( ) 2

4

11

( )

6

ENTER

23

b. 3(-4) + [(12 ÷ 6) - 5(-8)] KEYSTROKES:

12 ⫼ 6

( ) 4

3

5

( ) 8

ENTER

Evaluate each expression. 1. 8 + [7(12 ÷ 4)]

2. 5[(5 + 14) - 2(7)]

3. [6(8 ÷ 12)] × 3

4. 5 + [(8 × 2) - 7]

5. 4[3(21 - 17) + 3]

6. 7[5 + (13 - 4) ÷ 3]

EXAMPLE

owers and Exponents

Evaluate each expression. a. 52 × 23 KEYSTROKES:

⫻ 2

5

3 ENTER

b. (-4)3 ÷ 25 3 ⫼ 2

( ) 4

KEYSTROKES:

5 ENTER

Evaluate each expression. 7. 38

8. 2 · 63

10. 54 - 33

EXAMPLE

9. 43 · 27

11. 3 · 25 · 45

12. 3 · 53 + 4 · 25

oots

Evaluate each expression. 625 + 5 a. - √ KEYSTROKES:

( )

[ √ ] 625

5 ENTER

b. √ 62 + 82 KEYSTROKES:

[ √ ] 6

8

ENTER

Evaluate each expression. 13. √ 144 + 26

14. √ 324 - 12

15. - √ 225 + 6

16. - √ 169 - 3

17. √ 32 + 7

18. √ 142 + 27

The Graphing Calculator

11

Choose the Best Method of Computation Solving problems is more than using paper and pencil. Follow the path to choose the best method of computation.

12 Start Smart (tl)Image100 Ltd, (tr)Monica Lau/Getty Images, (bl)RubberBall/PictureQuest, (br)Digital Vision/PunchStock

Start Smart

Practice Choose the best method of computation to solve each problem. Then solve. 1. Kimi paid $677.48, including tax, for a surfboard. She then decided to have

a protective spray applied to her board for $47.98. What was the total cost of the surfboard and the spray? 2. An average-sized orca whale eats about 551 pounds of food a day. How

many pounds of food will an orca whale eat in one year? For Exercises 3-5, use the information in the table that shows the sales of Café Mocha’s coffee of the month.

Coffee of the Month Sales

3. How many more cups of this kind of

coffee did Café Mocha sell in January than in April? 4. If Café Mocha charges $1.98 for each

Month

Number of Cups Sold

January

850

February

765

March

587

April

500

May

387

cup of coffee, about how much money did Café Mocha earn in March? 5. In the month of May, Café Mocha had to raise their prices for each cup

of coffee to $2.25. How much money did Café Mocha earn in the month of May? 6. The Student Council is making pizzas to sell at the football game on

1 Friday. Each pizza needs 2_ cups of cheese. If the student council members 4

make 25 pizzas, how many cups of cheese will they need? 7. The length of Fun Center’s go-kart track is 843 feet. If Nadia circled the

track 9 times, about how many feet did she travel? 8. Use the table below to find the total area of the Great Lakes. Great Lakes Great Lake

Area (mi2)

Lake Superior

31,698

Lake Huron

23,011

Lake Michigan

22,316

Lake Erie

9922

Lake Ontario

7320

Source: worldatlas.com

9. Music Megastore was having a sale on blank CDs and DVDs. Ivan bought

a package of 300 blank CDs for $39 including tax. What is the individual cost of each CD? 10. An art supply store sells 5 different sized canvases. The surface area of the

middle-size canvas is 3.5 times larger than the surface area of the extrasmall canvas. If the surface area of the extra-small canvas is 81 square inches, what is the surface area of the middle-sized canvas? Choose the Best Method of Computation

13

Why do I need my math book? Have you ever been in class and not understood all of what was presented? Or, you understood everything in class, but at home, got stuck on how to solve a couple of problems? Maybe you just wondered when you were ever going to use this stuff?

These next few pages are designed to help you understand everything your math book can be used for … besides homework problems! Before you read, have a goal. • What information are you trying to find? • Why is this information important to you? • How will you use the information? Have a plan when you read. • • • •

Read the Main Idea at the beginning of the lesson. Look over photos, tables, graphs, and opening activities. Locate words highlighted in yellow and read their definitions. Find Key Concept and Concept Summary boxes for a preview of what’s important. • Skim the example problems. Keep a positive attitude. • Expect mathematics reading to take time. • It is normal to not understand some concepts the first time. • If you don’t understand something you read, it is likely that others don’t understand it either.

14 Start Smart John Evans

Start Smart

Doing Your Homework Regardless of how well you paid attention in class, by the time you arrive at home, your notes may no longer make any sense and your homework seems impossible. It’s during these times that your book can be most useful. • Each lesson has example problems, solved step-by-step, so you can review the day’s lesson material. •

has extra examples at pre-alg.com to coach you through solving those difficult problems.

• Each exercise set has HOMEWORK HE H LPP boxes that show you which examples may help with your homework problems. • Answers to the odd-numbered problems are in the back of the book. Use them to see if you are solving the problems correctly. If you have difficulty on an even problem, do the odd problem next to it. That should give you a hint about how to proceed with the even problem.

Doing Your Homework John Evans

15

Studying for a Test You may think there is no way to study for a math test! However, there are ways to review before a test. Your book can help! • Review all of the new vocabulary words and be sure you understand their definitions. These can be found on the first page of each lesson or highlighted in yellow in the text. • Review the notes you’ve taken on your questions that you still need answered.

and write down any

• Practice all of the concepts presented in the chapter by using the chapter Study Guide and Review. It has additional problems for you to try as well as more examples to help you understand. You can also take the Chapter Practice Test. • Take the Self-check Quizzes at pre-alg.com.

16 Start Smart John Evans

Start Smart

Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1. What is the title of Chapter 1? 2. How can you tell what you’ll learn in Lesson 1-1? 3. What is the key concept presented in Lesson 1-2? 4. List the new vocabulary words that are presented in

Lesson 1-3. 5. In the margin of Lesson 1-3, there is a Vocabulary Link.

What can you learn from that feature? 6. How many examples are presented in Lesson 1-3? 7. What is the web address where you could find extra

examples? 8. Suppose you’re doing your homework on page 40 and you

get stuck on Exercise 19. Where could you find help? 9. What is the title of the feature in Lesson 1-5 that tells you

how to read the ≠ symbol? 10. What is the title of the feature in Lesson 1-6 that tells you

about the units on the x- and y-axes? 11. Sometimes you may ask “When am I ever going to use

this”? Name a situation that uses the concepts from Lesson 1-7. 12. There is a Real-World Career mentioned in Lesson 1-7.

What is it? 13. What is the web address that would allow you to take a

self-check quiz to be sure you understand the lesson? 14. On what pages will you find the Study Guide and Review

for Chapter 1? 15. Suppose you can’t figure out how to do Exercise 35 in the

Study Guide on page 71. Where could you find help?

Scavenger Hunt

17

The following pages contain data that you’ll use throughout the book. JEFFERSON CIT Y, MISSOU

RI

-*

i`>À ÌÞ 7AIN 3

TREET

-ISSOURI "LVD

University of Akron ege Columbus State Community Coll Ohio University Sinclair University University of Toledo Bowling Green State University

FF 2

OAD

,' 2E

X7

5NIO N

HIT

TON

0ACIF IC

2AILR

%X PY

OA D

#F

2E

D7

HA

LEY

%X PY

Universities 2004 Ten Largest Ohio Colleges and Enrollment College or University

Cuyahoga Community College Kent State University

"LU

*EFFERSON #ITY -EMORIAL !IRPORT

ivviÀÃ ÌÞ

HIGHER EDUCATION

Ohio State University University of Cincinnati

0+

50,995

,+

27,178 25,214 24,347 23,282 21,872 20,096 19,563 19,480 18,989

INDEPENDENCE SQUARE

Source: infoplease.com

Congress Hall

BOSTO N CREA M PIE

Recipe for:

Boston Cream 1(2 laye Pie r size) p kg yello 1 pkg in w cake mix stant va nilla pu 1 (1 oz ) dding m ix sq unsw eetened 1 tables chocolate poon bu tter 1 cup sif _1 ted pow dered su 2 teaspo gar on vanil la Source 3 – 4 : cooks.c teaspoo om ns wate r

18

United States Data File

Independence Hall

Philosophical Hall

Great Essentials exhibit West Wing Sansom Street

Old City Hall

Independence Hall tours start in East Wing

Library Hall

INDEPENDENCE 500 ft

SQUARE 375 ft

The Declaration of Independence was first publicly read at Independence Square in Philadelphia, Pennsylvania.

N

BOOMERANG


EMPIRE STATE BUILDING

Cost

Ticket?

......................... $16.00 Adults (18–61) ............... ........................ $14.00 Youth (12–17).................. ...................... $10.00 Child (6–11) .................... ...................... $14.00 ..... ..... Seniors (62+) .......... ......................... $14.00 Military w/ID ............... ...................... Free Military in Uniform .......... ) ......................... Free Toddlers (5 or younger .....

Source: esbnyc.com

BIGHORN SHEEP

Opening Date: Spring 199 9 Length: 938 feet Height: 125 feet Top Speed: 60 mph Ride Time: 1 minute, 44 seconds Hourly Ride Capacity: 750 passengers per hour Source: sixflags.com

Weight: 115–280 lb Horn Weight: 40–60 lb Length with Tail: 50–62 in. Shoulder Height: 32–40 in. Source: desertusa.com

MINNESOTA VIKINGS

Rushing Statistics Yards Rushed

Average Yards per Rush

Length of Longest Rush

662

4.3

33

155

473

3.8

61

126

147

6.1

18

24

62

1.9

15

32

53

2.9

16

18

28

9.3

11

3

27

6.8

13

4

Number of Rushes

Mewelde Moore Michael Bennett

Player

Daunte Culpepper Ciatrick Fason Brad Johnson Troy Williamson Koren Robinson

The official state mammal of Colorado is the Bighorn Sheep.

Source: espn.com

United States Data File (l)Jeremy Murphy/www.lonestarthrills.com, (r)John Conrad/CORBIS

19

SUNFLOWER

MOUNT RUSHM

OR E

ica.com

Source: britann

er of Kansas is The state flow 1–4.5 cm Stem Height: 7.5–30 cm Leaf Length: 15–30 cm Head Width:

PUBLIC LIBRARY

Source: mountru

shmoreinfo.com

Mount Rushm ore is located in Black Hills, South D akota. • The length of each presid ent’s face is 60 feet from th e top of the h ead to the chin. • A total of 80 0 million pou nds of stone was carved o ut to complete the sculpture. • Each carvin g is scaled to men who would be 465 feet tall. • The monum ent can be seen from 60 miles away .

The Seattle Public Library’s new building, designed by architect Joshua Ramos, opened to the public in 2004. His 362,876 square foot design is comprised of 8 horizontal layers. Each layer varies in size to fit its function. At one corner of the library, the wedge-shaped base nearly diminishes to a point. The total cost of the building was about $1.7 million and its construction required 2050 tons of concrete and 4644 tons of steel.

GREAT LAKES Ontario Quebec

Su pe rio r ron Hu

Michigan

Minnesota Wisconsin

Michigan ie Er

Illinois Indiana

Source: greatbuildings.com

io Ontar

New York

Pennsylvania Ohio Image © GLIN


20

. the sunflower


(l)Garry Black/Masterfile; (tr)David C. Tomlinson/Getty Images; (br)Ron Wurzer/Getty Images

Lake

Area (mi2)

Erie Huron Michigan Ontario Superior

9940 23,010 22,178 7540 31,010

Maximum Surface Maximum Elevation (ft) Depth (ft) 572 580 581 246 602

210 750 923 778 1302

U.S. CELLULAR FIELD


G R AN

D CAN

YON

Source: ballparks.com

The 40,615-seat U.S. Cellular Field is home to the Chicago White Sox baseball team. The 1,300,00 square foot complex features a 15,000 square foot baseball and softball instruction area for kids. The baseball diamond is a square measuring 90 feet on each side.

The G ra is hom nd Canyon e Natio n seven to the Gra nd Ca al Park in natura A nyon l won one of rizona ders. Lengt t he h Width : 227 miles : Eleva 0.25–15 mil tion: 2 e 400–7 s Source 000 fe : nps.g et ov

TEMPERATURE Wisconsin: Temperature Extremes Month

Maximum °F

Year

Place

Minimum °F

Year

Place

January

66

1897

Prairie DuChien

-54

1922

Danbury

February

69

2000

Afton/Beloit/Broadhead

-55

1996

Couderay

March

86

1986

Dodge

-48

1962

Couderay

April

97

1980

Lone Rock

-20

1924

Rest Lake

May

109

1934

Prairie DuChien

7

1966

Gorden

June

106

1934

Racine

20

1964

Danbury

July

114

1936

Wisconsin Dells

27

1972

Jump River

August

108

1988

University of Wisconsin–Arboret

22

1950

Coddington Exp Farm

September

104

1939

Prairie DuChien

10

1949

Coddington Exp Farm

October

95

1897

Gratiot

-7

1925

Long Lake

November

84

1904

Prairie DuChien

-34

1898

Osceola

December

67

1998

La Crosse

-52

1983

Couderay


United States Data File (l)Ron Vesely/MLB Photos via Getty Images; (r)J. A. Kraulis/Masterfile

21

Algebra and Integers Focus Build a foundation of basic understandings of numbers, operations, and algebraic thinking. Use these understandings to solve linear equations.

CHAPTER 1 The Tools of Algebra Select and use appropriate operations to solve problems and justify solutions. Use graphs, tables, and algebraic representations to make predictions and solve problems.

CHAPTER 2 Integers Understand that different forms of numbers are appropriate for different situations. Use graphs, tables, and algebraic representations to make predictions and solve problems.

CHAPTER 3 Equations Select and use appropriate operations to solve problems and justify solutions. Use graphs, tables, and algebraic representations to make predictions and solve problems. 22 Unit 1 Algebra and Integers Laurence Parent

Algebra and Geography I Need a Vacation! Does your family like to go camping? Eastern Kentucky is home to the Daniel Boone National Park, which covers over 704,000 acres of rugged terrain. This forest provides excellent opportunities for outdoor recreation. In this project, you will explore how graphs and formulas can help you plan a family vacation. Log on to pre-alg.com to begin.

Unit 1 Algebra and Integers

23

The Tools of Algebra

1 •

Select and use appropriate operations to solve problems and justify solutions

•

Solve problems connected to everyday experiences and other subjects.

Key Vocabulary algebra (p.37) evaluate (p.32) solving the equation (p.49) variable (p.37)

Real-World Link Water Parks You can use the expression 2a + 2c to find price of admission for a family of 2 adults and 2 children to a water park where a is the price of an adult and c is the price of a child.

Problem Solving Make this Foldable to help you organize your strategies for solving problems. Begin with a sheet of paper.

1 Fold the short sides so they meet in the middle.

2 Fold the top to the bottom.

3 Unfold. Cut along the

4 Label each of the

second fold to make four tabs.

tabs as shown.

24 Chapter 1 The Tools of Algebra Rio Aventura, Schlitterbahn Beach Waterpark, South Padre Island, TX

%XPLORE

0LAN

#HECK

3OLVE

GET READY for Chapter 1 Diagnose Readiness You have two options for checking Prerequisite Skills.

Option 2 Take the Online Readiness Quiz at pre-alg.com.

Option 1

Take the Quick Check below. Refer to the Quick Review for help.

Find each sum or difference. (Prerequisite Skills, p. 745)

1. 6.6 + 8.2 3. 2.65 + 0.3 5. 4.25 - 0.7

2. 5.4 - 2.3 4. 1.08 + 1.2 6. 4.3 - 2.89

7. LUNCH Chad has $8.60. He spends $4.90 on lunch. How much money does Chad have left? (Prerequisite Skills, p. 745)

Example 1

Find 11.9 - 2.15. 8 10

11.9 0 Annex a zero to align the decimal points - 2.1 5 and subtract. 9.7 5 11.9 - 2.15 = 9.75

8. MONEY Jamal has $10.50. His sister Syreeta gives him the $5.75 she borrowed from him yesterday. How much money does Jamal have now? (Prerequisite Skills, p. 745)

Estimate each sum, difference, product, or quotient. (Prerequisite Skill) 9. 328 + 879 10. 22,431 - 13,183 11. 189 × 89 12. 21,789 ÷ 97 13. ANIMALS The peregrine falcon reaches horizontal cruising speeds of 55 miles per hour. At that rate, about how far can it travel in 3 hours? (Prerequisite Skill)

Example 2

Estimate 117 + 51. 117 + 51

→

120 Round to the nearest ten + 50 and add. 170

117 + 51 ≈ 170

Estimate each sum, difference, product, or quotient. (Prerequisite Skills, p. 744) 14. 8.8 + 5.3 15. $7.34 - $2.16 16. 4.2 × 29.3 17. 18.8(5.3) 18. 7.8 ÷ 2.3 19. 21.3 ÷ 1.7

Example 3

20. WEATHER Annual precipitation in Cape Hatteras, North Carolina, is about 56.1 inches. The city of Raleigh receives about 41.4 inches annually. About how much more precipitation does Cape Hatteras receive than Raleigh? (Prerequisite

3.2 × 61.5 ≈ 186

Estimate 3.2 × 61.5. 61.5 × 3.2

→

62 Round to the nearest whole × 3 number and multiply. 186

Skills, p. 744) Chapter 1 Get Ready for Chapter 1

25

1-1

Using a Problem-Solving Plan

Main Ideas • Use a four-step plan to solve problems. • Choose an appropriate method of computation.

New Vocabulary conjecture inductive reasoning

The first postage stamp ever issued by the United States Government was issued on July 1, 1847. The rate for a half-ounce letter was 5¢ for any distance up to 300 miles and 10¢ for any distance over 300 miles. The table shows first-class mail rates in 2006. a. Describe the pattern in the costs. b. How can you determine the cost to mail a 6-ounce letter? c. Suppose you were asked to find the cost of mailing a letter that weighs 8 ounces. What steps would you take to solve the problem?

Weight (oz)

Cost

1

$0.39

2

$0.63

3

$0.87

4

$1.11

5

$1.35

U.S. MAIL

Source: www.usps.gov

Four-Step Problem-Solving Plan It is often helpful to have an organized plan to solve math problems. The following four steps can be used to solve any math problem. 1. Explore

• Read the problem quickly to gain a general understanding of it. • Ask, “What facts do I know?” • Ask, “What do I need to find out?” • Ask, “Is there enough information to solve the problem? Is there extra information?”

2. Plan

• Reread the problem to identify relevant facts. • Determine how the facts relate to one another. • Make a plan and choose a strategy for solving it. There may be several strategies that you can use. • Estimate the answer.

3. Solve

• Use your plan to solve the problem. • If your plan does not work, revise it or make a new plan.

4. Check

• Reread the problem. Is there another solution? • Ask, “Is my answer reasonable and close to my estimate?” • Ask, “Does my answer make sense?” • If not, make a new plan and solve the problem another way.

26 Chapter 1 The Tools of Algebra

POSTAL SERVICE Refer to page 26. How much would it cost to mail a 9-ounce letter first class? Explore The table shows the weight of a letter and the cost to mail it first class. We need to find the cost to mail a 9-ounce letter. Plan

Use the information in the table. Look for a pattern in the costs. Extend the pattern to find the cost for a 9-ounce letter.

Solve

First, find the pattern. Weight (oz)

1

Cost

2

3

$0.39 $0.63 $0.87

4

5

$1.11

$1.35

+ 0.24 + 0.24 + 0.24 + 0.24

Each consecutive cost increases by $0.24. Next, extend the pattern. Weight (oz)

5

Cost

6

7

8

9

$1.35 $1.59 $1.83 $2.07 $2.31 + 0.24 + 0.24 + 0.24 + 0.24

It would cost $2.31 to mail a 9-ounce letter. Reasonableness

Always check to be sure your answer is reasonable. If the answer seems unreasonable, solve the problem again.

Check

To mail a 9-ounce letter, it would cost $0.39 for the first ounce and 8 × $0.24 or $1.92 for the eight additional ounces. Since $0.39 + $1.92 = $2.31, the answer is correct.

1. It costs $0.80 per ounce to mail a letter to England. How much would it cost to mail an 8-ounce letter to England? Personal Tutor at pre-alg.com

A conjecture is an educated guess. When you make a conjecture based on a pattern of examples or past events, you are using inductive reasoning.

EXAMPLE

Use Inductive Reasoning to Solve Problems

a. Find the next term in 1, 3, 9, 27, 81, …. 1

3 ×3

9 ×3

27 ×3

81 ×3

? ×3

Assuming the pattern continues, the next term is 81 × 3 or 243. b. Draw the next figure in the pattern.

The shaded square moves counterclockwise. Assuming the pattern continues, the shaded square will be at the bottom left of the figure.

2. Find the next term in 71, 59, 47, 35, . . . . Extra Examples at pre-alg.com

Lesson 1-1 Using a Problem-Solving Plan

27

Choose the Method of Computation Choosing the method of computation is Look Back

To review problem solving strategies, see pages 2–13.

also an important step in solving problems. In addition to using paper and pencil to solve problems, you can use number sense, estimation, mental math, and a calculator.

Do you need an exact answer?

Do you see a pattern or number fact?

yes

yes

Use mental math.

yes

Use paper and pencil.

no no Estimate.

Use a calculator.

no

Are there simple calculations to do?

Explore

You know the seating capacities of Comerica Park and Fenway Park. You need to find how many more seats Comerica Park has than Fenway Park.

Plan

The question uses the word about, so an exact answer is not needed. We can solve the problem using estimation.

Real-World Link Fenway Park was built in 1912 and is the oldest stadium in Major League Baseball. Source: Major League Baseball

Solve

40,950 40,000

TICKET

41,503 TICKET

TICKET

39,345 TICKET

33,871 TICKET

Pa Come rk (De rica Fe tro nw it) ay Pa rk (Bo sto n) Mi Pa nute rk (Ho Maid ust on ) (Sa SB C nF P ran ark Wr cis igl co) ey Fie ld (Ch ica go )

BASEBALL The graph shows the seating capacity of certain baseball stadiums. About how many more seats does Comerica Park have than Fenway Park?

Seating capacity

Ballpark Seating Capacity 42,000 41,000 40,000 39,000 38,000 37,000 36,000 35,000 34,000 33,000 32,000 31,000 30,000 0

Ballpark Source: Major League Baseball

Comerica Park: 40,000 → 40,000 Round to the nearest thousand. Fenway Park: 33,871 → 34,000 40,000 - 34,000 = 6000 Subtract 34,000 from 40,000. So, Comerica Park has about 6000 more seats than Fenway Park.

Check

Add 33,871 and 6000. Since 39,871 ≈ 40,000, the answer is reasonable.

3. About how many more seats does Minute Maid Park have than Wrigley Field? 28 Chapter 1 The Tools of Algebra Icon SMI/CORBIS

Example 1 (p. 27)

Example 2 (p. 27)

1. TIME The ferry schedule shows that the ferry departs at regular intervals. Use the four-step plan to find the earliest time Brady can catch the ferry if he cannot leave until 1:30 P.M.

South Bass Island Ferry Schedule

Find the next term in each list. 2. 10, 20, 30, 40, 50, …

Departures

Arrivals

8 : 4 5 A.M.

9:

9 : 3 3 A.M.

1

10:21 A.M.

3. 37, 33, 29, 25, 21, …

11:09 A.M.

4. 12, 17, 22, 27, 32, … 5. 3, 12, 48, 192, 768, … GEOMETRY Draw the next pattern in the figure. 6.

Example 3 (p. 28)

HOMEWORK

HELP

For See Exercises Examples 8–9 1 10–19 2 20–21 3

7. MONEY In 2003, the average U.S. household spent $13,432 on housing, $2060 on entertainment, $5340 on food, and $7781 on transportation. How much was spent on housing each month? Round to the nearest cent.

ANALYZE TABLES For Exercises 8 and 9, use the table that gives the approximate heart rate for a person exercising at 85% intensity. Age

20

25

30

35

40

45

Heart Rate (beats/min)

174

170

166

162

158

154

8. Assume the pattern continues. Use the four-step plan to find the heart rate a 15-year-old should maintain while exercising at this intensity. 9. What heart rate should a 55-year-old maintain while exercising at this intensity? Find the next term in each list. 10. 2, 5, 8, 11, 14, …

11. 4, 8, 12, 16, 20, …

12. 0, 5, 10, 15, 20, …

13. 2, 6, 18, 54, 162, …

14. 54, 50, 46, 42, 38, …

15. 67, 61, 55, 49, 43, …

16. 2, 5, 9, 14, 20, …

17. 3, 5, 9, 15, 23, …

GEOMETRY Draw the next figure in each pattern. 18. 19.

20. SAVINGS Juan needs to save $125 for a ski trip. He has $68 in his bank. He receives $15 for an allowance and earns $20 delivering newspapers and $16 shoveling snow. Does he have enough money for the trip? Explain. 21. COINS Using eight coins, how can you make change for 65 cents that will not make change for a quarter? Lesson 1-1 Using a Problem-Solving Plan

29

22. MEDICINE The numbers of different types of transplants that were performed in the United States in a recent year are shown in the table. About how many transplants were performed?

EXTRA

PRACTIICE

See pages 761, 794. Self-Check Quiz at pre-alg.com

H.O.T. Problems

23. CANDY A gourmet jelly bean company can produce 100,000 pounds of jelly beans a day. One ounce of these jelly beans contains 100 Calories. If there are 800 jelly beans in a pound, how many jelly beans can be produced in a day?

Transplant Heart Heart-Lung Intestine Kidney Kidney-Pancreas Liver Lung Pancreas

Number 2155 33 107 14,775 905 5329 1042 554

Source: The World Almanac

24. WATER A water tank is draining at a rate of 12 gallons every 8 minutes. If there are 234 gallons in the tank, when will it have just 138 gallons left? 25. OPEN ENDED Write a list of numbers in which 4 is added to get each succeeding term. CHALLENGE For Exercises 26 and 27, think of a 1-to-9 multiplication table. 26. Are there more odd or more even products? How can you determine the answer without counting? 27. Is this different from a 1-to-9 addition facts table? Explain. 28.

Writing in Math Explain why it is helpful to use a plan to solve problems. Include an explanation of the importance of performing each step of the four-step problem-solving plan.

29. What is the relationship between the number of cuts and the number of pieces in each circle?

A The number of pieces is half the number of cuts. B The number of pieces is the same as the number of cuts.

30. Suppose you had 1200 sugar cubes. What is the largest cube you could build with the sugar cubes? F 8 by 8 by 8

H 11 by 11 by 11

G 10 by 10 by 10

J 12 by 12 by 12

31. Antonia bought a video game system for $323.96. She paid in 12 equal installments. Which is the best estimate for the amount of each payment?

C The number of cuts is twice the number of pieces.

A less than $20 B between $20 and $25

D The number of cuts is half the number of pieces.

C between $25 and $30 D greater than $30

PREREQUISITE SKILL Round each number to the nearest whole number. (p. 743) 32. 2.8

33. 5.2


34. 35.4

35. 49.6

36. 109.3

Translating Expressions into Words Chinese, English, French, Russian, Spanish, and Arabic are the official languages of the United Nations. All formal meetings and all official documents, in print or online, are interpreted in all six languages. Translating numerical expressions into verbal phrases is an important skill in algebra. Key words and phrases play an essential role in this skill. The following table lists some words and phrases that suggest addition, subtraction, multiplication, and division. Addition

Subtraction

plus sum more than increased by in all

minus difference less than subtract decreased by less

Multiplication times product multiplied each of factors

Division divided quotient per rate ratio separate

A few examples of how to write an expression as a verbal phrase are shown. Expression 5×8 2+4 16 ÷ 2 8-6 2×5 5-2

Key Word times sum quotient less than product less

Verbal Phrase 5 times 8 the sum of 2 and 4 the quotient of 16 and 2 6 less than 8 the product of 2 and 5 5 less 2

Reading to Learn 1. Refer to the table above. Write a different verbal phrase for each expression. Choose the letter of the phrase that best matches each expression. 2. 9 - 3 a. the sum of 3 and 9 3. 3 ÷ 9 b. the quotient of 9 and 3 4. 9 · 3 c. 3 less than 9 5. 3 + 9 d. 9 multiplied by 3 6. 9 ÷ 3 e. 3 divided by 9 Write two verbal phrases for each expression. 7. 5 + 1 8. 8 + 6 9. 9 × 5 11. 12 ÷ 3

12. 20 4

13. 8 - 7

10. 2(4) 14. 11 - 5 Reading Math Translating Expressions into Words

Mike Segar/Reuters/CORBIS

31

1-2

Numbers and Expressions

Main Ideas • Use the order of operations to evaluate expressions. • Translate verbal phrases into numerical expressions.

New Vocabulary numerical expression evaluate order of operations

Scientific calculators are programmed to find the value of an expression in a certain order. Expression Value

1+2×5

8-4÷2

10 ÷ 5 + 14 × 2

11

6

30

a. Study the expressions and their respective values. For each expression, tell the order in which the calculator performed the operations. b. For each expression, does the calculator perform the operations in order from left to right? c. Based on your answer to parts a and b, find the value of each expression below. Check your answer with a scientific calculator. 12 - 3 × 2 16 ÷ 4 - 2 18 + 6 - 8 ÷ 2 × 3 d. Make a conjecture as to the order in which a scientific calculator performs operations.

Order of Operations Expressions like 1 + 2 × 5 and 10 ÷ 5 + 14 ÷ 2 are numerical expressions. Numerical expressions contain numbers and operations such as addition, subtraction, multiplication, and division. When you evaluate an expression, you find its numerical value. To avoid confusion, mathematicians have agreed upon the following order of operations. Order of operations are rules to follow when more than one operation is used in an expression. Order of Operations

Reading Math Grouping Symbols Grouping symbols include: • parentheses ( ), • brackets [ ], and • fraction bars, as in 6+4 _ , which means 2 (6 + 4) ÷ 2.

Step 1 Evaluate the expressions inside grouping symbols. Step 2 Multiply and/or divide in order from left to right. Step 3 Add and/or subtract in order from left to right.

Numerical expressions have only one value. Consider 6 + 4 × 3. 6 + 4 × 3 = 6 + 12 = 18

Multiply, then add.

6 + 4 × 3 = 10 × 3 = 30

Add, then multiply.

Using the order of operations, the correct value of 6 + 4 × 3 is 18. 32 Chapter 1 The Tools of Algebra

EXAMPLE

Evaluate Expressions

Find the value of each expression. a. 18 ÷ 3 × 2 18 ÷ 3 × 2 = 6 × 2 Divide 18 by 3. = 12

Reading Math Multiplication and Division Notation

A raised dot or parentheses represents multiplication. A fraction bar represents division.

Multiply 6 and 2.

b. 6(2 + 9) - 3 · 8 6(2 + 9) - 3 · 8 = 6(11) - 3 · 8

Evaluate (2 + 9) first.

= 66 - 3 · 8

6(11) means 6 × 11.

= 66 - 24

3 · 8 means 3 times 8.

= 42

Subtract 24 from 66.

c. 4[(15 - 9) + 8(2)] 4[(15 - 9) + 8(2)] = 4[6 + 8(2)]

Evaluate (15 - 9).

= 4(6 + 16)

Multiply 8 and 2.

= 4(22)

Add 6 and 16.

= 88

Multiply 4 and 22.

53 + 15 d. _ 17 - 13 53 + 15 _ = (53 + 15) ÷ (17 - 13) 17 - 13

Rewrite as a division expression.

= 68 ÷ 4

Evaluate 53 + 15 and 17 - 13.

= 17

Divide 68 by 4.

1A. 6 - 3 + 5

1B. 24 ÷ 3 × 9

1C. 2[(10 - 3) + 6(5)]

19 - 7 1D. _ 25 - 22

Personal Tutor at pre-alg.com

Translate Verbal Phrases into Numerical Expressions You have learned to translate numerical expressions into verbal phrases. It is often necessary to translate verbal phrases into numerical expressions.

EXAMPLE

Translate Phrases into Expressions

Reading Math

Write a numerical expression for each verbal phrase.

Differences and Quotients In this book,

a. the product of eight and seven

the difference of 9 and 3 means to start with 9 and subtract 3, so the expression is 9 - 3. Similarly, the quotient of 9 and 3 means to start with 9 and divide by 3, so the expression is 9 ÷ 3.

Words Expression

the product of eight and seven 8×7

b. the difference of nine and three Words Expression

the difference of nine and three 9-3

2A. the sum of 10 and 3 Extra Examples at pre-alg.com

2B. the quotient of 14 and 7 Lesson 1-2 Numbers and Expressions

33

CELL PHONES A cell phone company charges $20 per month and $0.10 for each call made or received. Write and then evaluate an expression to find the cost for 44 calls during one month. Words

and

Expression

+

20 + 0.10 × 44 = 20 + 4.40 = $24.40

f

ll × 44

Multiply. Add.

3. MONEY A taxi charges $4 for the first mile and $2 for each additional mile. Write and evaluate an expression for the fare for a 10-mile trip.

Example 1 (p. 33)

Example 2 (p. 33)

Example 3 (p. 34)

HOMEWORK

HELP


Find the value of each expression. 1. 32 - 24 ÷ 2

2. 18 + 2 × 4

3. 2 × 9 ÷ 3

4. 5(8) + 7

5. 6(15 - 4)

6. 2[3 + 7(4)]

7. 3[(20 - 7) + 1]

10 - 4 8. _ 1+2

34 + 18 9. _ 27 - 14

Write a numerical expression for each verbal phrase. 10. the quotient of fifteen and five 11. the product of six and eight 12. the difference of twelve and nine 13. the sum of eleven and sixteen 14. MUSIC Tyler purchased 3 CDs for $13 each and 2 digital songs for $0.99 each. Write and then evaluate an expression for the total cost.

Find the value of each expression. 15. 3 · 6 - 4

16. 12 - 3 × 3

17. 12 ÷ 3 + 21

18. 9 + 18 ÷ 3

19. 8 + 5(6)

20. 12(11) - 56

15 + 9 21. _ 32 - 20

45 - 18 22. _ 9÷3

23. 11(6 - 1)

24. (9 - 7) · 13

25. 56 ÷ (7 · 2) × 6

26. 75 ÷ (7 + 8) - 3

Write a numerical expression for each verbal phrase. 27. 29. 31. 33. 34.

seven increased by two 28. six minus three nine multiplied by five 30. eleven more than fifteen twenty-four divided by six 32. four less than eighteen the cost of 3 notebooks at $6 each the total amount of CDs if Sancho has 4 and Brianna has 5


ZOO For Exercises 35 and 36, use the information in the table about the price of admission to a zoo. 35. Write an expression that can be used to find the total cost of admission for 4 adults, 3 children, and 1 senior. 36. Find the total cost.

:OO !DMISSION 4ICKET !DULTS #HILDREN 3ENIORS

#OST

37. TRAVEL Joshua is packing for a trip. The total weight of his luggage cannot exceed 70 pounds. He has 3 suitcases that weigh 16 pounds each and 2 sport bags that weigh 9 pounds each. Is Joshua’s luggage within the 70-pound limit? Explain your reasoning. ANALYZING TABLES For Exercises 38 and 39, use the table and the following information. A national poll ranks college football teams using votes from sports reporters. Each vote is worth a certain number of points. Suppose that Penn State University receives 50 first-place votes, 7 second-place votes, 4 fourth-place votes, and 3 tenth-place votes. 38. Write an expression for the number of points that Penn State receives. 39. Find the total number of points.

Number of Points for Each Vote Vote

Points

1st place

25

2nd place

24

3rd place

23

4th place

22

5th place

21

25th place

1

40. Find the value of six added to the product of four and eleven. 41. What is the value of sixty divided by the sum of two and ten? Copy each sentence. Then insert parentheses to make each sentence true.

EXTRA

PRACTICE


H.O.T. Problems

42. 61 - 15 + 3 = 43

43. 12 × 3 ÷ 1 + 2 = 12

44. 56 ÷ 2 + 6 - 4 = 3

45. 5 + 2 · 9 - 3 = 42

46. PUBLISHING An International Standard Book Number (ISBN) is used to identify a published book. To determine if an ISBN is correct, multiply each digit in order by 10, 9, 8, 7, and so on. If the sum of the products can be divided by 11, with no remainder, the number is correct. Find the 10-digit ISBN on the back cover of this book. Is the number correct? Explain why or why not. 47. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would evaluate an expression. 48. OPEN ENDED Give an example of an expression involving multiplication and subtraction in which you would subtract first. 49. FIND THE ERROR Emily and Marcus are evaluating 24 ÷ 2 × 3. Who is correct? Explain your reasoning. Emily 24 ÷ 2 x 3 = 12 x 3 = 36

Marcus 24 ÷ 2 × 3 = 24 ÷ 6 =4

50. REASONING Do 2 × 4 + 3 and 2 × (4 + 3) have the same value? Explain. Lesson 1-2 Numbers and Expressions

35

51. CHALLENGE Suppose only the 1, , , ⫻ , ⫼ , ( , ) , and ENTER keys on a calculator are working. How can you get a result of 75 if you are only allowed to push these keys fewer than 20 times? 52.

Writing in Math

Explain why there should be an agreement on the

order of operations.

53. A bag of potting soil sells for $2, and a bag of fertilizer sells for $13. What is the expression for the total cost of 4 bags of soil and 2 bags of fertilizer? A (4 × 2) + (2 × 13)

54. GRIDDABLE The final standings of a hockey league are shown. A win is worth three points, and a tie is worth 1 point. Zero points are given for a loss. How many points did the Wildcats have?

B (4 × 13) + (2 × 2) C 4(2 + 13) D (2 + 13)(4 + 2)

Team

Wins

Losses

Ties

Knights Huskies Wildcats Mustangs Panthers

14 11 10 9 10

9 9 9 10 14

7 10 11 11 6

Find the next term in each list. (Lesson 1-1) 55. 2, 4, 8, 16, 32, . . .

56. 45, 42, 39, 36, 33, . . .

57. 20, 33, 46, 59, 72, . . .

58. 1, 3, 6, 10, 15, 21, . . .

59. 15, 18, 22, 25, 29, . . .

60. 1215, 405, 135, 45, . . .

Solve each problem. (Lesson 1-1) 61. BUSINESS Mr. Armas is a sales associate for a computer company. He receives a salary plus a bonus for any computer package he sells. Find Mr. Armas’ bonus if he sells 16 computer packages. 62. SPACE SHUTTLE The space shuttle can carry a payload of about 65,000 pounds. If a compact car weighs about 2450 pounds, about how many compact cars can the space shuttle carry? 63. TRAVEL The graph shows the number of travelers to the top five tourist destinations in a recent year. About how many more people traveled to France than to the United States?

Packages

Bonus

2

$100

4

$125

6

$150

8

$175

Top Five Tourism Destinations 90

Arrivals (millions)

76.5 70

49.5

50

45.5 39.1 33.2

PREREQUISITE SKILL Find each sum. 64. 18 + 34

65. 85 + 41

66. 78 + 592

67. 35 + 461

30 0 France

United States

Country Source: infoplease.com


Spain

Italy

China

1-3

Variables and Expressions

algebra variable algebraic expression defining a variable

Money Earned

2 5 8 11 h

5 · 2 or 10 5 · 5 or 25 5 · 8 or 40 5 · 11 or 55 ?

a. How much would the baby-sitter earn for working 10 hours?

5

5

New Vocabulary

Number of Hours

F

• Translate verbal phrases into algebraic expressions.

A baby-sitter earns $5 per hour. The table shows several possibilities for number of hours and earnings.

F

• Evaluate expressions containing variables.

5

Main Ideas

5

b. What is the relationship between the number of hours and earnings? c. If h represents any number of hours, what expression could you write to represent the amount of money earned?

Evaluate Expressions Algebra is a branch of mathematics dealing with symbols. One symbol that is frequently used is a variable. A variable is a placeholder for any value. As shown above, h represents some unknown number of hours. Any letter can be used as a variable. Notice the special notation for multiplication and division with variables. The letter x is most often used as a variable.

x+2

_y means y ÷ 3.

4h means 4 × h. mn means m × n.

4h - 5

3

mn

y 3

An expression like x + 2 is an algebraic expression because it contains sums and/or products of variables and numbers. To evaluate an algebraic expression, replace the variable or variables with known values and then use the order of operations.

EXAMPLE


Evaluate x + y - 9 if x = 15 and y = 26. x + y - 9 = 15 + 26 - 9

Replace x with 15 and y with 26.

= 41 - 9

Add 15 and 26.

= 32

Subtract 9 from 41.

READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.

1A. Evaluate 6 - e + f if e = 3 and f = 9. 1B. Evaluate 7k + h if k = 4 and h = 10. Lesson 1-3 Variables and Expressions

37

EXAMPLE


Evaluate each expression if k = 2, m = 7, and n = 4. a. 6m - 3k 6m - 3k = 6(7) - 3(2) = 42 - 6 or 36

Replace m with 7 and k with 2. Multiply. Then subtract.

b. mn 2 mn = mn ÷ 2 2

Rewrite as a division expression.

= (7 · 4) ÷ 2

Replace m with 7 and n with 4.

= 28 ÷ 2 or 14 Multiply. Then divide. c. n (k 5m) n + (k + 5m) = 4 + (2 + 5 · 7)

Replace n with 4, k with 2, and m with 7.

= 4 + (2 + 35)

Multiply 5 and 7.

= 4 + 37 or 41

Add 2 and 35. Then add 4 and 37.

Evaluate each expression if r = 1, s = 5, and t = 8. st 2A. 6s + 2r 2B. _ 20

2C. r + (40 - 3t)

Translate Verbal Phrases The first step in translating verbal phrases into algebraic expressions is to choose a variable and a quantity for the variable to represent. This is called defining a variable.

EXAMPLE Vocabulary Link Variable Everyday Use likely to change or vary Math Use a letter representing a value that can vary

Translate Verbal Phrases into Expressions

Translate each phrase into an algebraic expression. a. twelve points more than the Falcons scored Words

twelve points more than the Falcons scored.

Variable

Let p represent the points the Falcons scored.

Expression

p + 12

b. four times a number decreased by 6 Words Variable

four times a number decreased by 6 Let n represent the number.

Expression 4n

3A. two miles less than the athlete ran 3B. five more than three times a number 38 Chapter 1 The Tools of Algebra

Sometimes problems include more than one unknown quantity. You must define a variable for each unknown. Then you can write an expression to represent the situation.

SOCCER The Johnstown Soccer League ranks each team in its league using points. A team gets three points for a win and one point for a tie. a. Write an expression that can be used to find the total number of points a team receives. Words

three points times number of wins plus one point times numbers of ties

Variables

Let w = number of wins and t = number of ties.

Expression 3w + 1t

The expression is 3w + 1t or 3w + t. b. Suppose in one season, the North Rockets had 17 wins and 4 ties. How many points did they receive? 3w + 1t = 3(17) + 1(4)

Real-World Link Soccer is the most popular sport in the world. It is estimated that more than 240,000,000 people play soccer around the world. Source: The World Almanac for Kids

Replace w with 17 and t with 4.

= 51 + 4

Multiply.

= 55

Then add.

4. PHOTOGRAPHY A studio charges a sitting fee of $25 plus $4 for each 4-inch by 6-inch print. Write an expression that can be used to find the total cost to have photographs taken. Then find the cost of purchasing twelve 4-inch by 6-inch prints. Personal Tutor at pre-alg.com

Examples 1, 2 (pp. 37–38)

Example 3 (pp. 38)

Example 4 (pp. 39)

ALGEBRA Evaluate each expression if a = 5, b = 12, and c = 4. 1. b + 6

2. a - 3

3. 20 - c + a

4. 18 - 3c

2b 5. _ 8

6. 5a - (b - c)

ALGEBRA Translate each phrase into an algebraic expression. 7. 8. 9. 10.

eight dollars more than the amount Taimi saved five goals less than the Pirates scored the quotient of a number and four, minus five seven increased by the quotient of a number and eight

CAPACITY For Exercises 11 and 12, use the following information. One pint of liquid is the same as 16 fluid ounces. 11. Suppose the number of pints of liquid is represented by p. Write an expression to find the number of fluid ounces. 12. How many fluid ounces is 5 pints?

Extra Examples at pre-alg.com Najlah Feanny/CORBIS

Lesson 1-3 Variables and Expressions

39

HOMEWORK

HELP

For See Exercises Examples 13–14 1 15–24 2 25–30 3 31–34 4

ALGEBRA Evaluate each expression if x = 7, y = 3, and z = 9. 13. z + 2

14. 5 + x

15. 2 + 4z

16. 15 - 2x

6y 17. _ z

19. 3x - 2y

20. 4z - 3y

xz 21. 10 - _

23. 2x + 3z + 5y

24. 5z - 3x - 2y

xy 22. _ + 2 3

9x 18. _ y

9

ALGEBRA Translate each phrase into an algebraic expression. 25. Bianca’s salary plus a $200 bonus 26. three more than the number of cakes baked 27. six feet shorter than the mountain’s height 28. two seconds slower than Joseph’s time 29. three times as many balloons 30. the product of 12 and a number SCIENCE For Exercises 31 and 32, use the following information. The number of times a cricket chirps can be used to estimate the temperature in degrees Fahrenheit. Use c ÷ 4 + 37, where c is the number of chirps in 1 minute. 31. Find the approximate temperature if a cricket chirps 136 times in a minute. 32. What is the temperature if a cricket chirps 100 times in a minute?

Real-World Link

SHOPPING For Exercises 33 and 34, use the following information. The selling price of a sweater is the cost of the sweater plus the markup minus the discount. 33. Write an expression to show the selling price s of a sweater. Use c for cost, m for markup, and d for discount. 34. Suppose the cost of a sweater is $25, the markup is $20, and the discount is $6. What is the selling price of the sweater?

To convert cricket chirps to degrees Celsius, count the number of chirps in 25 seconds, divide by 3, then add 4 to get temperature.

ALGEBRA Evaluate each expression if x 9, y 4, and z 12.

Source: almanac.com

ANALYZE TABLES Write an algebraic expression that represents the relationship in each table.

35. 7z - (y + x)

36. (8y + 5) - 2z

37. (5z - 4x) + 3y

38. 6x - (z - 2y)

39. 2x + (4z - 13) - 5

40. (29 - 3y) + 4z - 7

41.

EXTRA

PRACTIICE


Age in Three Years

10

13

5

12

15

of Items

Total Cost

43. Regular Price

Sale Price

$25

$12

6

$30

$15

$11

$40

$18

$14

$8

15

18

8

20

23

10

$50

$24

$20

x

?

n

?

$p

?

ALGEBRA Translate each phrase into an algebraic expression. 44. seven less than the product of a number and eight 45. twice a number decreased by the quotient of eight and twice the number

40 Chapter 1 The Tools of Algebra Dennis Johnson/Papilio/CORBIS

42. Number

Age Now

H.O.T. Problems

46. OPEN ENDED Give two examples of algebraic expressions. Then give two examples of expressions that are not algebraic. 47. Which One Doesn’t Belong? Suppose a = 2 and b = 5. Identify the expression that does not belong with the other three. Explain your reasoning. a + 3b

6a – b

4b – (a + 1)

12 + b

48. CHALLENGE What value of t makes the expressions 6t, t + 5, and 2t + 4 equal?

Writing in Math Explain how variables are used to show relationships. Include an example to illustrate your reasoning.

49.

50. After the included minutes have been exhausted, a cell phone company charges an additional $0.08 per minute. Plan A uses a flat rate of $0.10 per minute. Which plan is the least costly if a person uses 750 minutes per month? Plan

Monthly Fee

Included Minutes

A B C D

$0 $29.99 $39.99 $49.99

None 500 1,000 1,500

A Plan A

C Plan C

B Plan B

D Plan D

51. Suppose Benito is selling 10 of his music CDs on the Internet, and it costs $1.25 per CD to send them to a buyer. If he decides to sell each CD for the same price p, which expression would you use to find how much money he will receive after sending all 10 CDs? F 22.75p G 22.5p 10 H 10 1.25 J 10p 12.5

Find the value of each expression. (Lesson 1-2) 52. 3 + (6 × 2) - 8

53. 5(16 - 5 × 3)

8 ÷ 8 + 11 54. _

55. 36 ÷ (9 · 2) + 7

56. 70 - (16 ÷ 2 + 21)

57. 4(20 - 13) + 4 × 5

15 - 4(3)

58. FOOD The table shows the amount in pounds of certain types of pasta sold in a recent year. About how many million pounds of these types of pasta were sold? (Lesson 1-1)

59. ANIMALS A Beluga whale’s heart beats about 16 times per minute. Find the number of times a Beluga whale’s heart beats in one hour. (Lesson 1-1)

PREREQUISITE SKILL Find each difference. 60. 53 - 17

61. 97 - 28

62. 104 - 82

Pasta

Amount (millions)

Spaghetti

308

Elbow

121

Noodles

70

Twirl

52

Penne

51

Lasagna

35

Fettuccine

24

Source: National Pasta Association

63. 152 - 123 Lesson 1-3 Variables and Expressions

41

EXTEND

Spreadsheet Lab

1-3

Expressions and Spreadsheets

One of the most common computer applications is a spreadsheet program. A spreadsheet is a table that performs calculations. It is organized into boxes called cells, which are named by a letter and a number. In the spreadsheet below, cell B2 is highlighted. An advantage of using a spreadsheet is that values in the spreadsheet are recalculated when a number is changed. You can use a spreadsheet to investigate patterns in data.

EXAMPLE

Interactive Lab pre-alg.com

Here’s a mind-reading trick! Think of a number. Then double it, add six, divide by two, and subtract the original number. What is the result? You can use a spreadsheet to test different numbers. Suppose we start with the number 10.

-IND 2EADING 4RICK !

"

4HINK OF A NUMBER $OUBLE IT !DD $IVIDE BY

3UBTRACT THE ORIGINAL NUMBER

" " "

" "

3HEET

3HEET

#

3HEET

4HE SPREADSHEET TAKES THE VALUE IN " DOUBLES IT AND ENTERS THE VALUE IN " .OTE THE IS THE SYMBOL FOR MULTIPLICATION 4HE SPREADSHEET TAKES THE VALUE IN " DIVIDES BY AND ENTERS THE VALUE IN " .OTE THAT IS THE SYMBOL FOR DIVISION

The result is 3.

EXERCISES To change information in a spreadsheet, move the cursor to the cell you want to access and click the mouse. Then type in the information and press Enter. Find the result when each value is entered in B1. 1. 6 2. 8 3. 25 4. 100 5. 1500 6. MAKE A CONJECTURE What is the result if a decimal is entered in B1? a negative number? 7. Explain why the result is always 3. Write an expression that describes your answer. 8. Make up your own mind-reading trick. Enter it into a spreadsheet to show that it works. Write an expression to describe the trick. 42 Chapter 1 The Tools of Algebra

1-4

Properties

Main Ideas • Identify and use properties of addition and multiplication. • Use properties of addition and multiplication to simplify algebraic expressions.

Abraham Lincoln delivered the Gettysburg Address more than 130 years ago. The table lists the number of words in certain historic documents.

Historical Document

Words

Preamble to The U.S.Constitution

52

Mayflower Compact

196

Atlantic Charter

375

Gettysburg Address (Nicolay Version)

238

Source: U.S. Historical Documents Archive

New Vocabulary properties counterexample simplify deductive reasoning

a. Suppose you read the Preamble to The U.S. Constitution first and then the Gettysburg Address. Write an expression for the total number of words read. b. Suppose you read the Gettysburg Address first and then the Preamble to the U.S. Constitution. Write an expression for the total number of words read. c. Find the value of each expression. What do you observe? d. Does it matter in which order you add any two numbers? Why or why not?

Vocabulary Link Commute Everyday Use to change or exchange Commutative Math Use property that allows you to change the order in which numbers are added or multiplied

Properties of Addition and Multiplication In algebra, properties are statements that are true for any numbers. For example, the expressions 3 + 8 and 8 + 3 have the same value, 11. This illustrates the Commutative Property of Addition. Likewise, 3 8 and 8 3 have the same value, 24. This illustrates the Commutative Property of Multiplication.

Commutative Property of Addition Words

The order in which numbers are added does not change the sum.

Symbols

For any numbers a and b, a + b = b + a.

Example

2+3=3+2 5=5

Commutative Property of Multiplication Words

The order in which numbers are multiplied does not change the product.

Symbols

For any numbers a and b, a · b = b · a.

Example

2·3=3·2 6=6 Lesson 1-4 Properties

43

When evaluating expressions, it is often helpful to group or associate the numbers. The Associative Property says that the way in which numbers are grouped when added or multiplied does not change the sum or the product. Associative Property of Addition

Vocabulary Link Associate Everyday Use to join together, connect, or combine Associative Math Use property that allows you to change the groupings in which numbers are added or multiplied

Words

The way in which numbers are grouped when added does not change the sum.

Symbols

For any numbers a, b, and c, (a + b) + c = a + (b + c).

Example

(5 + 8) + 2 = 5 + (8 + 2) 13 + 2 = 5 + 10 15 = 15 Associative Property of Multiplication

Words

The way in which numbers are grouped when multiplied does not change the product.

Symbols

For any numbers a, b, and c, (a · b) · c = a · (b · c).

Example

(4 · 6) · 3 = 4 · (6 · 3) 24 · 3 = 4 · 18 72 = 72

The following properties are also true. Properties of Numbers Property

Words

Symbols

Examples

Additive Identity

When 0 is added For any number a, to any number, the a + 0 = 0 + a = a. sum is the number.

5+0=5 0+9=9

Multiplicative Identity

When any number is multiplied by 1, the product is the number.

For any number a, a · 1 = 1 · a = a.

7·1=7 1·6=6

Multiplicative Property of Zero

When any number is multiplied by 0, the product is 0.

For any number a, a · 0 = 0 · a = 0.

4·0=0 0·2=0

EXAMPLE

Identify Properties

Name the property shown by each statement. a. 3 + 7 + 9 = 7 + 3 + 9 The order of the numbers changed. This is the Commutative Property of Addition.

b. (a · 6) · 5 = a · (6 · 5) The grouping of the numbers and variables changed. This is the Associative Property of Multiplication.

1A. 5 × 7 × 2 = 7 × 2 × 5

1B. 14 + (9 + 10) = (14 + 9) + 10

1C. 8 · 1 = 8

1D. 0 · 12 = 0


EXAMPLE Mental Math

Look for sums or products that end in zero.

Mental Math

Find 4 · (25 · 11) mentally. Group 4 and 25 because 4 · 25 = 100. It is easy to multiply by 100 mentally. 4 · (25 · 11) = (4 · 25) · 11 Associative Property of Multiplication = 100 · 11

Multiply 4 and 25 mentally.

= 1100

Multiply 100 and 11 mentally.

Find each sum or product mentally. 2B. (97 + 25) + 3

2A. 40 · (6 · 5)

Counterexample

You can disprove a statement by finding only one counterexample.

One way to find out if these properties apply to subtraction is to look for a counterexample. A counterexample is an example that shows a conjecture is not true.

EXAMPLE

Find a Counterexample

Is subtraction of whole numbers associative? If not, give a counterexample. Write two subtraction expressions using the Associative Property, and then check to see whether they are equal. 9 - (5 - 3) (9 - 5) - 3 9-24-3 7≠1

State the conjecture. Simplify within the parentheses. Subtract.

We found a counterexample. So, subtraction is not associative.

3. Is subtraction of decimals associative? If not, give a counterexample. Personal Tutor at pre-alg.com

Simplify Algebraic Expressions To simplify algebraic expressions means to write them in a simpler form.

EXAMPLE

Simplify Algebraic Expressions

Simplify each expression. a. (k + 2) + 7

b.

(k + 2) + 7 = k + (2 + 7)

5 · (d · 9) 5 · (d · 9) = 5 · (9 · d)

=k+9

= (5 · 9)d = 45d

4A. 12 · (10 · z)

4B. 10 + (p + 18)

Using facts, properties, or rules to reach valid conclusions is called deductive reasoning. Extra Examples at pre-alg.com

Lesson 1-4 Properties

45

Example 1 (p. 44)

Example 2 (p. 45)

Name the property shown by each statement. 1. 7 + 5 = 5 + 7

2. 8 + 0 = 8

3. 8 · 4 · 13 = 4 · 8 · 13

4. 1 × 6 = 6

5. 13 × 12 = 12 × 13

6. 6 + (1 + 9) = (6 + 1) + 9

MENTAL MATH Find each sum or product. Explain your reasoning. 7. 13 + 8 + 7

8. 6 · 9 · 5

9. 8 + 11 + 22 + 4

10. Is division of whole numbers commutative? If not, give a counterexample. Example 3 (p. 45)

Example 4 (p. 45)

HOMEWORK

HELP


ALGEBRA Simplify each expression. 11. 6 + (n + 7)

12. (3 + k) + 8

13. (3 · w) · 9

14. 10 · (r · 5)

15. SHOPPING Clara purchased a pair of jeans for $26, a T-shirt for $12, and a pair of socks for $4. What is the total cost of the items without tax? Explain how the Commutative Property of Addition can be used to find the total mentally.

Name the property shown by each statement. 16. 5 · 3 = 3 · 5

17. 12 · 8 = 8 · 12

18. 6 · 2 · 0 = 0

19. 1 · 4 = 4

20. 0 + 13 = 13 + 0

21. (4 + 5) + 15 = 4 + (5 + 15)

22. 1h = h

23. 7k + 0 = 7k

24. (5 + x) + 6 = 5 + (x + 6)

25. 4(mn) = (4m)(n)

MENTAL MATH Find each sum or product. Explain your reasoning. 26. 11 + 8 + 19

27. 17 + 5 + 33

28. 11 · 9 · 10

29. 2 · 7 · 30

30. 15 · 0 · 2

31. 125 · 4 · 0

32. 74 + 22 + 6

33. 23 + 8 + 27

State whether each conjecture is true. If not, give a counterexample. 34. Division of whole numbers is associative. 35. Subtraction of whole numbers is commutative. 36. The sum of two whole numbers is always greater than either addend. 37. The sum of two odd numbers is always odd. ALGEBRA Simplify each expression. 38. (m + 8) + 4

39. 15 + (12 + a)

40. (17 + p) + 9

41. 21 + (k + 16)

42. 6 · (y · 2)

43. 7 · (d · 4)

44. (6 · c) · 8

45. (3 · w) · 5

46. 25s(3)

47. FOOD In food preparation, chefs marinate meat before they cook it because meat absorbs the marinade during the cooking process. Is marinating and cooking meat commutative? Explain. 46 Chapter 1 The Tools of Algebra

EXTRA

PRACTIICE


48. BASKETBALL The Denver Nuggets made the following baskets during the 2005–2006 season. Write an expression that shows how many total baskets the team made during the season.

.UGGETS "ASKETS &REE 4HROWS

0OINT &IELD 'OALS

d

0OINT &IELD 'OALS

3OURCE NBACOM

H.O.T. Problems

49. OPEN ENDED Write a numerical sentence that illustrates the Commutative Property of Multiplication. 50. FIND THE ERROR Kimberly and Carlos are using the Associative Properties of Addition and Multiplication to rewrite expressions. Who is correct? Explain your reasoning. Carlos (2 + 7) · 5 = 2 + (7 · 5)

Kimberly (4 + 3) + 6 = 4 + (3 + 6)

51. CHALLENGE The Closure Property states that because the sum or product of two whole numbers is also a whole number, the set of whole numbers is closed under addition and multiplication. Is the set of whole numbers closed under subtraction and division? If not, give counterexamples. 52.

Writing in Math Explain how real-life situations can be commutative. Give an example of a real-life situation that is commutative and one that is not commutative.

53. How can you find 2 · 198 · 5 mentally? A Use the Associative Property.

54. Which property can NOT be used to show that 10 + 6 + 8 = 10 + 8 + 6?

B Use the Commutative Property.

F Associative Property of Addition

C Use the Additive Identity.

G Associative Property of Multiplication

D Use the Multiplicative Identity.

H Commutative Property of Addition J Multiplicative Identity

ALGEBRA Evaluate each expression if a = 6, b = 4, and c = 5. (Lesson 1-3) 55. a + c - b

56. 8a - 3b

57. 4a - (b + c)

58. 10a ÷ c

59. ALGEBRA Find the value of the expression 4 · (8 + 9) + 6. (Lesson 1-2) 60. MUSIC During a spring concert, the jazz band has 15 minutes to perform. If each of the songs they are considering performing is about 4 minutes long, about how many songs can they play? (Lesson 1-1)

PREREQUISITE SKILL Find each product. 61. 48 × 5

62. 8 × 37

63. 16 × 12

64. 25 × 42

65. 106 × 13

Lesson 1-4 Properties

47

CH

APTER

1

Mid-Chapter Quiz Lessons 1-1 through 1-4

1. GEOMETRY Draw the next two figures in the pattern. (Lesson 1-1)

Find the value of each expression. (Lesson 1-2) 5. 5 + 13 × 2 7. 28 ÷ 4 × 2

6. 7 + 8 - 4 8. 7(3 + 10) - 2 · 6

6(15 + 3) 9. 3[6(12 - 3)] - 17 10. _ 6(9 - 6)

2. MULTIPLE CHOICE The table shows the costs of four weekly magazines. Which magazine saves you the most money if you purchase a yearly subscription instead of an equivalent number of single copies? (Lesson 1-1) Magazine

Cost of Yearly Subscription

Cost of Single Copy

A

$129.99

$2.99

B

$99.95

$2.29

C

$200.95

$3.95

D

$160.00

$3.50

Write a numerical expression for each verbal phrase. (Lesson 1-2) 11. fourteen increased by forty-two 12. six less than the product of seven and nine Evaluate each expression if x = 4 and y = 2. (Lesson 1-3) 13. 5y 15. 7x - 3y

14. x + 10y 16. 9y + 4 - x

16 17. _ x

3x 18. _ x+y

A A

C C

B B

D D

19. SPACE Due to gravity, objects weigh three times as much on Earth as they do on Mercury. How much would an object weigh on Earth if it weighs 25 pounds on Mercury? (Lesson 1-3)

3. MULTIPLE CHOICE The distance between the school and the museum is 24 miles. If the bus driver averages 36 miles per hour, about how long would it take to travel from the school to the museum? (Lesson 1-1)

20. MULTIPLE CHOICE A taxi charges $1.25 for the first mile and then $0.75 for each additional mile m. Which expression can be used to find the total cost Grace would pay for a ride in a taxi? (Lesson 1-3)

F 30 min

H 45 min

A 0.75 + 1.25m

G 40 min

J

50 min

B 1.25 + 0.75m C 0.75m

4. WATER PARKS The table shows the price of admission to a water park. Write an expression to find the cost of admission for 3 adults, 4 children under 8, and 2 senior citizens. (Lesson 1-2) Ticket

D 1.25m Simplify each expression. (Lesson 1-4) 21. (7 + a) + 9

22. 8 · (h · 3)

23. 10 + (g + 20)

24. (12 · p) · 6

Price

Adult

$10

Senior Citizen

$6

Children (ages 8–13)

$8

Children under 8

$5


25. SCIENCE In chemistry, water is used to dilute acid. Since pouring water into acid could cause spattering and burns, it is important to pour the acid into the water. Is combining acid and water commutative? Explain. (Lesson 1-4)

1-5

Variables and Equations

Main Ideas • Identify and solve open sentences. • Translate verbal sentences into equations.

New Vocabulary equation open sentence solution solving the equation

The table shows the -ENS M &REESTYLE top four places of the men’s 3WIM 4IME 3WIMMER 1500-meter freestyle swimming 4IME S "EHIND B final in the 2004 Olympics. (ACKETT !53 n a. How far behind Hackett was *ENSEN 53! each swimmer? $AVIES '"2 b. Write a rule to describe how you found the time behind for 0RILUKOV 253 each swimmer. -ÕÀVi\ >ÌiÃÓää{°V c. Let s represent the swim time and b represent the amount of time behind Hackett. Rewrite your rule using numbers and variables.

Equations and Open Sentences A mathematical sentence that contains an equals sign (=) is called an equation. A few examples are shown. 5 + 9 = 14

2(6) - 3 = 9

x + 7 = 19

2m - 1 = 13

An equation that contains a variable is an open sentence. An open sentence is neither true nor false. When the variable in an open sentence is replaced with a number, you can determine whether the sentence is true or false. x + 7 = 19

x + 7 = 19

Reading Math Math Symbols The symbol ≠ means is not equal to.

11 + 7 19 Replace x with 11.

12 + 7 19 Replace x with 12. 19 = 19 true

18 ≠ 19 false

When x = 12, this sentence is true.

When x = 11, this sentence is false.

A value for the variable that makes an equation true is called a solution. For x + 7 = 19, the solution is 12. The process of finding a solution is called solving the equation.

EXAMPLE

Solve an Equation

Find the solution of 12 - m = 8. Is it 2, 4, or 7? Replace m with each value. Therefore, the solution of 12 - m = 8 is 4.

Value for m

12 - m = 8

True or False?

2

12 - 2 8

false

4

12 - 4 8

true

7

12 - 7 8

false

1. Find the solution of 18 = n + 7. Is it 8, 9, or 11? Lesson 1-5 Variables and Equations

49

Which value of x makes the equation 2x + 1 = 7 true? A 6

B 5

C 4

D 3

Read the Test Item The solution is the value that makes the equation true. Solve the Test Item Test each value. 2x + 1 = 7 Backsolving The strategy of testing each value is called backsolving. You can also use this strategy with complex equations.

Original equation

2x + 1 = 7

Original equation

2(6) + 1 = 7 Replace x with 6.

2(5) + 1 = 7

Replace x with 5.

13 ≠ 7 False 2x + 1 = 7

11 ≠ 7

False

Original equation

2x + 1 = 7

Original equation

2(4) + 1 = 7 Replace x with 4.

2(3) + 1 = 7

Replace x with 3.

9≠7

7 = 7 True

False

Since 3 makes the equation true, the answer is D.

2. Which value is the solution of 5x - 6 = 14? F3

G4

H5

J6


Translate Verbal Sentences into Equations Just as verbal phrases can be translated into algebraic expressions, verbal sentences can be translated into equations and then solved.

EXAMPLE

Translate Sentences into Equations

The difference of a number and ten is seventeen. Find the number. Words Variable Equation

The difference of a number and ten is seventeen. Let n = the number. n - 10 = 17

n - 10 = 17 Write the equation. 27 - 10 = 17 Solve mentally: What number minus 10 is 17? n = 27 The solution is 27.

3. The sum of a number and nine is twenty-one. Find the number. As with expressions, equations can also have two variables. The value of one variable changes as a change is made to the other variable. The value of one variable depends on the value of the other variable. A good way to see this relationship is with a table. 50 Chapter 1 The Tools of Algebra

APPLE CIDER A bushel of apples will make approximately 3 gallons of apple cider. The table shows the relationship between the number of bushels of apples and the number of gallons of apple cider.

!PPLE "USHELS L

a. Given b, the number of bushels needed, write an equation that can be used to find g, the number of gallons of apple cider. Words Variables Equation Reasonableness To check the equation, substitute values from the data table and verify that the equation works.

is Let = n

the

'ALLONS OF !PPLE #IDER }

. . Let =

=

b. How many bushels are needed to make 54 gallons of cider? g = 3b

Write the equation.

54 = 3b

Replace g with 54.

54 = 3(18) Solve: What number times 3 is 54? 18 = b

4. PART-TIME JOB Raul’s lawn mower runs for 1.5 hours on one gallon of gas. Given g gallons of gas, write an equation to find h, the number of hours the mower can run. Then find the number of gallons used in 6 hours.

Example 1 (p. 49)

Example 2 (p. 50)

Example 3 (p. 50)

Example 4 (p. 51)

ALGEBRA Find the solution of each equation from the list given. 1. h + 15 = 21; 5, 6, 7 3. k - 25 = 12; 36, 37, 38

2. 13 - m = 4; 7, 8, 9 4. 22 + n = 41; 18, 19, 20

48 5. MULTIPLE CHOICE Find the value of k that makes 6 = _ true. k

A6

B7

C8

D 12

ALGEBRA Define a variable. Then write an equation and solve. 6. A number increased by 8 is 23.

7. Twenty-five is 10 less than a number.

TRAVEL For Exercises 8 and 9, use the following information. The Geiger family is driving at an average speed of 55 miles per hour. The table shows the relationship between the distance driven and the time. 8. Given t, the time in hours, write an equation that can be used to find d, the distance driven. 9. How long would it take them to drive 495 miles?

Extra Examples at pre-alg.com

Time t (hours)

Distance d (miles)

1 2 4 5

55 110 220 275

Lesson 1-5 Variables and Equations

51

HOMEWORK

HELP


ALGEBRA Find the solution of each equation from the list given. 10. c + 12 = 30; 8, 16, 18

11. g + 17 = 28; 9, 11, 13

12. 23 - m = 14; 7, 9, 11

13. 18 - k = 6; 8, 10, 12

14. 14k = 42; 2, 3, 4

15. 75 = 15n; 3, 4, 5

51 16. _ z = 3; 15, 16, 17

60 17. _ p = 4; 15, 16, 17

ALGEBRA Define a variable. Then write an equation and solve. 18. 19. 20. 21. 22. 23.

The sum of 7 and a number is 23. The sum of 9 and a number is 36. A number minus 10 is 27. The difference between a number and 12 is 54. Twenty-four is the product of 8 and a number. A number times 3 is 45.

PLUMBING For Exercises 24 and 25, use the following information. A standard showerhead uses about 6 gallons of water per minute. The table shows the relationship between time and the water used. 24. Given m, the number of minutes, write an equation that can be used to find g, the number of gallons used. 25. How many minutes elapsed if 72 gallons of water were used? Real-World Link In 1990, the total number of indoor movie screens was about 23,000. Today, there are over 35,000 indoor movie screens. Source: National Association of Theatre Owners

CURRENCY For Exercises 26 and 27, use the following information. In a recent year, 1 U.S. dollar could be exchanged for 0.78 euros. The table shows the relationship between U.S. dollars and euros. 26. Given d, the number of U.S. dollars, write an equation that can be used to find c, the number of euros. 27. How many U.S. dollars can you receive for 7.8 euros?

4AKING A 3HOWER

4IME M MINUTES

7ATER 5SED G GALLONS

U.S. Dollars d

Euros c

1

0.78

2

1.56

3

2.34

5

3.90

Source: exchangerate.com

28. MOVIES Mariko purchased three movie tickets for $24. Define a variable. Then write an equation that can be used to find how much Mariko paid for each ticket. What was the cost of each ticket? EXTRA

PRACTICE


29. HEIGHT During the summer, Ana grew from a height of 65 inches to a height of 68 inches. Define a variable. Then write an equation that can be used to find the increase in height. How many inches did Ana grow? 30. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you write and solve an equation.

52 Chapter 1 The Tools of Algebra RNT Productions/CORBIS

H.O.T. Problems

31. OPEN ENDED Write two different equations whose solutions are each 5. 32. CHALLENGE Write three different equations for which there is no solution that is a whole number. 33. SELECT A TECHNIQUE Student Council had a budget of $650 for the winter dance. It had already spent $439. Which of the following technique(s) might Student Council use to determine how much money it has left? Justify your selection(s). Then use the technique(s) to solve the problem. make a model

34.

calculator

paper/pencil

Writing in Math Explain how solving an open sentence is similar to evaluating an expression. How are they different?

35. If the perimeter of the X V pentagon is 54 centimeters, find V the equation that will n V allow you to find the length of the missing side x. A 2x - 15 + 11 + 8 + 9 = 54 B x - (15 + 11 + 8 + 9) = 54 C x(15 + 11 + 8 + 9) = 54

£x V ££ V

36. Mr. Farley is running a race at a speed of 3.5 miles per hour. Which equation can be used to find the number of miles m he can run in h hours? F h = 3.5 + m G m = 3.5h H h = 3.5m J m = 3.5 + h

D 15 + 11 + 8 + 9 + x = 54

Simplify each expression. (Lesson 1-4) 37. 16 + (7 + d)

38. (4 · p) · 6

39. (23 + k) + 34

40. 10 · (z · 9)

BUSINESS For Exercises 41 and 42, use the following information. (Lesson 1-3) Cornet Cable charges $32.50 a month for basic cable television. Each premium channel selected costs an additional $4.95 per month. 41. Write an expression to find the cost of a month of cable service. 42. How much does cable service cost per year if Abby subscribes to 3 premium channels? Evaluate each expression. (Lesson 1-2) 43. 2 + 3 · 5

44. 8 ÷ 2 · 4

45. 10 - 2 · 4

46. (3 · 4) + (9 · 5)

47. What is the next term in 67, 62, 57, 52, 47, …? (Lesson 1-1)

PREREQUISITE SKILL Evaluate each expression for the given value. (Lesson 1-3) 48. 4x; x = 3

49. 3m; m = 6

50. 12p; p 11

51. 19u; u = 5

Lesson 1-5 Variables and Equations

53

1-6

Ordered Pairs and Relations

Main Ideas • Use ordered pairs to locate points.

Elisa and Nhu are playing a game. The player who gets four Xs or Os in a row wins. 1st move

• Use tables and graphs to represent relations.

X O O

2nd move Nhu places an O at 2 over and 2 up.

New Vocabulary coordinate system y-axis coordinate plane origin x-axis ordered pair x-coordinate y-coordinate graph relation domain range

Elisa places an X at 1 over and 3 up.

3rd move 4th

move

Elisa places an X at 1 over and 1 up.

X

Nhu places an O at 1 over and 2 up. Starting Position

a. Where should Elisa place an X now? Explain your reasoning. b. Suppose (1, 2) represents 1 over and 2 up. How could you represent 3 over and 2 up? c. How are (5, 1) and (1, 5) different? d. Where is a good place to put the next O? e. Work with a partner to finish the game.

Ordered Pairs In mathematics, a coordinate system is used to locate points. The coordinate system is formed by the intersection of two number lines that meet at right angles at their zero points. The vertical number line is called the y-axis.

The origin is at (0, 0), the point at which the number lines intersect.

8 7 6 5 4 3 2 1 O

The coordinate system is also called the coordinate plane.

y

1 2 3 4 5 6 7 8x

The horizontal number line is called the x-axis.

An ordered pair of numbers is used to locate any point on a coordinate plane. The first number is called the x-coordinate. The second number is called the y-coordinate. The x-coordinate corresponds to a number on the x-axis.


(3, 2)

The y-coordinate corresponds to a number on the y-axis.

To graph an ordered pair, draw a dot at the point that corresponds to the ordered pair. The coordinates are your directions to locate the point. Coordinate System Unless they are marked otherwise, you can assume that each unit on the xand y-axis represents 1 unit. Axes is the plural of axis.

EXAMPLE

Graph Ordered Pairs

Graph each ordered pair on a coordinate system. a. (4, 1)

y

Step 1

Start at the origin.

Step 2

Since the x-coordinate is 4, move 4 units to the right.

Step 3

Since the y-coordinate is 1, move 1 unit up. Draw a dot.

(4 , 1) x

O

b. (3, 0) Step 1


Step 2

The x-coordinate is 3. So, move 3 units to the right.

Step 3

Since the y-coordinate is 0, you will not need to move up. Place the dot on the axis.

y

(3 , 0) x

O

Graph each ordered pair on a coordinate system. 1 1A. (2, 3) 1B. (0, 2) 1C. 3, 1_

2

Sometimes a point on a graph is named by using a letter. To identify its location, you can write the ordered pair that represents the point.

EXAMPLE

Identify Ordered Pairs

Write the ordered pair that names each point. a. M Step 1


y

Step 2

Move right on the x-axis to find the x-coordinate of point M, which is 2.

Q

Step 3

M N

Move up the y-axis to find the y-coordinate, which is 5.

The ordered pair for point M is (2, 5).

P O

x

b. P The x-coordinate of P is 7, and the y-coordinate is 0. The ordered pair for point P is (7, 0).

2A. N

2B. Q Personal Tutor at pre-alg.com Lesson 1-6 Ordered Pairs and Relations

55

Relations A set of ordered pairs such as {(1, 2), (2, 4), (3, 0), (4, 5)} is a relation. A relation can also be shown in a table or a graph. The domain of the relation is the set of x-coordinates. The range of the relation is the set of y-coordinates. Ordered Pairs Table Graph y (1, 2) x y (2, 4) 1 2 (3, 0) 2 4 (4, 5) The domain is {1, 2, 3, 4}.

The range is {2, 4, 0, 5}.

EXAMPLE Interactive Lab pre-alg.com

3

0

4

5

x

O

Relations as Tables and Graphs

Express the relation {(0, 0), (2, 1), (1, 3), (5, 2)} as a table and as a graph. Then determine the domain and range. x

y

0

0

2

1

1

3

5

2

The domain is {0, 2, 1, 5}, and the range is {0, 1, 3, 2}.

y

O

x

3. Express the relation {(2, 4), (0, 3), (1, 4), (1, 1)} as a table and as a graph. Then determine the domain and range.

PLANTS Some species of bamboo grow 3 feet in one day.

Real-World Link Bamboo is a type of grass. It can vary in height from 1-foot dwarf plants to 100-foot giant timber plants. Source: American Bamboo Society

x

y

(x, y)

1

3

(1, 3)

2

6

(2, 6)

3

9

(3, 9)

4

12

(4, 12)

Bamboo Growth 14 12 10 8 6 4 2 0

y

1 2 3 4 5 6

x

Days

c. Describe the graph. The points appear to fall in a line.

4. CAPACITY One quart is the same as two pints. Make a table of ordered pairs in which the x-coordinate represents the number of quarts and the y-coordinate represents the number of pints for 1, 2, 3, and 4 quarts. Graph the ordered pairs and then describe the graph.

56 Chapter 1 The Tools of Algebra Michael Boys/CORBIS

b. Graph the ordered pairs.

Growth (ft)

a. Make a table of ordered pairs in which the x-coordinate represents the number of days and the y-coordinate represents the amount of growth for 1, 2, 3, and 4 days.


Example 1 (p. 55)

Example 2 (p. 55)

Example 3 (p. 56)

Graph each ordered pair on a coordinate system. 1. H(5, 3)

2. D(6, 0)

3. W(4, 1)

4. Z(0, 1)

Refer to the coordinate system shown at the right. Write the ordered pair that names each point. 5. Q

6. P

7. S

8. R

y

P Q

S

Express each relation as a table and as a graph. Then determine the domain and range.

R x

O

9. {(2, 5), (0, 2), (5, 5)} 10. {(1, 6), (6, 4), (0, 2), (3, 1)} Example 4 (p. 56)

HOMEWORK

HELP


ENTERTAINMENT For Exercises 11 and 12, use the following information. It costs $4 to buy a student ticket to the movies. 11. Make a table of ordered pairs in which the x-coordinate represents the number of student tickets and the y-coordinate represents the cost for 2, 4, and 5 tickets. 12. Graph the ordered pairs (number of tickets, cost).

Graph each ordered pair on a coordinate system. 13. A(3, 3)

14. D(1, 8)

15. G(2.5, 7)

16. X(7, 2)

17. P(0, 6)

1 18. N 4 _ , 0

Refer to the coordinate system at the right. Write the ordered pair that names each point. 19. C

20. J

21. N

22. T

23. Y

24. B

Express each relation as a table and as a graph. Then determine the domain and range.

2

y

C

T B J

N Y

x

O

25. {(4, 5), (5, 2), (1, 6)}

26. {(6, 8), (2, 9), (0, 1)}

27. {(7, 0), (3, 2), (4, 4), (5, 1)}

28. {(2, 4), (1, 3), (5, 6), (1, 1)}

29. {(0, 1), (0, 3), (0, 5), (2, 0)}

30. {(4, 3), (3, 4), (1, 2), (2, 1)}

AIR PRESSURE For Exercises 31–33, use the table and the following information.

Height (mi)

Pressure (lb/in2)

0 (sea level)

14.7

1

10.2

31. Write a set of ordered pairs for the data.

2

6.4

32. Graph the data.

3

4.3

33. State the domain and the range of the relation.

4

2.7

5

1.6

The air pressure decreases as the distance from Earth increases. The table shows the air pressure for certain distances.

Lesson 1-6 Ordered Pairs and Relations

57

SCIENCE For Exercises 34–36, use the following information. Elizabeth is conducting a physics experiment. She drops a tennis ball from a height of 100 centimeters and then records the height after each bounce. The results are shown in the table. Bounce Height (cm)

0

1

2

3

4

100

50

25

13

6

34. Write a set of ordered pairs for the data. 35. Graph the data. 36. How high do you think the ball will bounce on the fifth bounce? Explain. SCIENCE For Exercises 37– 40, use the following information and the information at the left. Water boils at sea level at 100°C. The boiling point of water decreases about 5°C for every mile above sea level. Real-World Link Salt Lake City, Utah, is 4330 feet above sea level. Anderson, South Carolina, is 772 feet above sea level. Source: The World Almanac

37. Make a table that shows the boiling point at sea level and at 1, 2, 3, 4, and 5 miles above sea level. 38. Show the data as a set of ordered pairs. 39. Graph the ordered pairs. 40. At about what temperature does water boil in Anderson, South Carolina? in Salt Lake City, Utah? (Hint: 1 mile = 5280 feet) Graph each ordered pair on a coordinate system. 41. W(0.25, 4)

3 43. Y 2_ , 0

42. X(1, 1.3)

1 _ 44. Z 3_ , 31

4

5

4

45. Where are all of the possible locations for the graph of (x, y) if y = 0? If x = 0? EXTRA

PRACTICE

See pages 762, 794.

Graph each relation on a coordinate system. Then find the coordinates of another point that follows the pattern in the graph. 46.

Self-Check Quiz at pre-alg.com

H.O.T. Problems

x

1

3

5

7

y

2

4

6

8

47.

x

0

2

4

6

y

10

8

6

4

48. OPEN ENDED Give an example of an ordered pair, and identify the x- and y-coordinate. GEOMETRY For Exercises 49–53, draw a coordinate system. 49. Graph (2, 1), (2, 4), and (5, 1). 50. Connect the points with line segments. Describe the figure formed. 51. Multiply each coordinate in the set of ordered pairs by 2. 52. Graph the new ordered pairs. Connect the points with line segments. What figure is formed? 53. MAKE A CONJECTURE How do the figures compare? Write a sentence explaining the similarities and differences of the figures. 54. CHALLENGE Where are all of the possible locations for the graph of (x, y) if x = - 2? 55.

Writing in Math Use the information about ordered pairs found on pages 54–56 to explain how they are used to graph real-life data. Include an example of a situation where ordered pairs are used to graph data.

58 Chapter 1 The Tools of Algebra age fotostock/SuperStock

7ÀÕÌ /i ÕÌiÃ®

56. Felipe drew a graph that shows his daily workout times for the past five days. Find the range of the relation.

57. What relationship exists between the x- and y-coordinates of each of the data points shown on the graph? y

>Þ 7ÀÕÌÃ Îä

Y

Ón ÓÈ Ó{ ÓÓ Óä ä

£

Ó

Î { >Þ

x

x

O

È X

F The y-coordinate varies, and the x-coordinate is always 4.

A {5, 29} B {1, 2, 3, 4, 5}

G The y-coordinate is 4 more than the x-coordinate.

C {20, 21, 26, 28, 29}

H The sum of the x- and y-coordinate is always 4.

D {1, 20}, {2, 21}, {3, 26}, {4, 28}, {5, 29}

J The x-coordinate varies, and the y-coordinate is always 4.

ALGEBRA Solve each equation. (Lesson 1-5) 58. a + 6 = 17

59. 28 = j - 13

54 61. _ n =6

60. 7t = 42

62. Name the property shown by 4 · 1 = 4. (Lesson 1-4) ALGEBRA Evaluate each expression if a = 5, b = 1, and c = 3. (Lesson 1-3) 65 63. _ a

64. a + bc

65. ca - cb

66. 5a - 6c

Write a numerical expression for each verbal phrase. (Lesson 1-2) 67. fifteen less than twenty-one

68. the product of ten and thirty

69. twelve divided into sixty

70. the total of fourteen and nine

71. MANUFACTURING A wagon manufacturer can produce 8000 wagons a day at peak production. Explain how you can find the maximum number of wagons that can be produced in a year. Then find the total. (Lesson 1-1)

PREREQUISITE SKILL Find each quotient. 72. 74 ÷ 2

73. 96 ÷ 8

74. 102 ÷ 3

75. 112 ÷ 4

76. 80 ÷ 16

77. 91 ÷ 13

78. 132 ÷ 22

79. 153 ÷ 17

Lesson 1-6 Ordered Pairs and Relations

59

EXPLORE

1-7

Algebra Lab

Scatter Plots Sometimes, it is difficult to determine whether a relationship exists between two sets of data by simply looking at them. To determine whether a relationship exists, we can write the data as a set of ordered pairs and then graph the ordered pairs on a coordinate system.

ACTIVITY Collect data to investigate whether a relationship exists between height and arm span. Step 1 Work with a partner. Use a meterstick to measure your partner’s height and the length of your partner’s arm span to the nearest centimeter. Record the data in a table like the one shown. Name

Height (cm)

Arm Span (cm)

y

Step 3 Make a list of ordered pairs in which the x-coordinate represents height and the y-coordinate represents arm span. Step 4 Draw a coordinate plane like the one shown and graph the ordered pairs (height, arm span).

Arm Span (cm)

Step 2 Extend the table. Combine your data with that of your classmates.

O

Height (cm)

x

ANALYZE THE RESULTS 1. Does there appear to be a trend in the data? If so, describe the trend. 2. Using your graph, estimate the arm span of a person whose height is 60 inches. 72 inches. 3. How does a person’s arm span compare to his or her height? 4. MAKE A CONJECTURE Suppose the variable x represents height and the variable y represents arm span. Write an expression for arm span. 5. Collect and graph data to determine whether a relationship exists between height and shoe length. Explain your results. 60 Chapter 1 The Tools of Algebra

1-7

Scatter Plots

Main Ideas

New Vocabulary scatter plot

a. What appears to be the trend in sales of movies on videocassette?

Videocassette Sales Number Sold

• Analyze trends in scatter plots.

Suppose you work at a video store. The number of movies on videocassettes sold in a five-year period is shown in the graph.

200 160 120 80 40 0 ’02

’03

b. Estimate the number of movies on videocassette sold for 2008.

’04 Year

’05

’06

Construct Scatter Plots A scatter plot is a graph that shows the relationship between two sets of data. In a scatter plot, two sets of data are graphed as ordered pairs on a coordinate system.

EXAMPLE

Construct a Scatter Plot

TEST SCORES Make a scatter plot of the average SAT math scores from 1995–2004.

Year

Score

‘95

506

‘96

508

‘97

511

‘98

512

‘99

511

‘00

514

‘01

514

‘02

516

‘03

519

‘04

518

Let the horizontal axis, or x-axis, represent the year. Let the vertical axis, or y-axis, represent the score. Then graph ordered pairs (year, score). ÛiÀ>}i -/ -VÀiÃ] £xqÓää{

-VÀi

• Construct scatter plots.

Source: The College Board

9i>À

1. Make a scatter plot of the average ACT scores from 1995 to 2004. Year

‘95

‘96

‘97

‘98

‘99

‘00

‘01

‘02

‘03

‘04

Score

20.8

20.9

21.0

21.0

21.0

21.0

21.0

20.8

20.8

20.9


Lesson 1-7 Scatter Plots

61

Analyze Scatter Plots The following scatter plots show the types of relationships or patterns of two sets of data.

Types of Relationships Scatter Plots Data that appear to go uphill from left to right show a positive relationship. Data that appear to go downhill from left to right show a negative relationship.

Positive Relationship

Negative Relationship

y

O

As x increases, y increases.

EXAMPLE

No Relationship

y

x

O

y

x

x

O

As x increases, y decreases.

No obvious pattern.

Interpret Scatter Plots

CAR VALUE Determine whether a scatter plot of the age of a car and the value of a car might show a positive, negative, or no relationship. Explain your answer.

Value (thousands of dollars)

Car Value

As the age of a car increases, the value of the car decreases. So, a scatter plot of the data would show a negative relationship.

Real-World Link

27 y 24 21 18 15 12 9 6 3 0

x 1 2 3 4 5 6 7 8 9 Age (years)

A car loses 15–20% of its value each year.

2. Determine whether a scatter plot of the birth month and birth weight data might show a positive, negative, or no relationship. Explain your answer.

Birth Weight

Birth Weight (lb)

Source: bankrate.com

10 y 9 8 7 6 5 4 3 2 1 0

x J F M A M J J A S O N D Birth Month

You can also use scatter plots to spot trends, draw conclusions, and make predictions about the data. 62 Chapter 1 The Tools of Algebra Toyota

BIOLOGY A biologist recorded the lengths and weights of some largemouth bass. The table shows the results. Length (in.)

9.2

10.9 12.3 12.0 14.1 15.5 16.4 16.9 17.7 18.4 19.8

Weight (lb)

0.5

0.8

0.9

1.3

1.7

2.2

2.5

3.2

a. Make a scatter plot of the data.

Biologist A biologist uses math to study animal populations and monitor trends of migrating animals. For more information, go to pre-alg.com.

4.1

4.8

Largemouth Bass

Weight (lb)

Let the horizontal axis represent length, and let the vertical axis represent weight. Graph the data.

Real-World Career

3.6

b. Does the scatter plot allow you to draw a conclusion about a relationship between the length and weight of a largemouth bass? Explain. As the length of the bass increases, so does its weight. So, the scatter plot shows a positive relationship.

5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 9 10 11 12 13 14 15 16 17 18 19 20 Length (in.)

c. Predict the weight of a bass that measures 22 inches. By looking at the pattern in the graph, we can predict that the weight of a bass measuring 22 inches would be between 5 and 6 pounds.

KEYBOARDING The table shows keyboarding speeds in words per minute (wpm) of 12 students. Experience (weeks)

4

7

8

1

6

3

5

2

9

6

7

10

Speed (wpm)

38

46

48

20

40

30

38

22

52

44

42

55

3A. Make a scatter plot of the data. 3B. Draw a conclusion about the type of relationship the data shows. 3C. Predict the keyboarding speed of a student with 12 weeks of experience. Personal Tutor at pre-alg.com

Example 1 (p. 61)

Example 2 (p. 62)

1. HEALTH CARE The table shows the number of physicians and hospital beds for nine rural counties. Make a scatter plot of the data. Physicians

11

26

10

19

22

9

15

7

1

Hospital Beds

85

67

32

69

49

43

90

49

18

2. Determine whether a scatter plot of hours worked and weekly earnings of a person on the wait staff of a restaurant would show a positive, negative, or no relationship. Explain your answer.

Extra Examples at pre-alg.com David Hiser/Stone/Getty Images

Lesson 1-7 Scatter Plots

63

COMMUNICATION The table shows the number of people in a family and the number of telephone calls made per week.

Example 3 (p. 63)

Number in Family

5

1

4

2

4

6

3

4

7

3

5

8

2

Number of Calls

31

8

26

9

18

34

13

10

25

15

20

36

15

3. Make a scatter plot of the data. 4. Does the scatter plot show a relationship between the number of people in a family and the number of telephone calls made per week? Explain. 5. If a relationship exists, predict the number of calls made during the week for a family of 10.

HOMEWORK

HELP


6. MUSIC The table shows the number of songs and the total number of minutes on different CDs. Make a scatter plot of the data. Number of Songs

15

18

20

13

12

15

16

17

14

18

20

19

11

14

Total Minutes

64

78

63

70

59

61

77

75

72

71

78

75

63

69

7. OLYMPICS The table shows the winning times for the women’s Olympic 100-meter run. Make a scatter plot of the data. Year

‘28

‘32

‘36

‘48

‘52

‘56

‘60

‘64

‘68

Winning Times (s)

12.2

11.9

11.5

11.9

11.5

11.5

11.0

11.4

11.08

Year

‘72

‘76

‘80

‘84

‘88

‘92

‘96

‘00

‘04

11.07

11.08

11.06

10.97

10.54

10.82

10.94

10.75

10.93

Winning Times (s) Source: olympic.org

Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. 8. size of household and amount of water bill 9. hair color and height 10. temperature and heating costs 11. speed and distance traveled Determine whether the scatter plot of the data shows a positive, negative, or no relationship. Explain your answer. 12.

13.

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X n £Ó £È Óä Ó{ Ón ÎÓ

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È x { Î Ó £ ä

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BASKETBALL For Exercises 14–16, use the information and table below. The number of minutes played and the number of field goal attempts for certain players of the Los Angeles Sparks for the 2004 season are shown. Player

Leslie Mabika Teasley Dixon Whitmore

Minutes Played

Field Goal Attempts

Player

Minutes Played

Field Goal Attempts

1150 965 1105 913 595

451 383 278 269 173

Milton-Jones Thomas Macchi Hodges Masciadri

604 547 410 245 116

161 143 106 52 25

Source: wnba.com

14. Make a scatter plot of the data. 15. Does the scatter plot allow you to draw a conclusion about the relationship between minutes played and field goal attempts? Explain. 16. Suppose a player played 1500 minutes. If a relationship exists, predict the number of field goal attempts for that player. 17. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you draw a scatter plot.

EXTRA

PRACTICE


H.O.T. Problems

The high and low temperatures for your vacation destinations can be shown in a scatter plot. Visit pre-alg.com to continue work on your project.

19. What appears to be the trend in the number of hatchlings between 1965 and 1972? 20. What appears to be the trend between 1972 and 1985?

Average Number of Bald Eagles per Breeding Area 1.4 Number of Eagles

ANIMALS For Exercises 18–21, use the scatter plot shown. 18. Do the data show a positive, negative, or no relationship between the year and the number of bald eagle hatchlings?

1.2 1.0 0.8 0.6 0.4 0 ’65

’70

’75

’80

’85

Year

Source: CHANCE 21. What factors could contribute to the trends displayed in the scatter plot? Predict the number of eagles in years after the data points.

22. OPEN ENDED Draw a scatter plot with ten ordered pairs that shows a negative relationship. 23. CHALLENGE Refer to Example 1 on page 61. Do you think the upward trend in the test scores will continue indefinitely? Why or why not? Explain. NUMBER SENSE What type of relationship is shown on a graph that shows the following values? 24. As x increases, y decreases. 25. As x decreases, y decreases. 26. As x decreases, y increases. 27.

Writing in Math

Explain how you can use scatter plots to help you spot trends. Include real-life examples to illustrate each type of scatter plot. Lesson 1-7 Scatter Plots

65

The scatter plot shows the study time and test scores for the students in Ms. Flores’ math class.

29. Which statement best describes the relationship in the scatter plot? F The longer students studied, the better they did on the test.

Test Score

Study Time and Test Scores

G The shorter students studied, the better they did on the test.

100 95 90 85 80 75 70 65 60 0

H The longer students studied, the worse they did on the test. J There is no relationship between study time and test scores. 10

30

50

70

90 110 20 40 60 80 100 120 Study Time (min)

30. Based on the results, which of the following is a reasonable amount of study time for a student who scores a 75 on the test?

28. Based on the results, which of the following is a reasonable score for a student who studies for 1 hour? A 68

C 87

B 72

D 98

A 10 min

C 61 min

B 32 min

D 88 min

Graph each ordered pair on a coordinate system. (Lesson 1-6)

31. M(3, 2)

32. X(5, 0)

33. K(0, 2)

34. Determine the domain and range of the relation {(0, 9), (4, 8), (2, 3), (6, 1)}. (Lesson 1-6) ANIMALS One year of a dog’s life is equivalent to 7 years of human life, as shown in the table. (Lesson 1-5) 35. Given d, a dog’s age, write an equation to find h, the equivalent human age. 36. What is the age of a dog, if the equivalent human age is 42?

! $OGS ,IFE %QUIVALENT $OGS !GE D (UMAN !GE H

37. ALGEBRA Simplify 15 + (b + 3). (Lesson 1-4) ALGEBRA Evaluate each expression if m = 8 and y = 6. (Lesson 1-3) 38. (2m + 3y) - m

39. 3m + (y - 2) + 3

40. 16 + (mn - 12)

41. COMMUNICATION A telephone tree is set up so that every person calls three other people. Jeffrey needs to tell his co-workers about a time change for a meeting. Suppose it takes 2 minutes to call 3 people. In 10 minutes, how many people will know about the change of time? (Lesson 1-1) 66 Chapter 1 The Tools of Algebra

Graphing Calculator Lab

EXTEND

1-7

Scatter Plots You have learned that graphing ordered pairs as a scatter plot on a coordinate plane is one way to make it easier to “see” if there is a relationship. You can use a TI-83/84 Plus graphing calculator to create scatter plots.

ACTIVITY SCIENCE A zoologist studied extinction times (in years) of island birds. The zoologist wanted to see if there was a relationship between the average number of nests and the time needed for each bird to become extinct on the islands. Use the table of data below to make a scatter plot. Bird Name

Bird Size

Average Number of Nests

Extinction Time

Buzzard

Large

2.0

5.5

Quail

Large

1.0

1.5

Curlew

Large

2.8

3.1

Cuckoo

Large

1.4

2.5

Magpie

Large

4.5

10.0

Swallow

Small

3.8

2.6

Robin

Small

3.3

4.0

Stonechat

Small

3.6

2.4

Blackbird

Small

4.7

3.3

Tree-sparrow

Large

2.0

5.5

Step 1 Enter the data.

The first data pair is (2, 5.5).

• Clear any existing list. KEYSTROKES:

STAT

CLEAR

• Enter the average number of nests as L1 and extinction times as L2. KEYSTROKES:

2

STAT

2.2

5.5

1

…

1.5

… 1.9 Step 2 Format the graph. • Turn on the statistical plot. KEYSTROKES:

2nd [STAT PLOT]

• Select the scatter plot, L1 as the Xlist and L2 as the Ylist. KEYSTROKES:

2nd [L1]

Other Calculator Keystrokes at pre-alg.com Jim Zipp/Photo Researchers

(continued on the next page) Extend 1-7 Graphing Calculator Lab: Scatter Plots

67

Step 3 Graph the data. • Display the scatter plot. KEYSTROKES:

ZOOM 9

• Use the TRACE feature and the left and right arrow keys to move from one point to another.

ANALYZE THE RESULTS RACE . Use the left and right arrow keys to move from 1. Press TRACE one point to another. What do the coordinates of each data point represent?

2. Describe the scatter plot. 3. Is there a relationship between the average number of nests and extinction times? If so, write a sentence or two that describes the relationship. 4. Are there any differences between the extinction times of large birds versus small birds? 5. Separate the data by bird size. Enter average number of nests and extinction times for large birds as lists L1 and L2 and for small birds as lists L3 and L4. Use the graphing calculator to make two scatter plots with different marks for large and small birds. Does your scatter plot agree with your answer in Exercise 4? Explain. For Exercises 6–8, make a scatter plot for each set of data and describe the relationship, if any, between the x- and y-values. 6.

8.

x 70 80 40 50 30 80 60 60 50 40

y 323 342 244 221 121 399 230 200 215 170

7.

x 8 5 9 10 3 4 10 7 6 7

y 89 32 30 18 26 72 51 34 82 60

x

5.2

5.8

6.3

6.7

7.4

7.6

8.4

8.5

9.1

y

12.1

11.9

11.5

9.8

10.2

9.6

8.8

9.1

8.5

9. RESEARCH Find two sets of data on your own. Then determine whether a relationship exists between the data. 68 Chapter 1 The Tools of Algebra

[0, 5] scl:1 by [0, 12] scl:1

CH

APTER

1

Study Guide and Review

wnload Vocabulary view from pre-alg.com

Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.

%XPLORE

0LAN

#HECK

3OLVE

Key Concepts

• Check

(Lesson 1-1)

↓

• Explore ↑

• Plan ↓

↑

Problem-Solving Plan

• Solve

Order of Operations

(Lesson 1-2)

• Step 1 Evaluate the expressions inside grouping symbols. • Step 2 Multiply and/or divide in order from left to right. • Step 3 Add and/or subtract in order from left to right.

Properties

(Lessons 1-4 and 1-5)

For any numbers a, b, and c, the following are true. • a+b=b+a • a·b=b·a • (a + b) + c = a + (b + c) • (a · b) · c = a · (b · c) • a+0=0+a=a • a·1=1·a=a • a·0=0·a=0

Coordinate Plane

(Lesson 1-6)

• x- and y-coordinates are used to indicate a point’s position in a coordinate system. • The domain of a relation is the set of x-coordinates and the range of a relation is the set of y-coordinates.

algebra (p. 37) algebraic expression (p. 37) conjecture (p. 27) coordinate plane (p. 54) counterexample (p. 45) deductive reasoning (p. 45) defining a variable (p. 38) domain (p. 56) equation (p. 49) evaluate (p. 32) inductive reasoning (p. 27) numerical expression (p. 32)

open sentence (p. 49) ordered pair (p. 54)

order of operations (p. 32) origin (p. 54) properties (p. 43) range (p. 56) relation (p. 56) scatter plot (p. 61) simplify (p. 45) solution (p. 49) solving the equation (p. 49) variable (p. 37) x-axis (p. 54) x-coordinate (p. 54) y-axis (p. 54) y-coordinate (p. 54)

Vocabulary Check State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. m + 3n - 4 is an example of a numerical expression. 2. To find the value of a numerical expression, you evaluate that expression. 3. The set of all y-coordinates of a relation is called the domain. 4. 20 + 12 ÷ 4 - 1 × 12 is an example of a numerical expression. 5. The set of all x-coordinates of a relation is called the domain. 6. An ordered pair names a point on a coordinate plane. 7. A counterexample is an example that shows an equation is not true. 8. A relation is a graph that shows the relationship between two sets of data.

Vocabulary Review at pre-alg.com

Chapter 1 Study Guide and Review

69

CH

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1


Lesson-by-Lesson Review 1–1

Using a Problem-Solving Plan

(pp. 26–30)

9. FOOD The table below shows the cost of various-sized hams. How much will it cost to buy a ham that weighs 7 pounds? Weight (lb) Cost

1

2

3

4

5

$4.38 $8.76 $13.14 $17.52 $21.90

10. MONEY The cash register drawer has $267 in bills. None of the bills is greater than $10. The drawer has eleven $10 bills and seven fewer $5 bills than $1 bills. How many $5 and $1 bills are in the drawer? Find the next term in each list. 11. 2, 4, 6, 8, 10, …

Example 1 A pay phone at the mall requires 40 cents for a local call. It takes quarters, dimes, and nickels and does not give change. How many combinations of coins could be used to make exact change for a local call? Use the four-step plan to solve the problem. Four-Step Problem-Solving Plan Explore We need to find the number of combinations of quarters, dimes, and nickels that make 40 cents. Plan Make a table showing the different combinations of coins. Solve

12. 5, 8, 11, 14, 17, … 13. 2, 6, 18, 54, 162, … 14. 1, 2, 4, 7, 11, 16, ….

Check

1–2

Numbers and Expressions

2(17 + 4)

18. 3

19. 4[9 + (1 · 16) - 8] 20. 18 ÷ (7 - 4) + 6 21. PROFITS Fuyu, Collin, and Sydney spent $284 to buy supplies to make bracelets. They sold the bracelets for $674. If they split the profits evenly, how much did each person earn?


Dimes 1 0 4 3 2 1 0

Nickels 1 3 0 2 4 6 8

All of the combinations equal 40 cents and there are no other combinations possible.

(pp. 32–36)

Find the value of each expression. 15. 7 + 3 · 5 16. 36 ÷ 9 - 3 17. 5 · (7 - 2) - 9

Quarters 1 1 0 0 0 0 0

Example 2 Find the value of 3[(10 - 7) + 2]. 3[(10 - 7) + 2] = 3(3 + 2)

Evaluate (10 - 7).

= 3(5)

Add 3 and 2.

= 15

Multiply 3 and 5.

Mixed Problem Solving

For mixed problem-solving practice, see page 794.

1–3

Variables and Expressions

(pp. 37–41)

Evaluate each expression if x = 3, y = 8, and z = 5. 22. y + 6 23. 17 - 2x 6y 24. _ x +9

25. 6x - 2z + 7

26. Translate the phrase nine less than a number into an algebraic expression. 27. PHYSICAL EDUCATION Arturo’s time for climbing the rope was 5 seconds more than half of Brandon’s time. Define the variables and represent this situation as an algebraic expression.

1–4

Properties

5a + 2 = 5(7) + 2

Replace a with 7.

= 35 + 2

Multiply 5 and 7.

= 37

Add 35 and 2.

Example 4 Translate 8 more than 3 divided by a number into an algebraic expression. Words 8 more than 3 divided by a number Variable Let n represent the number. Expression

3 8+_ n

(pp. 43–47)

Name the property shown by each statement. 28. 1 + 9 = 9 + 1 29. 6 + 0 = 6 30. 15 × 0 = 0

31. (x · 8) · 2 = x · (8 · 2)

ALGEBRA Simplify each expression. 32. 3 + (b + 4) 33. 8 · (9 · d) 34. COLLECTIONS Gloria has 58 dolls. If she does not add any dolls to her collection, write a sentence that represents the situation. Then name the property that is illustrated.

1–5

Example 3 Evaluate 5a + 2 if a = 7.

Variables and Equations

Example 5 Name the property shown by the statement. a. (2 + 3) + 6 = 2 + (3 + 6) Associative Property of Addition b. 1 · 6 · 9 = 6 · 1 · 9 Commutative Property of Multiplication Example 6 Simplify (k + 7) + 9. (k + 7) + 9 = k + (7 + 9) = k + 16

Assoc. (+) Add 7 and 9.

(pp. 49–53)

ALGEBRA Solve each equation mentally. 35. n + 3 = 13 36. 24 = 7 + g

Example 7 Find the solution of 26 = 33 - w. Is it 5, 6, or 7?

37. 38 = g + 16

38. 6x = 48

Replace w with each value.

39. 54 = 9h

56 40. _ a = 14

41. BICYCLES A bicycle wheel travels 72 inches in one revolution. Given r, the number of revolutions of the wheel, write an equation to find d, the distance traveled.

Value for w

26 = 33 - w True or False?

5

26 33 - 5

false

6

26 33 - 6

false

7

26 33 - 7

true

Therefore, the solution of 26 = 33 - w is 7.


71

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1 1–6


Ordered Pairs and Relations

(pp. 54–59)

Express each relation as a table and as a graph. Then determine the domain and range. 42. {(2, 3), (6, 1), (7, 5)}

Example 8 Express the relation {(1, 4), (3, 2), (4, 3), (0, 5)} as a table and as a graph. Then determine the domain and range.

43. {(0, 2), (1, 7), (5, 2), (6, 5)}

x

y

44. FAIRS It costs $2 per person to ride the Ferris wheel. Graph the ordered pairs in which the x-coordinate represents the number of people and the y-coordinate represents the cost for 1, 2, and 4 people to ride the Ferris wheel.

1

4

3

2

4

3

0

5

y

x

O

The domain is {1, 3, 4, 0}, and the range is {4, 2, 3, 5}.

Scatter Plots

(pp. 61–66)

SLEEP The table shows the amount of sleep students received the night before a standardized test and their score on the test. Number of Sleep Hours

8

9.5

8

5

9

7

Score

89

91

94

68

81

77

45. Make a scatter plot of the data. 46. Does the scatter plot allow you to draw a conclusion between sleep time and test score? Explain your reasoning. 47. Predict the score of a person who got 4 hours of sleep the night before the test.

Example 9 TREES The scatter plot shows the approximate heights and circumferences of various giant sequoia trees. Height and Circumference of Giant Sequoia Trees Circumference (ft)

1–7

105 100 95 90 85 80 75 0

220 240 260 280 300 Height (ft)

a. Does the scatter plot allow you to draw a conclusion about the heights and circumferences of Giant Sequoia trees? Explain your reasoning. Yes. As the heights of the trees increase, so do their circumferences. b. If a relationship exists, predict the circumference of a 245-foot Giant Sequoia. The circumference is about 93 ft.


A PT ER

1

Practice Test

1. STRAWBERRIES Five people can pick 10 baskets of strawberries in one hour. How many baskets of strawberries can 20 people pick in one-half hour? 2. MONEY Mrs. Adams rents a car for a week and pays $79 for the first day and $49 for each additional day. Mr. Lowe rents a car for $350 a week. Which was the better price for a seven-day rental? Explain. ALGEBRA Evaluate each expression if a = 7, b = 3, and c = 5. 3. 42 ÷ [a(c - b)] 4. 5c + (a + 2b) - 8 Name the property shown by each statement. 5. (5 · 6) · 8 = 5 · (6 · 8) 6. x + y = y + x 7. 20 · 1 = 20 8. MULTIPLE CHOICE The table shows the point values of different scoring plays in football. Which set of scoring plays does not result in 30 points? Use the equation 6t + x + 2c + 3f + 2s = 30 to help you. Scoring Play

Points

touchdown (t)

6

extra point (x)

1

two-point conversion (c)

2

field goal (f )

3

safety (s)

2

9. SHOPPING Jacob paid $12 for 6 loaves of bread at the grocery store. Find an equation that can be used to find how much Jacob paid for each loaf of bread. 10. What is the domain of the function shown in the table? ⫺1

1

4

5

y

3

7

13

15 y

E D

11. C 12. D 13. E

C x

O

Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. 14. outside temperature and air conditioning bill 15. number of siblings and height 16. MULTIPLE CHOICE The scatter plot shows semester grades and school days missed for the students in Mr. Hernandez’s math class. Which of the following is a reasonable score for a student who missed 3 days? Attendance 100

A 2 touchdowns, 1 two-point conversion, 5 field goals

x

Refer to the coordinate system at the right. Write the ordered pair that names each point.

y

95 Semester Score

CH

90 85 80 75 70

B 3 touchdowns, 2 extra points, 2 field goals, 2 safeties

0

2

4

6 8 10 12 14 16 x Days Missed

C 4 touchdowns, 4 extra points, 1 safety D 3 touchdowns, 2 extra points, 1 twopoint conversion, 2 field goals, 1 safety

Chapter Test at pre-alg.com

F 85 G 83

H 91 J 95

Chapter 1 Practice Test

73

A PT ER


1

Chapter 1

Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The equation c = 0.95t represents c, the cost of t tickets on a subway. Which table can be used to find the values that fit this equation? A Cost of Subway Tickets t

1

2

3

4

c

$1.95

$2.95

$3.95

$4.95

B

3. The manager of an ice cream shop has recorded the average number of snow cones sold per hour based on the outside temperature. The results are shown in the scatter plot. Which description best represents the relationship of the data? -Ü iÃ -` *iÀ ÕÀ

CH

Cost of Subway Tickets t

1

2

3

4

c

$1.95

$2.90

$3.85

$4.80

C


1

2

3

4

c

$0.95

$1.95

$2.95

$3.95

D


1

2

3

4

c

$0.95

$1.90

$2.85

$3.80

2. James, Kyle, and Tommy scored a total of 39 points in their last basketball game. James scored three times as many points as Kyle, and Tommy scored 3 fewer points than James. Which is a reasonable conclusion about the number of points scored by the basketball players? F James scored the most points. G Tommy and Kyle scored the same number of points. H James scored exactly half the total number of points. J Tommy had the fewest points.

Question 2 Eliminate any answers that do not

make sense. For example, in this problem, you can eliminate H because half of 39 points would be 19.5 points and it is impossible to score half a point.


A B C D

Ü xä

Y

{x {ä Îx Îä Óx ä

X Çä Çx nä nx ä x "ÕÌÃì /i«iÀ>ÌÕÀi ®

Negative trend No trend Positive trend Cannot be determined

4. Edward and his sister agreed to split their lunch bill evenly. The subtotal came to $24.70, the sales tax was $1.48, and they left a tip of $5.00. How much did each person owe? F $22.04 G $18.73 H $15.59 J $14.20 5. GRIDDABLE Find the sum of the whole numbers mentally. 28 + 41 + 22 + 19 6. A plumber had a 5-meter length of pipe in his truck. He used 120 centimeters of the pipe on a new hot water heater and 240 centimeters to repair a floor drain. How many meters of the pipe are left? A 0.14 m B 1.4 m C 140 m D 280 m Standardized Test Practice at pre-alg.com

Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.

7. The expression below can be used to generate the terms of a pattern where n is the term number. 2n - 4

12. What are the coordinates of point D on the coordinate grid below?

What is the fifth term of the pattern?

A B C D

F 6 G8 H 10

(3, 2) (6, -2) (-2, 6) (-4, -3)

C

654321 1 2 3 E 4 5 6

J 12

y

F B x O 1 2 3 4 5 6

G

A

Pre-AP

8. GRIDDABLE An elevator began on the fifth floor of a hotel. The elevator then traveled 6 floors up, 3 floors down, 8 floors down, 5 floors up, 4 floors down, and then 7 floors up. What was the number of the floor where the elevator finally stopped?

Record your answers on a sheet of paper. Show your work. 13. The table below shows the results of a survey about the average time that individual students spend studying on weeknights.

9. Which property is illustrated by the equation below? 4+6=6+4 A B C D

6 5 4 3 2 1

D

Associative Property Commutative Property Distributive Property Identity Property

10. Suppose your sister has 3 more CDs than you do. Which equation represents the number of CDs that you have? Let y represent your CDs and s represent your sister’s CDs. F y=s+3 G y=3-s H y=s-3 J y = 3s

Grade

Time (min)

Grade

Time (min)

2

20

6

60

2

15

6

45

2

20

6

55

4

30

6

60

4

20

8

70

4

25

8

80

4

40

8

75

4

30

8

60

a. Make a scatter plot of the data. b. What are the coordinates of the point that represents the longest time spent on homework? c. Does a relationship exist between grade level and time spent studying? If so, write a sentence to describe the relationship. If not, explain why not.

11. GRIDDABLE What is the next term in the pattern 4, 12, 36, 108, . . . ?

NEED EXTRA HELP? If You Missed Question...

1

2

3

4

5

6

7

8

9

10

11

12

13

Go to Lesson...

1-6

1-1

1-7

1-2

1-4

1-2

1-3

1-1

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Chapter 1 Standardized Test Practice

75

Integers

2 •

Select and use appropriate operations to solve problems and justify solutions.

•

Communicate mathematics through informal and mathematical language, representations, and models.

Key Vocabulary absolute value (p. 80) integers (p. 78) negative number (p. 78) opposites (p. 88)

Real-World Link Golf The scoring system in golf is based on integers. A positive score is over par, a negative score is under par, and a score of 0 is par.

Operations with Integers Make this Foldable to help you organize your notes about operations with integers. Begin with a sheet of grid paper.

1 Fold in half.

2 Fold the top to the bottom twice.

3 Open and cut along the second fold to make four tabs.

76 Chapter 2 Integers Scott Halleran/Getty Images

4 Fold lengthwise. Draw a number line on the outside. Label each tab as shown.



Option 1


Example 1

Evaluate each expression if a = 4, b = 10, and c = 8. (Lesson 1-3) 1. a + b + c

2. bc - ab

3. b + ac

4. 4c + 3b

Evaluate a - 2b + 3c if a = 1, b = 2, and c = 3. a - 2b + 3c = 1 - 2(2) + 3(3) Replace a with 1, b with 2, and c with 3.

5. SALES Laura sold three times as many bottles of water on Sunday as on Saturday. How many bottles did she sell Saturday if she sold 120 bottles Sunday? (Lesson 1-3)

6. 34, 40, 46, 52, 58, …

13. (0, 2)

14. (6, 1)

15. (6, 4)

16. (4, 2)

17. (1, 1)

18. (2, 5)

y

N

V M

U T P L

Q S

11 +5

16 +5

21 +5

? +5

The next term is 21 + 5 or 26.

Example 3

Use the coordinate plane to name each ordered pair. (Prerequisite Skill) 12. (3, 4)

Simplify.

6

8. PHONE The telephone company charges $0.30 for the first minute and $0.15 for each additional minute. How much would it cost to talk for 10 minutes? (Lesson 1-1)

11. (5, 5)

=6

Find the next term in the list. 6, 11, 16, 21, …

7. 135, 120, 105, 90, 75, …

10. (5, 2)

Multiply.

Example 2

Find the next term in each list. (Lesson 1-1)

9. (1, 3)

=1-4+9

R

O

x

ANIMALS A mole can dig a tunnel 300 feet

long in one night. (Prerequisite Skill) 19. Make a table of ordered pairs in which the x-coordinate represents the number of nights and the y-coordinate represents the tunnel length for 1, 2, 3, and 4 nights.

Use the coordinate plane to write the ordered pair that names point A.

Y

Step 1 Start at the origin.

!

Step 2 Move right on " the x-axis to find the x-coordinate of point A, which is 4.

X

Step 3 Move up the y-axis to find the y-coordinate, which is 1. The ordered pair for point A is (4, 1).

20. Graph the ordered pairs. Chapter 2 Get Ready for Chapter 2

77

2-1

Integers and Absolute Value

Main Ideas

• Find the absolute value of an expression.

New Vocabulary negative number integers coordinate inequality absolute value

The western United States was unusually dry in 2002. In the graph, a value of -6 represents 6 inches below the normal rainfall.

Rainfall, 2002 Albuquerque, NM Normal Rainfall

a. What does a value of -3 represent?

Rainfall (in.)

• Compare and order integers.

b. Which city was farthest from its normal rainfall? c. How could you represent 5 inches above normal rainfall?

4 2 0

Denver, CO

Salt Lake City, UT

⫺2 ⫺4 ⫺6 ⫺8

Cities

Sources: weather.com, wonderground.com

Compare and Order Integers With normal rainfall as the starting point of 0, you can express 6 inches below normal as 0 - 6, or -6. A negative number is a number less than zero.

Reading Math Integers Read –6 as negative 6. A positive integer like 6 can be written as +6. It is usually written without the + sign, as 6.

Negative numbers like -6, positive numbers like +6, and zero are members of the set of integers. Integers can be represented as points on a number line. positive integers

negative integers Numbers to the left of zero are less than zero.

-6 -5 -4 -3 -2 -1 0

1

2

3

4

5

6

Numbers to the right of zero are greater than zero.

Zero is neither negative nor positive.

This set of integers can be written {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}, where … means continues indefinitely.

EXAMPLE

Write Integers for Real-World Situations

Write an integer for each situation. a. 500 feet below sea level

The integer is -500.

b. a temperature increase of 12°

The integer is +12.

c. a loss of $240

The integer is -240.

1A. a loss of 8 yards 78 Chapter 2 Integers

1B. a deposit of $15

To graph integers, locate the points named by the integers on a number line. The number that corresponds to a point is called the coordinate of that point. graph of a point with coordinate -4

-6

Reading Math Inequality Symbols Read the symbol < as is less than. Read the symbol > as is greater than.

-5

-4

-3 -2

graph of a point with coordinate 2

0

-1

1

2

3

4

5

6

Notice that the numbers on a number line increase as you move from left to right. This can help you determine which of two numbers is greater. Words

-4 is less than 2.

OR

-4 < 2

OR

Symbols

2 is greater than -4. 2 > -4

The symbol points to the lesser number.

Any mathematical sentence containing < or > is called an inequality. An inequality compares numbers or quantities.

EXAMPLE

Compare Two Integers

Use the integers graphed on the number line below. -6 -5 -4 -3 -2 -1

0

1

2

3

4

5

6

a. Write two inequalities involving -3 and 4. Since -3 is to the left of 4, write -3 < 4. Since 4 is to the right of -3, write 4 > -3. b. Replace the with or = in -5 -1 to make a true sentence. -1 is greater since it lies to the right of -5. So write -5 < -1.

2A. Write two inequalities involving -2 and -6. 2B. Replace the with < or > in 2 -1 to make a true sentence.

Real-World Link Annika Sorenstam won the 2004 LPGA Championship at 13 under par. She was the LPGA’s leading money winner from 2001 to 2004. Source: LPGA.com

GOLF The top ten fourth-round scores of the 2004 LPGA Championship tournament were 0, +1, -5, -2, -1, +4, +2, +3, +5, and -3. Order the scores from least to greatest. Graph each integer on a number line. -5

-4

-3

-2 -1

0

1

2

3

4

5

Write the numbers as they appear from left to right. The scores -5, -3, -2, -1, 0, +1, +2, +3, +4, and +5 are in order from least to greatest.

3. GOLF The top ten fourth-round scores of the 2004 PGA Championship were +4, -2, +6, +1, -4, -3, +5, -1, +2, and +3. Order the scores from least to greatest. Personal Tutor at pre-alg.com Lesson 2-1 Integers and Absolute Value Jonathan Daniel/Getty Images

79

Absolute Value On the number line, notice that -5 and 5 are each 5 units from 0, even though they are on opposite sides of 0. Numbers that are the same distance from zero have the same absolute value. 5 units

-6

Common Misconception It is not always true that the absolute value of a number is the opposite of the number. Remember that absolute value is always positive or zero.

-5

-4

-3 -2

5 units

0

-1

1

2

3

4

5

6

The symbol for absolute value is two vertical bars on either side of the number.

⎪5⎥ = 5 The absolute value of 5 is 5. ⎪-5⎥ = 5 The absolute value of -5 is 5. Absolute Value Words

The absolute value of a number is the distance the number is from zero on the number line. The absolute value of a number is always greater than or equal to zero.

Examples

⎪5⎥ = 5

EXAMPLE

⎪-5⎥ = 5

Expressions with Absolute Value

Evaluate each expression. a. ⎪-8⎥ 8 units

-10

-8

-6

-4

0

-2

2

⎪-8⎥ = 8 The graph of -8 is 8 units from 0. b. ⎪9⎥ + ⎪-7⎥

The absolute value of 9 is 9.

⎪9⎥ + ⎪-7⎥ = 9 + 7 The absolute value of -7 is 7. = 16

Simplify.

4B. ⎪-4⎥ - ⎪3⎥

4A. ⎪-3⎥

Since variables represent numbers, you can use absolute value notation with algebraic expressions involving variables.

EXAMPLE

Algebraic Expressions with Absolute Value

ALGEBRA Evaluate ⎪x⎥ - 3 if x = -5. ⎪x⎥- 3 = ⎪-5⎥ - 3

Replace x with -5.

=5-3

The absolute value of -5 is 5.

=2

Simplify.

5. Evaluate ⎪y⎥ + 8 if x = -7. 80 Chapter 2 Integers


Example 1 (p. 78)

Example 2 (p. 79)

Write an integer for each situation. Then graph on a number line. 1. 8° below zero

2. a 15-yard gain

Write two inequalities using the numbers in each sentence. Use the symbols < or >. 3. -7° is colder than 3°. Replace each 5. -18

Example 3 (p. 79)

HELP

For See Exercises Examples 16–21 1 22–33 2 34–39 3 40–51 4 52–57 5

{ä

xä

Èä

VViiÀ>Ì ÉÃ ® £È Î£ £ä È

Î{

ä

-9

£ä

Óä

Îä

Çä

nä

ä

ää

£ä £Óä «

HOMEWORK

/i Ã®

nä

{ä

(p. 80)

7. 9

-3

9. TEST DRIVES The table shows the recorded acceleration for a new car at regular intervals. Order the accelerations from least to greatest. Îä

Example 5

6. 0

8. Order the integers {28, -6, 0, -2, 5, -52, 115} from least to greatest.

£ä ä

(p. 80)

with , or = to make a true sentence.

-8

Óä

Example 4

4. -6 is greater than -10.

Ó

ÓÎ £ Ó

Evaluate each expression. 11. ⎪10⎥ - ⎪-4⎥

10. ⎪-10⎥

12. ⎪16⎥ + ⎪-5⎥

ALGEBRA Evaluate each expression if a = -8 and b = 5. 13. 9 + ⎪a⎥

14. ⎪a⎥ - b

15. 2⎪a⎥

Write an integer for each situation. Then graph on a number line. 16. a bank withdrawal of $100

17. a loss of 6 pounds

18. a salary increase of $250

19. a gain of 9 yards

20. 12° above zero

21. 5 seconds before liftoff

Write two inequalities using the numbers in each sentence. Use the symbols < or >. 22. 3 meters is taller than 2 meters. 23. A temperature of -5°F is warmer than a temperature of -10°F. 24. 55 miles per hour is slower than 65 miles per hour. 25. A 4-yard loss is less than no gain. Replace each 26. -6

-2

30. -18

8

with , or = to make a true sentence. 27. -10 31. 5

-13

-23

28. 0 32. ⎪9⎥

-9

⎪-9⎥

29. 14

0

33. ⎪-20⎥

⎪-4⎥

Order the integers in each set from least to greatest. 34. {5, 0, -8}

35. {-15, -1, -2, -4}

36. {19, -16, 4, 62, -80}

37. {41, -14, 50, -23, -20}

38. {24, 5, -46, 9, 0, -3}

39. {98, -57, -60, 38, 188} Lesson 2-1 Integers and Absolute Value

81

Evaluate each expression. 40. ⎪-15⎥

41. ⎪46⎥

42. -⎪20⎥

43. -⎪5⎥

44. ⎪0⎥

45. ⎪7⎥

46. ⎪-5⎥ + ⎪4⎥

47. ⎪0⎥ + ⎪-2⎥

48. ⎪15⎥ - ⎪-1⎥ 49. ⎪0 + 9⎥

50. ⎪9 - 5⎥ - ⎪6 - 8⎥ 51. -⎪-6 + 1⎥ - ⎪5 - 6⎥

ALGEBRA Evaluate each expression if a = 0, b = 3, and c = -4.

Real-World Link The Marianas Trench in the Pacific Ocean is the deepest part of all of the oceans at 35,840 feet.

52. 14 + ⎪b⎥

53. ⎪c⎥ - a

54. a + b + ⎪c⎥

55. ab + ⎪-40⎥

56. ⎪c⎥ - b

57. ⎪ab⎥ + b

58. GEOGRAPHY The Caribbean Sea has an average depth of 8448 feet below sea level. Use an integer to express this depth. ANALYZE TABLES For Exercises 59–62, use the table. Record Lowest Temperatures by State

Source: U.S. Department of Defense

State

Station

Date

Temperature (°F)

Alaska Montana Wisconsin

Prospect Creek Camp Rogers Pass Danbury

Jan. 23, 1971 Jan. 20, 1954 Jan. 24, 1922

-80 -70 -54

59. Graph the temperatures on a number line. 60. Compare the lowest temperature in Alaska and the lowest temperature in Wisconsin using the < symbol. 61. Compare the lowest temperature in Montana and the lowest temperature in Wisconsin using the > symbol. 62. Write the temperatures in order from greatest to least. Graph each set of integers on a number line. 63. {0, -2, 4}

64. {-3, 1, 2, 5}

65. {-2, -4, -5, -8}

66. {-4, 0, 6, -7, -1}

67. Name the coordinates of each point graphed on the number line.

B -6

EXTRA

PRACTICE


H.O.T. Problems

-4

D -2

A 0

C 2

4

6

68. SOLAR SYSTEM The average temperature of Saturn is -218°F while the average temperature of Jupiter is -162°F. Which planet has the lower average temperature? Explain. 69. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would compare and order integers. 70. OPEN ENDED Write two inequalities using integers. 71. NUMBER SENSE Explain how to find the number of units apart -4 and 5 are on a number line. 72. Which One Doesn’t Belong? Identify the expression that does not belong with the other three. Explain your reasoning.

⎪12 – ⎪–4⎥⎥ 82 Chapter 2 Integers NOAA

⎪–2⎥ + ⎪6⎥

–⎪7 + 1⎥

⎪–8⎥

CHALLENGE Consider two numbers A and B on a number line. 73. Is it always, sometimes, or never true that the distance between A and B equals the distance between |A| and |B|? Explain. 74. Assume A > B. Is it always, sometimes, or never true that A - |B| ≤ A + B? Explain. 75.

Writing in Math Use the information about integers on page 78 to explain how they can be used to model real-world situations. Include an explanation of how integers are used to describe rainfall.

76. What is the temperature shown on the thermometer at the right?

5

78. Which of the following describes the absolute value of -2°? F It is the distance from -2 to 2 on the thermometer.

0

A 8°F

⫺5

B 7°F

⫺10

G It is the distance from -2 to 0 on the thermometer.

C -7°F

H It is the actual temperature outside when the thermometer reads -2°.

D -8°F

J None of these describes the absolute value of -2°.

77. GRIDDABLE How many units apart are -4 and 3 on a number line?

Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. (Lesson 1-7) 79. height and arm length 80. birth month and weight Express each relation as a table and as a list of ordered pairs. (Lesson 1-6)

81.

82.

y

O

x

y

x

O

Name the property shown by each statement. (Lesson 1-4)

83. 20(18) = 18(20)

84. 9(8)(0) = 0

PREREQUISITE SKILL Find each sum or difference. 86. 18 + 29 + 46 87. 232 + 156 89. 36 - 19 90. 479 - 281

85. 3ab = 3ba

88. 451 + 629 + 1027 91. 2011 - 962

Lesson 2-1 Integers and Absolute Value

83

EXPLORE

2-2

Algebra Lab

Adding Integers In a set of algebra tiles, £ represents the integer 1, and ⫺1 represents the integer -1. You can use algebra tiles and an integer mat to model operations with integers.

ACTIVITY 1 The following example shows how to find the sum -3 + (-2) using algebra tiles. Remember that addition means combining. The expression -3 + (-2) tells you to combine a set of 3 negative tiles with a set of 2 negative tiles. #OMBINE THE TILES ON THE MAT 3INCE THERE ARE NEGATIVE TILES ON THE MAT THE SUM IS

0LACE NEGATIVE TILES AND NEGATIVE TILES ON THE MAT

•

Therefore, -3 + (-2) = -5. There are two important properties to keep in mind when you model operations with integers. • When one positive tile is paired with one negative tile, the result is called a zero pair. • You can add or remove zero pairs from a mat because removing or adding zero does not change the value of the tiles on the mat. The following example shows how to find the sum -4 + 3. 2EMOVE THE ZERO PAIRS

0LACE NEGATIVE TILES AND POSITIVE TILES ON THE MAT

£

£

£

£

£

£

£

£ {

Î

Therefore, -4 + 3 = -1. 84 Chapter 2 Integers

3INCE THERE IS ONE NEGATIVE TILE REMAINING THE SUM IS

£

£

£

£

£

£

£ { Î

{ Î £

EXERCISES Use algebra tiles to model and find each sum. 1. -2 + (-4)

2. -3 + (-5)

3. -6 + (-1)

4. -4 + (-5)

5. -4 + 2

6. 2 + (-5)

7. -1 + 6

8. 4 + (-4)

ACTIVITY 2 The Addition Table was completed using algebra tiles. In the highlighted portions of the table, the addends are -3 and 1, and the sum is -2. So, -3 + 1 = -2. You can use the patterns in the Addition Table to learn more about integers. Addition Table

+

4

3

2

1

0

-1

-2

-3

addends

-4

⎫

4

8

7

6

5

4

3

2

1

0

3

7

6

5

4

3

2

1

0

-1

2

6

5

4

3

2

1

0

-1

-2

1

5

4

3

2

1

0

-1

-2

-3

0

4

3

2

1

0

-1

-2

-3

-4

-1

3

2

1

0

-1

-2

-3

-4

-5

-2

2

1

0

-1

-2

-3

-4

-5

-6

-3

1

0

-1

-2

-3

-4

-5

-6

-7

-4

0

-1

-2

-3

-4

-5

-6

-7

-8

⎭

⎬ sums

addends

ANALYZE THE RESULTS 9. MAKE A CONJECTURE Locate all of the positive sums in the table. Describe the addends that result in a positive sum. 10. MAKE A CONJECTURE Locate all of the negative sums in the table. Describe the addends that result in a negative sum. 11. MAKE A CONJECTURE Locate all of the sums that are zero. Describe the addends that result in a sum of zero. 12. The Identity Property says that when zero is added to any number, the sum is the number. Does it appear that this property is true for addition of integers? If so, write two examples that illustrate the property. If not, give a counterexample. 13. The Commutative Property says that the order in which numbers are added does not change the sum. Does it appear that this property is true for addition of integers? If so, write two examples that illustrate the property. If not, give a counterexample. 14. The Associative Property says that the way numbers are grouped when added does not change the sum. Is this property true for addition of integers? If so, write two examples that illustrate the property. If not, give a counterexample. Explore 2-2 Algebra Lab: Adding Integers

85

2-2

Adding Integers

Main Ideas • Add two integers. • Add more than two integers.

In football, forward progress is represented by a positive integer. Being pushed back is represented by a negative integer. On the first play a team loses 5 yards and on the second play they lose 2 yards. ⫺2

New Vocabulary opposites additive inverse

⫺5

⫺9 ⫺8⫺7⫺6⫺5 ⫺4⫺3 ⫺2 ⫺1 0 1 2

40

50

a. What integer represents the total yardage on the two plays? b. Write an addition sentence that describes this situation.

Add Integers The equation -5 + (-2) = -7 is an example of adding two integers with the same sign. Notice that the sign of the sum is the same as the sign of the addends. Recall that the numbers you add are called addends. The result is called the sum.

EXAMPLE

Add Integers on a Number Line

Find -2 + (-3) ⫺3

⫺2

-7 -6-5-4 -3 -2 -1 0 1 2

Start at zero. Move 2 units to the left. From there, move 3 more units to the left.

-2 + (-3) = -5

1A. -3 + (-4)

1B. -6 + (-14)

This example suggests a rule for adding integers with the same sign. Adding Integers with the Same Sign


86 Chapter 2 Integers

Words

To add integers with the same sign, add their absolute values. The sum is: • positive if both integers are positive. • negative if both integers are negative.

Examples -5 + (-2) = -7

6+3=9

EXAMPLE

Add Integers with the Same Sign

Find -4 + (-5). -4 + (-5) = -9 Add ⎪-4⎥ and ⎪-5⎥. The sum is negative. Find each sum. 2B. -1 + (-12)

2A. -8 + (-2)

A number line can also help you understand how to add integers with different signs.

EXAMPLE

Add Integers on a Number Line

Find each sum. b. 2 + (-3)

a. 7 + (-4) Adding Integers on a Number Line Always start at zero. Move right to model a positive integer. Move left to model a negative integer.

⫺4

⫺3

7 -2 -1 0 1 2 3 4 5 6 7

2 -4 -3 -2-1 0 1 2 3 4 5

Start at zero. Move 7 units to the right. From there, move 4 units to the left.

Start at zero. Move 2 units to the right. From there, move 3 units to the left.

7 + (-4) = 3

2 + (-3) = -1

3A. 5 + (-2)

3B. 4 + (-8)


Notice how the sums in Example 3 relate to the addends. The sign of the sum is the same as the sign of the addend with the greater absolute value. Adding Integers with Different Signs To add integers with different signs, subtract their absolute values. The sum is:

• positive if the positive integer’s absolute value is greater. • negative if the negative integer’s absolute value is greater.

EXAMPLE

Add Integers with Different Signs

Find each sum. a. -8 + 3

b. 10 + (-4)

-8 + 3 = -5

10 + (-4) = 6

To find -8 + 3, subtract ⎪3⎥ from ⎪-8⎥. The sum is negative because ⎪-8⎥ > ⎪3⎥.

To find 10 + ⎪-4⎥, subtract ⎪-4⎥ from ⎪10⎥. The sum is positive because ⎪10⎥ > ⎪-4⎥.

4A. -9 + 4 Extra Examples at pre-alg.com

4B. 12 + (-5) Lesson 2-2 Adding Integers

87

ASTRONOMY During the night, the average temperature on the moon is -140°C. By noon, the average temperature has risen 252°C. What is the average temperature on the moon at noon? Temperature at night

Words Variable

plus

increase by noon

equals

252

=

temperature at noon

Let x = the temperature at noon.

Equation

-140

+

x

Solve the equation. Estimate -140 + 250 = 110.

Real-World Link The temperatures on the moon are so extreme because the moon does not have any atmosphere to trap heat.

-140 + 252 = x To find the sum, subtract ⎪-140⎥ from 252. 112 = x The sum is positive because ⎪252⎥ > ⎪-140⎥. The average temperature at noon is 112°C. The solution is reasonable to the estimate.

5. SUBMARINES A submarine was at a depth of 103 feet below the surface of the water. It rose 68 feet. What is its current depth?

Add More Than Two Integers Two numbers with the same absolute value but different signs are called opposites. For example, -4 and 4 are opposites. An integer and its opposite are also called additive inverses. Additive Inverse Property Words

The sum of any number and its additive inverse is zero.

Symbols

x + (-x) = 0

EXAMPLE Adding Mentally One way to add mentally is to group the positive addends together and the negative addends together. Then add to find the sum. Also look for addends that are opposites. You can always add in order from left to right.

NASA

6 + (-6) = 0

Add Three or More Integers

Find each sum. a. 9 + (-3) + (-9) 9 + (-3) + (-9) = 9 + (-9) + (-3) Commutative Property = 0 + (-3) Additive Inverse Property: 9 + (-9) = 0 = -3 Identity Property of Addition b. -4 + 6 + (-3) + 9 Commutative Property -4 + 6 + (-3) + 9 = -4 + (-3) + 6 + 9 = [-4 + (-3)] + (6 + 9) Associative Property = -7 + 15 or 8 Simplify.

6A. 4 + (-2) + (-7) 88 Chapter 2 Integers

Example

6B. -10 + 3 + (-7) + 12

Examples 1– 4 (pp. 86–87)

Example 5 (p. 88)

Example 6 (p. 88)

HOMEWORK

HELP

For See Exercises Examples 13–22 1, 2 23–30 3, 4 31, 32 5 33–40 6

Find each sum. 1. -2 + (-4)

2. -10 + (-5)

3. -14 + (-4)

4. 7 + (-2)

5. 11 + (-3)

6. 8 + (-5)

7. 2 + (-16)

8. 9 + (-12)

9. -15 + 4

10. FOOTBALL A team gained 4 yards on one play. On the next play, they lost 5 yards. Write an addition sentence to find the total yardage. Find each sum. 11. 8 + (-6) + 2

12. -6 + 5 + (-10)

Find each sum. 13. -4 + (-1)

14. -5 + (-2)

15. -4 + (-6)

16. -3 + (-8)

1 7. -7 + (-8)

18. -12 + (-4)

19. -9 + (-14)

20. -15 + (-6)

21. -11 + (-15)

22. -23 + (-43)

23. 8 + (-5)

24. 6 + (-4)

25. 3 + (-7)

26. 4 + (-6)

27. -15 + 6

28. -5 + 11

29. 18 + (-32)

30. -45 + 19

Write an addition sentence for each situation. Then find the sum. 31. GAME SHOWS A contestant has -1500 points. Suppose he loses another 1250 points. 32. STOCKS A stock price increases $6. It then decreases $10. Find each sum. 33. 6 + (-9) + 9

34. 7 + (-13) + 4

35. -9 + 16 + (-10)

36. -12 + 18 + (-12)

37. 14 + (-9) + 6

38. 28 + (-35) + 4

39. -41 + 25 + (-10)

40. -18 + 35 + (-17)

41. MONEY The starting balance in a checking account was $50. What was the balance after checks were written for $25 and for $32? Use estimation to determine whether your answer is reasonable. 42. HIKING Sally starts hiking at an elevation of 324 feet. She descends to an elevation of 201 feet and then ascends to an elevation 55 feet higher than where she began. She descends 183 feet. Describe the overall change in elevation. Find each sum. 43. 18 + (-13)

44. -27 + 19

45. -25 + (-12) Lesson 2-2 Adding Integers

89

EXTRA

PRACTIICE

See pages 763, 795.

POPULATION For Exercises 46 and 47, use the table that shows the change in population of several cities from 2002 to 2003. #ITY %L 0ASO 48 3AN *OSE #! ,EXINGTON +9 #OLUMBIA 3#


0OPULATION

#HANGE AS OF

-ÕÀVi\ 4HE 7ORLD !LMANAC

46. What was the population in each city in 2003? 47. What was the total change in population of these cities?

H.O.T. Problems

48. OPEN ENDED Give an example of two integers that are additive inverses. 49. CHALLENGE True or false? ⫺n always names a negative number. If false, give a counterexample. CHALLENGE Name the property illustrated by each of the following. 50. a(b + (-b)) = (b + (-b))a 52.

Writing in Math

51. a(b + (-b)) = 0

Explain how a number line can help you add integers.

53. A Guadelupe bass was swimming underwater at a depth of 12 feet. It rose 3 feet, dropped 5 feet, rose 10 feet, and dropped 1 foot. What is the current depth of the fish?

54. Which expression is represented by the model?

A -7 ft x{ÎÓ £ ä £ Ó Î { x

B -5 ft C -3 ft

F -5 + -1

H -5 + 1

D 7 ft

G -5 + 0

J

-5 + 4

55. CHEMISTRY The freezing point of oxygen is 219 degrees below zero on the Celsius scale. Use an integer to express this temperature. (Lesson 2-1) Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 1-7)

56. age and family size

57. temperature and sales of mittens

ALGEBRA Find the solution of each equation from the list given. (Lesson 1-5) 58. 18 - n = 12; 3, 6, 30

59. 7a = 49; 7, 42, 343

PREREQUISITE SKILL Evaluate each expression if a = 6, b = 10, and c = 3. (Lesson 1-3) 60. a + 19 90 Chapter 2 Integers

61. 2b - 6

62. ab - ac

63. 3a - (b + c)

Learning Mathematics Vocabulary Some words used in mathematics are also used in English and have similar meanings. For example, in mathematics add means to combine. The meaning in English is to join or unite. Some words are used only in mathematics. For example, addend means a number to be added to another. Some words have more than one mathematical meaning. For example, an inverse operation undoes the effect of another operation, and an additive inverse is a number that when added to a given number gives zero. The list below shows some of the mathematics vocabulary used in Chapters 1 and 2.

Vocabulary

Meaning

Examples

algebraic expression

an expression that contains at least one variable and at least one mathematical operation

2 + x, _c , 3b

evaluate

to find the value of an expression

2+5=7

simplify

to find a simpler form of an expression

3b + 2b = 5b

integer

a whole number, its additive inverse, or zero

-3, 0, 2

factor

a number that is multiplied by another number

3(4) = 12 3 and 4 are factors.

product

the result of multiplying

3(4) = 12 ← product

quotient

the result of dividing two numbers

12 _ = 3 ← quotient

dividend

the number being divided

12← _ =3

dividend

divisor

the number being divided into another number

12 _ =3 ←

divisor

coordinate

a number that locates a point

(5, 2)

4

4 4

4

Reading to Learn 1. Name two of the words above that are also used in everyday English. Use the Internet, a dictionary, or another reference to find their everyday definition. How do the everyday definitions relate to the mathematical definitions? 2. Name two words above that are used only in mathematics. 3. Name two words above that have more than one mathematical meaning. List their meanings. Reading Math Learning Mathematics Vocabulary

91

EXPLORE

2-3

Algebra Lab

Subtracting Integers You can also use algebra tiles to model subtraction of integers. Remember one meaning of subtraction is to take away.

ACTIVITY

Use algebra tiles to find each difference.

a. 7 - 4

b. -8 - (-3)

Place 7 positive tiles on the mat. Remove 4 positive tiles.

Place 8 negative tiles on the mat. Remove 3 negative tiles.

£

£

£

£

£

£

£

£

£

£ £

£

£

£

£

So, 7 - 4 = 3 So, -8 - (-3) = -5. d. -6 - 2

c. 5 - (-2) Place 5 positive tiles on the mat and then remove 2 negative tiles. However, there are 0 negative tiles. First you must add 2 zero pairs to the set.

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

Then remove the 2 negative tiles. £

Place 6 negative tiles on the mat. Remove 2 positive tiles. Since there are no positive tiles, add 2 zero pairs to the mat.

£

£

£

£

Then remove the 2 positive tiles. £

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£

£ £

So, 5 - (-2) = 7.

£

£

So, -6 - 2 = -8.

ANALYZE THE RESULTS Apply what you learned to find each difference. 1. 9 - 7

2. 5 - (-3)

3. 6 - (-3)

4. 1 - (-5)

5. 3 - (-9)

6. -8 - 3

7. -8 - (-1)

8. -1 - 4

9. MAKE A CONJECTURE Write a rule that will help you determine the sign of the difference of two integers. 92 Chapter 2 Integers

2-3

Subtracting Integers BrainPOP® pre-alg.com

Main Ideas

You can use a number line to subtract integers. The model below shows how to find 6 - 8.

• Subtract integers. • Evaluate expressions containing variables.

Step 1 Step 2

⫺8

Start at 0. Move 6 units right to show positive 6.

6

From there, move 8 units left to subtract positive 8.

⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0

1 2

3 4 5 6

7

a. What is 6 - 8? b. What direction do you move to indicate subtracting a positive integer? c. What addition sentence is also modeled by the number line above?

Subtract Integers When you subtract 8, as shown on the number line above, the result is the same as adding -8. When you subtract 5, the result is the same as adding -5. These examples suggest a method for subtracting integers. additive inverses

6 - 8 = -2

additive inverses

6 + (-8) = -2

-3 - 5 = -8

-3 + (-5) = -8

Subtracting Integers Words

To subtract an integer, add its additive inverse.

Symbols

a - b = a + (-b)

EXAMPLE Subtracting a Positive Integer To subtract a positive integer, think about moving left on a number line from the starting integer. In Example 1a, start at 8, then move left 13. You’ll end at -5. In Example 1b, start at -4, then move left 10. You’ll end at -14.

Subtract a Positive Integer

Find each difference. a. 8 - 13 8 - 13 = 8 + (-13) To subtract 13, add -13. = -5

Simplify.

b. -4 - 10 -4 - 10 = -4 + (-10) To subtract 10, add -10. = -14

1A. 9 - 16


Simplify.

1B. -5 - 11 Lesson 2-3 Subtracting Integers

93

Review Vocabulary inductive reasoning making a conjecture based on a pattern of examples or past events (Lesson 1-1)

In Example 1, you subtracted a positive integer by adding its additive inverse. Use inductive reasoning to see if the method also applies to subtracting a negative integer. Subtracting an Integer ↔

Adding Its Additive Inverse

2-2=0 2-1=1 2-0=2 2 - (-1) = ?

2 + (-2) = 0 2 + (-1) = 1 2+0=2 2+1=3

Continuing the pattern in the first column, 2 - (-1) = 3. The result is the same as when you add the additive inverse.

EXAMPLE

Subtract a Negative Integer

Find each difference. a. 7 - (- 3)

b. -2 - (-4)

7 - (- 3) = 7 + 3 = 10

To subtract -3, add 3.

2A. 12 - (-4)

Real-World Link The hottest place in the world is Dallol, Ethiopia. High temperatures average 94.3°F throughout the year.

add 4.

=2

2B. -6 - (-15)

WEATHER The table shows the record high and low temperatures in selected states as of a recent year. What is the range, or difference between the highest and lowest temperatures, for Virginia? Explore

-2 - (- 4) = -2 + 4 To subtract -4,

State

Lowest Highest Temperature (˚F) Temperature (˚F)

Utah

-69

117

Vermont

-50

105

Virginia

-30

110

Washington -48 You know the West Virginia -37 highest and lowest temperatures. You Source: The World Almanac need to find the range for Virginia’s temperatures.

118

Plan

To find the range, or difference, subtract the lowest temperature from the highest temperature.

Solve

110 - (-30) = 110 + 30 To subtract -30, add 30. = 140

Source: Scholastic Book of World Records

Add 110 and 30.

The range for Virginia is 140°.

Check

Think of a thermometer. The difference between 110° above zero and 30° below zero must be 110 + 30 or 140°. The answer appears to be reasonable.

3. WEATHER What is the range of temperatures for Washington? Personal Tutor at pre-alg.com

94 Chapter 2 Integers Victor Englebert

112

Evaluate Expressions You can use the rule for subtracting integers to evaluate expressions.

EXAMPLE

Evaluate Algebraic Expressions

a. Evaluate x - (-6) if x = 12. x - (-6) = 12 - (- 6) Write the expression. Replace x with 12. = 12 + 6

To subtract -6, add its additive inverse, 6.

= 18

Add 12 and 6.

b. Evaluate a - b + c if a = 15, b = 5, and c = -8. a - b + c = 15 - 5 + (-8) Replace a with 15, b with 5, and c with -8. = 10 + (-8)

Order of operations

=2

Add 10 and -8.

Evaluate each expression if = 7, m = -3, and n = -10. 4B. - m + n 4A. n -

Examples 1, 2 (pp. 93–94)

Example 3 (p. 94)

Example 4 (p. 95)

HOMEWORK

HELP

For See Exercises Examples 14–21 1 22–31 2 32, 33 3 34–45 4

Find each difference. 1. 8 - 11

2. 10 - 15

3. - 10 - 14

4. -9 - 3

5. 7 - (-10)

6. 16 - (-12)

7. -6 - (-4)

8. -2 - (-8)

9. -15 - (- 18)

10. ANIMALS A gopher begins at 7 inches below the surface of a garden and digs down another 9 inches. Find an integer that represents the gopher’s position in relation to the surface of the garden. ALGEBRA Evaluate each expression if x = 10, y = -4, and z = -15. 11. x - (-10)

12. y - x

13. x + y - z

14. 3 - 8

15. 4 - 5

16. 2 - 9

17. 9 - 12

18. -3 - 1

19. -5 - 4

20. -6 - 7

21. -4 - 8

22. 6 - (-8)

23. 4 - (-6)

24. 7 - (-4)

25. 9 - (-3)

26. -9 - (-7)

27. -7 - (-10)

28. -11 - (-12)

29. -16 - (-7)

30. 10 - 24

31. 48 - (-50)

Find each difference.

32. MONEY Suppose you deposited $25 into your checking account and wrote a check for $38. What was the change in your account balance? 33. GEOGRAPHY The highest point in California is Mt. Whitney, with an elevation of 14,494 feet. The lowest point is Death Valley, with an elevation of -282 feet. Find the difference in the elevations. Lesson 2-3 Subtracting Integers

95

ALGEBRA Evaluate each expression if x = -3, y = 8, and z = -12.

Real-World Link Consumers spent a total of $38.4 billion on their lawns and gardens in 2003. Source: The National Gardening Association

34. y - 10

35. 12 - z

36. 3 - x

37. z - 24

38. x - y

39. z - x

40. y - z

41. z - y

42. x + y - z

43. z - y + x

44. x - y - z

45. z - y - x

ANALYZE TABLES For Exercises 46 and 47, use the table. 46. Describe the change in the sales related to each gardening activity from 2002 to 2003. 47. What was the total change in sales related to these gardening activities from 2002 to 2003?

3ALES 2ELATED TO 'ARDENING !CTIVITIES IN MILLIONS

!CTIVITY

)NDOOR HOUSEPLANTS ,ANDSCAPING ,AWN CARE 4REE CARE 6EGETABLE GARDENING 7ATER GARDENING

BUSINESS The formula P = I - E relates profit P to income I and expenses E. One month a small business has income of $19,592 and expenses of $20,345. 48. What is the profit for the month? 49. What does a negative profit mean?

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Find each difference. 50. 125 - (-114)

51. -320 - (-106)

52. -2200 - (-3500)

ANALYZE TABLES The daily closing prices for a company’s stock during one week are shown in the table. Date

Nov. 3

Nov. 4

Nov. 5

Nov. 6

Nov. 7

Closing Price

$33.30

$30.59

$31.04

$31.97

$30.15

—

?

?

?

?

Change

EXTRA

PRACTICE

53. Find the change in the closing price since the previous day. 54. What is the difference between the highest and lowest changes in closing price?


H.O.T. Problems

55. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would subtract integers. CHALLENGE Determine whether each statement is true or false. If false, give a counterexample. 56. Subtraction of integers is commutative. 57. Subtraction of integers is associative. 58. FIND THE ERROR José and Amy are finding 8 - (-2). Who is correct? Explain your reasoning. José 8 - (-2) = 8 + 2 = 10

Amy 8 - (- 2) = 8 + (-2) =6

59. OPEN ENDED Write examples of a positive and a negative integer and their additive inverses. 96 Chapter 2 Integers Colin Paterson/Getty Images

60. SELECT A TECHNIQUE Reiko is filling out her check register. Which technique(s) might Reiko use to find out if she spent more money than she had in her checking account? Justify your selection(s). Then use the technique(s) to find how much she has left in her account.

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mental math 61.

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number sense

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Writing in Math Use the information about subtracting integers on page 93 to explain how the addition and subtraction of integers are related.

62. The melting point of metal mercury is -39°C. The freezing point of alcohol is -114°C. How much warmer is the melting point of mercury than the freezing point of alcohol? A -153°C

C 75°C

B -75°C

D 153°C

63. The terms in a pattern are given in the table. What is the value of the fifth term? Term Value

F -7

1 13

2 8

3 3

G -5

4 -2

5 ?

H 5

J 7

64. OCEANOGRAPHY A submarine at 1300 meters below sea level descends an additional 1150 meters. What integer represents the submarine’s position with respect to sea level? (Lesson 2-2) 65. ALGEBRA Evaluate ⎪b⎥ - ⎪a⎥ if a = 2 and b = -4. (Lesson 2-1) ALGEBRA Translate each phrase into an algebraic expression. (Lesson 1-3) 66. a number divided by 5 68. the quotient of eighty-six and b

67. the sum of t and 9 69. s decreased by 8

Find the value of each expression. (Lesson 1-2)

70. 2 × (5 + 8) - 6

71. 96 ÷ (6 × 8) ÷ 2

PREREQUISITE SKILL Find each product. 73. 5 · 15 74. 8 · 12

75. 3 · 5 · 8

72. 17 - (21 + 13) ÷ 17

76. 4 · 9 · 12

Lesson 2-3 Subtracting Integers

97

CH

APTER

2


1. MULTIPLE CHOICE Choose the integer between 2 and -1. (Lesson 2-1) A -3 B -0.5 C 1 D 2.5 Replace each with , or = to make a true sentence. (Lesson 2-1) 2. 9 4. -8

3. -3

-5

5. 2

-6

0 -4

15. SPACE During night, the average temperature on Mars is -140°F. During the day, the average temperature rises 208°F. What is the average daytime temperature on Mars? (Lesson 2-2) 16. ACCOUNTING A small company had the following profits and losses for a six-month period. How much did the company earn during this time period? (Lesson 2-2)

6. MULTIPLE CHOICE Refer to the number line. Which statement is true? (Lesson 2-1) $ "

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x{ Î Ó £ ä £ Ó Î { x

Jan.

-$3674

Feb.

$4013

Mar.

-$1729

Apr.

-$1415

F ⎪B⎥ < ⎪C⎥

H B>C

May

$1808

G C>A

J ⎪D⎥ > ⎪A⎥

Jun.

-$547

7. TEMPERATURE Order the temperatures from least to greatest. (Lesson 2-1) Temperature (˚F)

Liquid helium

-452

Outer space

-457

Dry ice

-108

Source: The Sizesaurus

9. -5 + 11

10. -6 + 9 + (-8)

11. 12 + (-6) + (-15)

12. ⎪-33 + 19⎥

13. ⎪-23 + -20⎥

14. MULTIPLE CHOICE Which day had the greatest change in stock price? (Lesson 2-2) Day Mon.

Open Price

Close Price

$43.29

$48.55

Tues.

$48.55

$46.65

Wed. Thurs.

$46.65 $41.30

$41.30 $45.99

A Mon. B Tues. C Wed. D Thurs. 98 Chapter 2 Integers

18. -15 - 8

19. 25 - (-7)

20. -16 - (-11)

ALGEBRA Evaluate each expression if x = 5, y = -2, and z = -3. (Lesson 2-3) 21. x - y

Find each sum. (Lesson 2-2) 8. -5 + (-15)

17. 16 - 23

22. z - 6

23. x - y - z

24. WEATHER If the temperature is -9°F and it drops 5°F overnight, what is the new temperature? (Lesson 2-3) 25. ASTRONOMY The graph shows the highest and lowest points of three planets. (Lesson 2-3) i}ÌÉ i«Ì ®

Item

Find each difference. (Lesson 2-3)

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What is the range of each of the planets? Which planet has the greatest range?

EXPLORE

2-4

Algebra Lab

Multiplying Integers You can also use algebra tiles to model multiplication of integers. Remember that 2 × 3 means two sets of three items. So, you can show 2 × 3 by placing 2 sets of 3 positive tiles on a mat. Similarly, you can model 2 × (-3) by placing 2 sets of 3 negative tiles on the mat, as shown at the right.

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If the first factor is negative, you will need to remove tiles from the mat.

ACTIVITY Step 1 The expression -2 × (-3) means to remove 2 sets of 3 negative tiles. To do this, first place 2 × 3 or 6 zero pairs on the mat. £

£

£

£

£

£

£

£

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£

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£

Step 2 Then remove 2 sets of 3 negative tiles from the mat. There are 6 positive tiles remaining. So, -2 × (-3) = 6. Animation pre-alg.com £

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EXERCISES Use algebra tiles to model and find each product. 1. 6 × (-2)

2. 3 × (-5)

3. 3 × (-4)

4. 1 × (-8)

5. -4 × (-2)

6. -5 × (-2)

7. -7 × (-1)

8. -2 × (-2)

9. Explain the meaning of -2 × 3. Then find the product using algebra tiles. Use algebra tiles to model and find each product. 10. -4 × 2

11. -3 × 5

12. -2 × 6

13. -1 × 3

ANALYZE THE RESULTS 14. How are the operations -3 × 4 and 4 × (-3) the same? How do they differ? 15. MAKE A CONJECTURE Find a rule you can use to find the sign of the product of two integers given the sign of both factors. Explore 2- 4 Algebra Lab: Multiplying Integers

99

2-4

Multiplying Integers

Main Ideas • Multiply integers. • Simplify algebraic expressions.

The temperature drops 7°C for each 1 kilometer increase in altitude. A drop of 7°C is represented by -7. So, the temperature change equals the altitude times -7. a. Suppose the altitude is 4 kilometers. Write an expression to find the temperature change.

Altitude (km)

Altitude Rate of Change

Temperature Change (°C)

1

1(7)

7

2

2(7)

14

3

3(7)

21

…

…

…

11

11(7)

77

b. Use the pattern in the table to find 4(-7).

Multiply Integers Multiplication is repeated addition. So, 3(-7) means Reading Math

that -7 is used as an addend 3 times.

Parentheses Recall that a product can be written using parentheses. Read 3(-7) as 3 times negative 7.

3(-7) = (-7) + (-7) + (-7) = -21

-7 -21

-7

-7 -14

-7

0

7

By the Commutative Property of Multiplication, 3(-7) = -7(3). This example suggests the following rule. Multiplying Integers with Different Signs Words

The product of two integers with different signs is negative.

Examples 4(-3) = -12

EXAMPLE

-3(4) = -12

Multiply Integers with Different Signs

Find each product. a. 5(-6) 5(-6) = -30

The factors have different signs. The product is negative.

b. -4(16) -4(16) = -64 The factors have different signs. The product is negative.

1A. 7(-8) 100 Chapter 2 Integers

1B. -6(12) Extra Examples at pre-alg.com

The product of two positive integers is positive. What is the sign of the product of two negative integers? Use a pattern to find a rule. One positive and one negative factor: Negative product

Two negative factors: Positive product

(-4)(2)

= -8

(-4)(1)

= -4

(-4)(0)

=

0

(-4)(-1) =

4

(-4)(-2) =

8

+4 Each product is 4 more than the previous product.

+4 +4 +4

Multiplying Integers with the Same Sign Words

The product of two integers with the same sign is positive.

Examples 4(3) = 12

EXAMPLE

-4(-3) = 12

Multiply Integers with the Same Sign

Find each product. b. -4(-5)(-8)

a. -6(-12) -6(-12) = 72 The product is

-4(-5)(-8) = [(-4)(-5)](-8)

positive.

2A. -5(-11)

= 20(-8) = -160

2B. -3(-4)(-5)

A glacier receded at a rate of 300 feet per day. What was the glacier’s movement in 5 days? A -1500 ft Context Clues Read the problem. Try to picture the situation. Look for words that suggest mathematical concepts.

B -300 ft

C -60 ft

D 305 ft

Read the Test Item The word receded means moved backward, so the rate per day is represented by -300. Multiply 5 and -300 to find the movement in 5 days. Solve the Test Item 5(-300) = -1500

The product is negative.

The answer is A.

3. A scuba diver descended at a rate of 5 feet per minute. What was the scuba diver’s depth at 5 minutes? F -25 ft

G -10 ft

H 10 ft

J 25 ft

Personal Tutor at pre-alg.com Lesson 2-4 Multiplying Integers

101

Algebraic Expressions You can use the rules for multiplying integers to simplify and evaluate algeraic expressions.

EXAMPLE

Simplify and Evaluate Algebraic Expressions

a. Simplify -2x(3y). -2x(3y) = (-2)(x)(3)(y)

-2x = (-2)(x), 3y = (3)(y)

= (-2 · 3)(x · y) Commutative Property of Multiplication = -6xy

-2 · 3 = -6, x · y = xy

b. Evaluate 4ab if a = 3 and b = -5. 4ab = 4(3)(-5)

Replace a with 3 and b with -5.

= [4(3)](-5)

Associative Property of Multiplication

= 12(-5)

The product of 4 and 3 is positive.

= -60

The product of 12 and -5 is negative.

4A. Simplify -3(6y). 4B. Simplify -7a(3b). 4C. Evaluate 2rs if r = 5 and s = -10.

Examples 1, 2 (pp. 100–101)

Find each product. 1. -3 · 8

2. 5(-8)

3. -2(11)

4. 4 · 30

5. -7(-4)

6. -6 · -6

7. -4(-2)(-6)

8. 8(-3)(-5)

9. -5(-9)(-12)

10. FITNESS The table shows burned Calories per minute for a 120-pound person during different activities. What is the change in the number of Calories in a 120-pound person’s body if he runs for 20 minutes?

!CTIVITY

#ALORIES PER -INUTE

"ALLET $ANCING "ICYCLING MPH 'OLF CARRYING CLUBS (ANDBALL 2UNNING 3KATEBOARDING

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Example 3 (p. 101)

11. MULTIPLE CHOICE The research submarine Alvin, used to locate the wreck of the Titanic, descended at a rate of about 100 feet per minute. Which integer describes the distance Alvin traveled in 5 minutes? A -500 ft

Example 4 (p. 102)

B -100 ft

C -20 ft

D 100 ft

ALGEBRA Simplify each expression. 12. -4 · 3x

13. 7(-3y)

14. -8c(-3d)

ALGEBRA Evaluate each expression. 15. -6h, if h = -20 102 Chapter 2 Integers

16. -4st, if s = -9 and t = 3

HOMEWORK

HELP


Find each product. 17. -3 · 4

18. -7 · 6

19. 4(-8)

20. 9 · (-8)

21. -12 · 3

22. 14(-5)

23. 6 · 19

24. 4(32)

25. -8(-11)

26. -15(-3)

27. -5(-4)(6)

28. 5(-13)(-2)

29. -7(-8)(-3)

30. -6(-8)(11)

31. 2(-8)(-9)(10)

32. 4(-7)(-4)(-12)


34. -8 · 12y

35. 6(-8a)

36. 5(-11b)

37. -7s(-8t)

38. -12m(-9n)

39. 2ab(3)(-7)

40. 3x(5y)(-9)

41. -4(-p)(-q)

42. -8(-11b)(-c)

43. 9(-2c)(3d)

44. -6j(3)(5k)

ALGEBRA Evaluate each expression. 45. -7n, if n = -4

46. 9s, if s = -11

47. ab, if a = 9 and b = 8

48. -2xy, if x = -8 and y = 5

49. -16cd, if c = 4 and d = -5

50. 18gh, if g = -3 and h = 4

51. ELEVATORS An elevator takes students from the ground floor of a building down to an underground parking garage. Where will the elevator be in relation to the ground floor after 5 seconds if it travels at a rate of 3 feet per second? 52. TRAVEL A driver depresses the brake pedal of her car and begins decelerating at a rate of 2.3 meters per second per second. How much will the car’s speed change if the brake is applied for 6 seconds? 53. ANALYZE GRAPHS Write the product that is modeled on the number line.

EXTRA

PRACTICE


H.O.T. Problems

-5 -15

-5

-5

-12 -10 -8 -6 -4 -2

0

2

4

TIDES For Exercises 54 and 55, use the information below. In Wrightsville, North Carolina, during low tide, the beachfront in some places is about 350 feet from the ocean to the homes. High tide can change the width of a beach at a rate of -17 feet an hour. It takes 6 hours for the ocean to move from low to high tide. 54. What is the change in the width of the beachfront from low to high tide? 55. What is the distance from the ocean to the homes at high tide? 56. ALGEBRA Find the values that complete the table below for y = -4x. x

-2

-1

0

1

y

?

?

?

?

57. OPEN ENDED Give an example of three integers whose product is negative. 58. REASONING Calculate (-10)(5)(18)[7 + (-7)] mentally. Justify your answer. 59. CHALLENGE Positive integers A and C satisfy A(A - C) = 23. What is the value of C? Lesson 2-4 Multiplying Integers

103

The cost of a trip to a popular amusement park can be determined with integers. Visit pre-alg.com to continue work on your project.

60. SELECT A TOOL During a drought, the amount of water in a pond changes by -9 gallons per day due to evaporation. Which of the following tools might you use to find the number of days it takes for the amount of water in a pond to change by -108 gallons of water? Justify your selection(s). Then use the tool(s) to solve the problem. draw a model

61.

real objects

calculator

Writing in Math Explain how the signs of factors and products are related. Include an explanation of why the product of a positive and a negative integer must be negative.

62. An airplane descends at a rate of 200 feet per minute. Write a multiplication equation that tells the altitude of the airplane after 2 minutes.

63. GRIDDABLE At 8:00 P.M., a temperature of 78°F was recorded. The temperature then changed at an average rate of -2°F per hour for a 15-hour period. What was the temperature in degrees Fahrenheit at 7:00 A.M.?

A -200(2) = -400 C 200(2) = 400 B 200(-2) = -400 D -200(-2) = 400

ALGEBRA Evaluate each expression if a = -2, b = -6, and c = 14. (Lesson 2-3) 64. a - c 65. a - b 66. c - a + b 67. b - a + c 68. SWIMMING Lincoln High School’s swim team finished the 4 × 100-meter freestyle relay in 5 minutes 18 seconds. Prospect High School’s swim team finished the race in 5 minutes 7 seconds. Find an integer that represents Lincoln’s finish compared to Prospect’s finish. (Lesson 2-3) Find each sum. (Lesson 2-2) 69. -10 + 8 + 4

70. -4 + (-3) + (-7)

71. 9 + (-14) + 2 y

Refer to the coordinate system. Write the ordered pair that names each point. (Lesson 1-6) 72. E

73. C

74. B

75. F

76. D

77. A

B

D C

A

E F

O

PREREQUISITE SKILL Find each quotient. 78. 40 ÷ 8 79. 90 ÷ 15 104 Chapter 2 Integers

80. 45 ÷ 3

x

81. 91 ÷ 7

EXPLORE

2-5

Algebra Lab

Dividing Integers You can model division by separating algebra tiles into equal-sized groups.

ACTIVITY Use positive or negative tiles to find each quotient. a. 10 ÷ 2

b. -15 ÷ 5

Place 10 positive tiles on the mat to represent 10.

Place 15 negative tiles on the mat to represent -15.

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£

Separate the tiles into 2 equal-sized groups.

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Separate the tiles into 5 equal-sized groups. £

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£

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£

£

There are 5 positive tiles in each of the 2 groups.

There are 3 negative tiles in each of the 5 groups.

So, 10 ÷ 2 = 5.

So, -15 ÷ 5 = -3.

EXERCISES Apply what you learned to find each quotient. 1. 12 ÷ 6

2. 16 ÷ 2

3. 14 ÷ 7

4. -8 ÷ 2

5. -9 ÷ 3

6. -6 ÷ 2

7. -16 ÷ 4

8. -5 ÷ 5

9. -10 ÷ 2

ANALYZE THE RESULTS For Exercises 10–12, study the quotients in Exercises 1–9. 10. When the dividend and the divisor are both positive, is the quotient positive or negative? How does this compare to the sign of a product when both factors are positive? 11. When the dividend is negative and the divisor is positive, is the quotient positive or negative? How does this compare to the sign of a product when one factor is positive and one is negative? 12. MAKE A CONJECTURE Write a rule that will help you determine the sign of the quotient of two integers. Explore 2-5 Algebra Lab: Dividing Integers

105

2-5

Dividing Integers

Main Ideas • Divide integers. • Find the average of a set of data.

You can find the product 3 × (-4) on a number line. To find the product, start at 0 and then move -4 units three times. -12

New Vocabulary

-4

-4

-4

mean -12

-10

-8

-6

-4

-2

0

a. What is the product 3 × (-4)? b. What division sentence is also shown on the number line? c. Draw a number line and find the product 5 × (-2). Then find the related division sentence.

Divide Integers You can find the quotient of two integers by using the related multiplication sentence. Think of this factor . . . to find this quotient.

-4 × 3 = -12 → -2 × 5 = -10 →

Reading Math Parts of a Division Sentence In a division sentence, like 15 ÷ 5 = 3, the number you are dividing, 15, is called the dividend. The number you are dividing by, 5, is called the divisor. The result, 3, is called the quotient.

-12 ÷ (-4) = 3 -10 ÷ (-2) = 5

In the division sentences -12 ÷ (-4) = 3 and -10 ÷ (-2) = 5, notice that the dividends and divisors are both negative. In both cases, the quotient is positive. negative dividend and divisor

-12 ÷ (-4) = 3

-10 ÷ (-2) = 5 positive quotient

You already know that the quotient of two positive integers is positive. 12 ÷ 4 = 3

10 ÷ 2 = 5

These and similar examples suggest the following rule for dividing integers with the same sign. Dividing Integers with the Same Sign Words

The quotient of two integers with the same sign is positive.

Examples -12 ÷ (-3) = 4


12 ÷ 3 = 4

EXAMPLE

Divide Integers with the Same Sign

Find each quotient. 75 b. _

a. -32 ÷ (-8) -32 ÷ (-8) = 4

5 75 = 75 ÷ 5 5

The quotient is positive.

The quotient is positive.

= 15

-39 1B. _

1A. 35 ÷ 5

-3

What is the sign of the quotient of a positive and a negative integer? Look for a pattern in the following related sentences. Think of this factor … to find this quotient.

-4 × (-6) = 24 → 2 × (-9) = -18 →

24 ÷ (-4) = -6 -18 ÷ 2 = -9

Notice that the signs of the dividend and divisor are different. In both cases, the quotient is negative. different signs

24 ÷ (-4) = -6 -18 ÷ 2 = -9

negative quotient

different signs

These and other similar examples suggest the following rule.

Dividing Integers with Different Signs Words

The quotient of two integers with different signs is negative.

Examples

-12 ÷ 4 = -3

EXAMPLE

12 ÷ (-4) = -3

Divide Integers with Different Signs

Find each quotient. Check Your Work Always check your work after finding an answer. If -42 ÷ 3 = -14, does -14 × 3 = -42?

a. -42 ÷ 3 -42 ÷ 3 = -14 The quotient is negative.

2A. 63 ÷ (- 7) Extra Examples at pre-alg.com

48 b. _

-6 48 _ = 48 ÷ (-6) The quotient is negative. -6 = -8 Simplify.

-110 2B. _ 11

Lesson 2-5 Dividing Integers

107

EXAMPLE


Evaluate ab ÷ (-4) if a = -6 and b = -8. ab ÷ (-4) = -6(-8) ÷ (-4)

Replace a with -6 and b with -8.

= 48 ÷ (-4) or -12 Simplify.

3. Evaluate 12y ÷ x if x = -6 and y = -3.

Mean (Average) Division is used in statistics to find the average, or mean, of a set of data. To find the mean of a set of numbers, find the sum of the numbers and then divide by the number of items in the data set.

WEATHER The windchill temperatures in degrees Fahrenheit for the first six days in January were -2, 8, 5, -9, -12, and -2. Find the mean temperature for the six days. Checking Reasonableness

-2 + 8 + 5 + (-9) + (-12) + (-2) -12 ___ =_ 6

The average must be between the greatest and least numbers in the set. Is the average in Example 4 reasonable?

6

= -2

Find the sum of the set of integers. Divide by the number in the set. Simplify.

The mean temperature is -2°F.

4. GOLF Linda has scores of -3, -2, 1, and 0 during 4 rounds of golf. Find the mean of her golf scores. Personal Tutor at pre-alg.com

Operations with Integers Examples

Words Adding Two Integers To add integers with the same sign, add their absolute values. Give the result the same sign as the integers. To add integers with different signs, subtract their absolute values. Give the result the same sign as the integer with the greater absolute value.

-5 + (-4) = -9

5+4=9

-5 + 4 = -1

5 + (-4) = 1

5 - 9 = 5 + (-9) or -4 5 - (-9) = 5 + 9 or 14

Subtracting Two Integers To subtract an integer, add its additive inverse. Multiplying Two Integers The product of two integers with the same sign is positive. The product of two integers with different signs is negative. Dividing Two Integers The quotient of two integers with the same sign is positive. The quotient of two integers with different signs is negative.


5 · 4 = 20

-5 · (-4) = 20

-5 · 4 = -20

5 · (-4) = -20

20 ÷ 5 = 4

-20 ÷ (-5) = 4

-20 ÷ 5 = -4

20 ÷ (-5) = -4

Examples 1, 2 (p. 107)

Example 3 (p. 108)

Example 4 (p. 108)

HOMEWORK

HELP

For See Exercises Examples 10–15 1 16–21 2 22–27 3 28, 29 4

Find each quotient. 1. 88 ÷ 8

2. -20 ÷ (-5)

-36 3. _

4. -18 ÷ 6

70 5. _ -7

-81 6. _ 9

-4

ALGEBRA Evaluate each expression. 8. _s , if s = -45 and t = 5

7. x ÷ 4, if x = -52

t

9. WEATHER The low temperatures for 7 days in January in degrees Fahrenheit were -2, 0, 5, -1, -4, 2, and 0. Find the average for the 7-day period.

Find each quotient. 10. 54 ÷ 9

11. 45 ÷ 5

12. -27 ÷ (-9)

13. -64 ÷ (-8)

14. -72 ÷ (-9)

15. -60 ÷ (-6)

16. -77 ÷ 7

17. -300 ÷ 6

18. 480 ÷ (-12)

-150 19. _ 10

600 20. _ -20

-350 21. _ 70

ALGEBRA Evaluate each expression. x 22. _ , if x = -85

-5 24. _c , if c = -63 and d = -7 d

26. xy ÷ (-3), if x = 9 and y = -7

108 23. _ m , if m = -9

25. _s , if s = 52 and t = -4 t

27. ab ÷ 6, if a = -12 and b = -8

28. STATISTICS Find the mean of 4, -8, 9, -3, -7, 10, and 2. 29. BASKETBALL In their first five games, the Jefferson Middle School basketball team scored 46, 52, 49, 53, and 45 points. What was their average number of points per game? ENERGY For Exercises 30 and 31, use the information below. h+ The formula d = 65 - _ can be used to find degree days, where h is the 2 high and is the low temperature. 30. If Las Vegas, Nevada, had a high of 94° and a low of 80°, find the degree days. 31. If Charleston, South Carolina, had a high of 56° and a low of 32°, find the degree days. 32. RESEARCH Use the Internet or another resource to find the high and low temperature for your city for a day in January. Find the degree days. EXTRA

PRACTICE


33. SPACE The surface temperature on Mercury at night can fall to -300°F. 5(F - 32)

Use the expression _, where F represents the temperature in degrees 9 Fahrenheit, to find the temperature on Mercury in degrees Celsius. Round to the nearest tenth. Lesson 2-5 Dividing Integers

109

H.O.T. Problems

34. OPEN ENDED Write an equation with three integers that illustrates dividing integers with different signs. 35. CHALLENGE Find values for x, y, and z, so that all of the following statements are true. • y > x, z < y, and x < 0

• z ÷ 2 and z ÷ 3 are integers.

• x ÷ z = -z

• x÷y=z

36. CHALLENGE Addition and multiplication are said to be closed for whole numbers, but subtraction and division are not. That is, when you add or multiply any two whole numbers, the result is a whole number. Which operations are closed for integers? 37.

Writing in Math Use the information about dividing integers on pages 106–107 to explain how dividing integers is related to multiplying integers. Illustrate your answer with two related multiplication and division sentences.

38. The table shows the sales of a computer chip manufacturer in two recent years. What is the average change in sales per year? Year

Sales (millions)

2005 2000

$115 $128

39. Pedro has quiz scores of 8, 7, 8, and 9. What is the lowest score he can get on the remaining quiz to have a final average (mean) score of at least 8? F

7

G 8 H 9

A -$13 million

C $2.6 million

B -$2.6 million

D $13 million

Find each difference or product. (Lessons 2-3 and 2-4) 40. -8 - (-25) 41. 75 - 114

J 10

42. 2ab · (-2)

43. (-10c)(5d)

44. ANIMALS The height of an adult giraffe is 3 times the height of a newborn giraffe. Given n, the height of a newborn giraffe, write an equation that can be used to find a, the height of an adult giraffe. (Lesson 1-5) 45. PATTERNS Find the next two numbers in the pattern 5, 4, 2, -1, …. (Lesson 1-1)

PREREQUISITE SKILL Use the coordinate plane to name the point for each ordered pair. (Lesson 1-6) 46. (1, 5) 47. (6, 2) 48. (4, 5)

y

C B

D E

A F

49. (0, 3)

G H O


x

2-6

The Coordinate System

Main Ideas • Graph points on a coordinate plane. • Graph algebraic relationships.

New Vocabulary

A GPS, or Global Positioning System, can be used to find a location anywhere on Earth by identifying its latitude and longitude. Several cities are shown on the map below. For example, Brisbane, Australia, is located at approximately 30°S, 150°E. Èäc

Èäc

'REENWICH

quadrants

"EIJING

$ALLAS

Îäc

Îäc

äc

äc

Îäc-

"RISBANE

Îäc-

£xäc

£Óäc

äc

Èäc

Îäc

#APE 4OWN äc

Îäc7

Èäc7

äc7

£Óäc7

£xäc7

3ANTIAGO

a. Latitude is measured north and south of the equator. What is the latitude of Dallas? b. Longitude is measured east and west of the prime meridian. What is the longitude of Dallas? c. What does the location 32°N, 100°W mean?

Graph Points Latitude and longitude are a kind of coordinate system. The coordinate system, or coordinate plane you used in Lesson 1-6 can be extended to include points below and to the left of the origin. Review Vocabulary

4 3 2 1 4 321 O 1 2 P (4, 2) 3 4

1 2 3 4x

The x-axis extends to the right and left of the origin. Notice that the numbers to the left of zero on the x-axis are negative.

Recall that a point graphed on the coordinate system has an x-coordinate and a y-coordinate. The dot at the ordered pair (-4, -2) is the graph of point P. x-coordinate

y-coordinate

(-4, -2)

冦

Coordinate System a coordinate plane formed by the intersection of two number lines that meet at right angles at their zero points (Lesson 1-6)

origin

The y-axis extends above and below the origin. Notice that the numbers below zero on the y-axis are negative.

y

ordered pair Lesson 2-6 The Coordinate System

111

EXAMPLE

Write Ordered Pairs

Write the ordered pair that names each point. a. A Ordered Pairs Notice that the axes in an ordered pair (x, y) are listed in alphabetical order.

A

The x-coordinate is -3. The y-coordinate is 2. The ordered pair is (-3, 2).

y

-4 -3-2-1 O 1 2 3 4 x -1 B -2 C D -3 -4 5

b. B The x-coordinate is 4. The y-coordinate is -2. The ordered pair is (4, -2).

1A. C

4 3 2 1

1B. D

The x-axis and the y-axis separate the coordinate plane into four regions, called quadrants. The axes and points on the axes are not located in any of the quadrants. The quadrants are named I, II, III, and IV.

The coordinates are (negative, positive).

The coordinates are (negative, negative).


II

4 y 3 2 1

-4 -3-21 O -1 III -2 -3 -4

I

The coordinates are (positive, positive).

1 2 3 4x

IV

The coordinates are (positive, negative).

Graph Points and Name the Quadrant

Graph and label each point on a coordinate plane. Name the quadrant in which each point lies. y a. E(2, 4) ( Start at the origin. Move 2 units right. Then move 4 units up and draw a dot. Point E(2, 4) is in Quadrant I. b. F(-3, -2) Start at the origin. Move 3 units left. Then move 2 units down and draw a dot. Point F(-3,-2) is in Quadrant III.

4 3 2 1

-4 -3-2-1 O -1 -2 ( ) F -3, -2 -3 -4

E 2, 4)

G (4, 0) 1 2 3 4x

c. G(4, 0) Start at the origin. Move 4 units right. Since the y-coordinate is 0, the point lies on the x-axis. Point G(4, 0) is not in any quadrant.

2A. H(4, -3) 112 Chapter 2 Integers

2B. J(0, -2)

2C. I(-1, 4)

Reading Math Coordinate System Coordinate plane, coordinate grid, and coordinate graph are other names for the coordinate system.

Graph Algebraic Relationships You can use a coordinate graph to show relationships between two numbers.

EXAMPLE

Graph an Algebraic Relationship

The sum of two numbers is 5. If x represents the first number and y represents the second number, make a table of possible values for x and y. Graph the ordered pairs and describe the graph. First, make a table. Choose values for x and y that have a sum of 5. Then graph the ordered pairs on a coordinate plane. x+y=5 y

(x, y)

2

3

(2, 3)

1

4

(1, 4)

0

5

(0, 5)

-1

6

(-1, 6)

-2

7

(-2, 7)

x

y

O

x

The points on the graph are in a line that slants downward to the right. The line crosses the y-axis at y = 5.

3. The difference of two numbers is 4. If x represents the first number and y represents the second number, make a table of possible values for x and y. Graph the ordered pairs and describe the graph. Personal Tutor at pre-alg.com

Example 1 (p.112)

Name the ordered pair for each point graphed at the right. 1. A

2. C

3. G

4. K

y

A

D B

C

O

Example 2 (p.112)

Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. 5. J(3, -4)

6. K(-2, 2)

7. L(0, 4)

8. M(-1, -2)

x

G F

H

K

9. GEOMETRY Graph points A(-4, 3), B(1, 3), C(1, 2), and D(-4, 2) on a coordinate plane and connect them to form a rectangle. Name the quadrant in which each point is located. Example 3 (p.113)

10. ALGEBRA Make a table of values and graph six ordered integer pairs where x + y = 3. Describe the graph.


Lesson 2-6 The Coordinate System

113

HOMEWORK

HELP


Name the ordered pair for each point graphed at the right. 11. R

12. G

13. M

14. B

15. V

16. H

17. U

18. W

19. A

y

R H G

V W x

O

T

U

M B

A

20. T

Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. 21. A(4, 5)

22. H(0, -3)

23. M(4, -2)

24. B(-5, -5)

25. S(2, -5)

26. F(-4, 0)

27. E(0, 3)

28. K(-5, 1)

29. G(5, 0)

30. C(6, -1)

31. D(0, 0)

32. R(-3, 5)

ALGEBRA Make a table of values and graph six sets of ordered integer pairs for each equation. Describe the graph. 33. x + y = 4

34. x - y = -2

35. y = 2x

36. y = -2x

37. y = x + 2

38. y = x - 1

Graph each point. Then connect the points in alphabetical order and identify the figure. 39. A(0, 6), B(4, -6), C(-6, 2), D(6, 2), E(-4, -6), F(0, 6)

Reading Math Vertex, Vertices A vertex of a triangle is a point where two sides of a triangle meet. Vertices is the plural of vertex.

40. A(5, 8), B(1, 13), C(5, 18), D(9, 13), E(5, 8), F(5, 6), G(3, 7), H(3, 5), I(7, 7), J(7, 5), K(5, 6), L(5, 3), M(3, 4), N(3, 2), P(7, 4), Q(7, 2), R(5, 3), S(5, 1) GEOMETRY On a coordinate plane, draw triangle ABC with vertices at A(3, 1), B(4, 2), and C(2, 4). Then graph and describe each new triangle formed in Exercises 41–44. 41. Multiply each coordinate of the vertices in triangle ABC by 2. 42. Multiply each coordinate of the vertices in triangle ABC by -1. 43. Add 2 to each coordinate of the vertices in triangle ABC. 44. Subtract 4 from each coordinate of the vertices in triangle ABC. 45. RESEARCH Find a map of your school and draw a coordinate grid on the map with the library as the center. Locate the cafeteria, principal’s office, your math classroom, gym, counselor’s office, and the main entrance on your grid. Write the coordinates of these places. How can you use these points to help visitors find their way around your school? Graph and label each point on a coordinate plane. 46. A(-6.5, 3)

EXTRA

PRACTICE

47. B(-2, -5.75)

48. C(4.1, -1)

49. D(-3.4, 1.5)

See page 765, 795.

50. ALGEBRA Graph eight ordered integer pairs where ⎪x⎥ > 3. Describe the graph.


51. ALGEBRA Graph all ordered integer pairs that satisfy the condition ⎪x⎥ < 4 and ⎪y⎥ < 3.


H.O.T. Problems

52. OPEN ENDED Name two ordered pairs whose graphs are not located in one of the four quadrants. 53. FIND THE ERROR Keisha says that if you interchange the coordinates of any point in Quadrant I, the new point would be in Quadrant I. Jason says the new point would be in Quadrant III. Who is correct? Explain your reasoning. CHALLENGE If the graph of A(x, y) satisfies the given condition, name the quadrant in which point A is located. 54. x > 0, y > 0

55. x < 0, y < 0

56. x < 0, y > 0

57. NUMBER SENSE Graph eight sets of integer coordinates that satisfy ⎪x⎥ + ⎪y⎥ > 3. Describe the location of the points. 58.

Writing in Math Use the information on page 111 to explain how a coordinate plane is used to locate places on Earth. Include an explanation of how coordinates can describe a location and how latitude and longitude are related to the x- and y-axes on a coordinate plane.

For Exercises 59 and 60, refer to the graph at the right. 59. On the coordinate plane, what are the coordinates of the point that shows the location of the library? A (4, -2)

C (4, 2)

B (-2, -4)

D (-4, -2)

y Pool Park O x Library

Grocery Store

60. On the coordinate plane, what location has coordinates (5, -2)? F park

H library

G school

J

School

store

Find each quotient. (Lesson 2-5)

61. -24 ÷ (-8)

62. 105 ÷ (-5)

63. -400 ÷ (-50)

ALGEBRA Evaluate each expression if f = -9, g = -6, and h = 8. (Lesson 2-4) 64. -5fg

65. 2gh

66. -10fh

67. WEATHER In the newspaper, Ruben read that the low temperature for the day was expected to be -5ºF and the high temperature was expected to be 8ºF. What was the difference in the expected high and low temperatures? (Lesson 2-3) ALGEBRA Simplify each expression. (Lesson 1-4) 68. (a + 8) + 6

69. 4(6h)

70. (n · 7) · 8

71. (b · 9) · 5

72. (16 + 3y) + y

73. 0(4z) Lesson 2-6 The Coordinate System

115

CH

APTER

2




Key Concepts Integers and Absolute Value

(Lesson 2-1)

• Numbers on a number line increase as you move from left to right. • The absolute value of a number is the distance the number is from zero on the number line.

absolute value (p. 80) additive inverse (p. 88) coordinate (p. 79) inequality (p. 79) integers (p. 78) mean (p. 108) negative number (p. 78) opposites (p. 88) quadrants (p. 112)

Adding and Subtracting Integers (Lessons 2-2 and 2-3)

• To add integers with the same sign, add their absolute values. Give the result the same sign as the integers.

Vocabulary Check

• To add integers with different signs subtract their absolute values. Give the result the same sign as the integer with the greater absolute value.

Complete each sentence with the correct term. Choose from the list above. 1. A(n)_______ is a number less than zero.

• To subtract an integer, add its additive inverse.

Multiplying and Dividing Integers (Lessons 2-4 and 2-5)

• The product or quotient of two integers with the same sign is positive. • The product or quotient of two integers with different signs is negative.

The Coordinate Plane

(Lesson 2-6)

• The x-axis and the y-axis separate the coordinate plane into four quadrants. • The axes and points on the axes are not located in any of the quadrants.

2. The number that corresponds to a point on the number line is called the ________. 3. An integer and its opposite are also called _________ of each other. 4. The four regions separated by the axes on a coordinate plane are called ________. 5. The set of _________ includes positive whole numbers, their opposites, and zero. 6. The _________ of a number is the distance the number is from zero on the number line. 7. A(n) _______ is a mathematical sentence containing < or >. 8. To find the _______ of a set of numbers, find the sum of the numbers and then divide by the number of items in the data set. 9. Two numbers with the same absolute values but different signs are _______.






Integers and Absolute Value

(pp. 78–83)

Replace each with , or = to make a true sentence. 11. -3 -3 10. 8 -8 12. -2

13. -12

0

-21

Evaluate each expression. 15. ⎪25⎥ 14. ⎪-32⎥ 17. ⎪-8⎥ + ⎪-14⎥

16. -⎪15⎥

Example 1 Replace with , or = in -3 2 to make a true sentence. ⫺4⫺3 ⫺2⫺1 0 1 2 3 4

Since -3 is to the left of 2, -3 < 2. Example 2 Evaluate ⎪-4⎥. 4 units

18. BASEBALL CARDS Jamal traded away 7 shortstop cards for 5 outfielder cards. Find an integer that represents the change in the number of cards Jamal had after the trade.

2–2

Adding Integers

The graph of -4 is 4 units from 0. So, ⎪-4⎥ = 4.

(pp. 86–90)

Example 3 Find -3 + (-4).

Find each sum. 19. -6 + (-3)

20. -4 + (-1)

21. -2 + 7

22. 4 + (-8)

23. 4 + 7 + (-3)

24. -9 + 6 + (-8)

25. GOLF A golfer’s scores for the last five weeks are -5, +7, -2, -4, and +5. What is the sum of his scores?

2–3

⫺5⫺4⫺3 ⫺2 ⫺1 0 1 2

Subtracting Integers

-3 + (-4) = -7

The sum is negative.

Example 4 Find 5 + (-2). 5 + (-2) = 3 The sum is positive.

(pp. 93–97)

Find each difference. 26. 4 - 9 27. -3 - 5 28. 7 - (-2)

29. -1 - (-6)

30. -7 - 8

31. 6 - 10

32. ELEVATORS The postal carrier entered the elevator on floor 15. She rode down 6 floors. Then she rode up 10 floors and got off. What floor was she on when she left the elevator?

Example 5 Find -5 - 2. -5 - 2 = -5 + (-2) To subtract 2, add -2. = -7 Example 6 Find 8 - (-4). 8 - (-4) = 8 + 4 To subtract -4, add 4. = 12


117

CH

A PT ER

2 2–4


Multiplying Integers

(pp. 100–104)

Example 7 Find 6(-4).

Find each product. 33. -9(5)

34. 11(-6)

35. -4(-7)

36. -3(-16)

37. SNOWBOARDING For each trick he completes incorrectly in the half pipe event, Kurt receives -3 points. If Kurt incorrectly completes five tricks, what is his score?

2–5

Dividing Integers

40. -36 ÷ 9

41. 88 ÷ (-4)

42. RACING The number of seconds Elena is behind the leader for the first five legs of the bicycle race is shown. What is her average time behind the leader? +32 s, +5 s, +10 s, +8 s, +12 s

The Coordinate System

45. K(-1, 3)

46. R(3, 0)

47. GAMES The coordinate plane represents a board game. Name the quadrant in which each player’s game piece is located.

-8(-2) = 16 The factors have the same sign, so the product is positive.

Example 9 Find -30 ÷ (-5). -30 ÷ (-5) = 6

The dividend and divisor have the same sign, so the quotient is positive.

Example 10 Find 27 ÷ (-3). 27 ÷ (-3) = -9 The dividend and divisor have different signs, so the quotient is negative.

Example 11 Graph and label F(5, -3) on a coordinate plane. Name the quadrant in which the point is located. Point F(5, -3) is in Quadrant IV.

Y

"

X

&

Y

"Üi

`Þ

X

"

iÀÞ >L


Example 8 Find -8(-2).

(pp. 111–115)

Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. 43. A(4, 3) 44. J(-2, -5)

iÞVi

The factors have different signs, so the product is negative.

(pp. 106–110)

Find each quotient. 38. -14 ÷ (-2) 39. -52 ÷ (-4)

2–6

6(-4) = -24

CH

A PT ER

2

Practice Test

Write two inequalities using the numbers in each sentence. Use the symbols < and >. 1. -5 is less than 2. 2. 12 is greater than -15. 3. MULTIPLE CHOICE A scuba diver records her depth in the lake every minute. Choose the group of depths that is listed in order from least to greatest.

23. WEATHER The table shows the low temperatures during one week in Anchorage, Alaska. Find the average low temperature for the week. Day Temperature (°F)

S

M

T

W

T

F

S

-12

3

-7

0

-4

1

-2

A -13 ft, -12 ft, -9 ft, -3 ft, -1 ft, -5 ft B -5 ft, -3 ft, -1 ft, -9 ft, -12 ft, -13 ft C -12 ft, -13 ft, -3 ft, -1 ft, -9 ft, -5 ft D -13 ft, -12 ft, -9 ft, -5 ft, -3 ft, -1 ft 4. FOOTBALL During the first play of the game, the Brownville Tigers football team lost seven yards. On each of the next two plays, an additional four yards were lost. Express the total yards lost at the end of the first three plays as an integer. Find each sum or difference.

ALGEBRA Evaluate each expression if a = -5, b = 3, and c = -10. 24. ab - c

25. c ÷ a

bc 26. _ a -6

27. 4c + ⎪a⎥

28. MULTIPLE CHOICE A vertex of a triangle is a point where two sides of the triangle meet. Which ordered pair is not a vertex of ABC?

5. -4 + (-8)

6. -9 + 15

F (-1, 1)

7. 12 + (-15)

8. 14 + (-7) + -11

G (2, -3)

9. 4 - 13 11. -6 - (-10)

10. 8 - (-6)

H (1, 2)

12. -14 - (-7)

J (-1, -1)

13. STOCK MARKET On Thursday, a company’s stock closed at $67.24. On Friday, it closed at $64.27. What was the change in the closing price?

Y

"

#

Graph and label each point on a coordinate plane. Name the quadrant in which each point is located.

Find each product or quotient.

29. D(-2, 4)

30. E(3, -4)

14. 6(-8)

15. -9(8)

31. F(-1, -3)

32. G(3, 2)

16. -7(-5)

17. 2(-4)(11)

18. 54 ÷ (-9)

19. -64 ÷ (-4)

20. -250 ÷ 25

21. -144 ÷ (-6)

22. SWIMMING POOL The water in a swimming pool drains at a rate of 24 gallons per minute. Describe the change in the amount of water in the swimming pool after 1 hour.


X

"

!

33. MULTIPLE CHOICE Suppose Elan’s home represents the origin on a coordinate plane. If Elan leaves his home and walks two miles west and then four miles north, what is his current location as an ordered pair? A (-2, 4)

C (-2, -4)

B (2, 4)

D (4, -2)


119

CH

A PT ER

2

Standardized Test Practice Cumulative, Chapters 1–2

Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

5. What are the coordinates of the center of the circle below? £ £ " £ Ó Î { x È Ç

1. A scuba diver descends at a rate of 40 feet per minute. Which equation shows how far the scuba diver moves in 2 minutes? C 40(2) = 80 A -40(-2) = 80 B -40(2) = -80 D 40(-2) = -80 2. On Wednesday, the low temperature in Fairbanks, Alaska was -6°F, and the high temperature was 14°F. How much warmer was the high temperature than the low temperature? F -20°F H 8°F G -8°F J 20°F 3. GRIDDABLE Michael had $45 in his savings account at the beginning of the week. He made a withdrawal of $22 to buy a video game on Tuesday, and he made a deposit of $25 on Friday when he received some money for his birthday. How much money in dollars did Michael have in the account at the end of the week if he made no other withdrawals or deposits? 4. Tyrone’s long distance phone bills were $21.35, $11.14, $22.82, and $33.05 over the past four months. He estimated that the phone bill would cost $80 over these four months. Which statement best describes how reasonable his estimate is? A Less than the actual amount because he rounded to the nearest $10 B Less than the actual amount because he rounded to the nearest $100 C More than the actual amount because he rounded to the nearest $10 D More than the actual amount because he rounded to the nearest $100


F G H J

Y £ Ó Î { x È ÇX

(-3, 2) (-2, 3) (3, -2) (2, -3)

6. GRIDDABLE What is the twelfth term of the pattern given by the expression below where n is the term number? 3(n - 5)

7. Last week, Traci wrote checks for $32, $58, and $14. She also made two deposits totaling $189. What other information is needed in order to find the current balance in Traci’s account? A The amount of each deposit. B Traci’s balance last week. C To whom Traci wrote checks last week. D Traci’s deposit when she opened the account.

8. Of the six books in a mystery series, four have 200 pages and two have 300 pages. Which expression represents the total number of pages in the series? F 200 + 300 H 4(200) + 2(300) G 6(200 + 300) J 8(200 + 300)

Standardized Test Practice at pre-alg.com


9. A pattern of equations is shown below. 1% of 2,000 = 20 2% of 1,000 = 20 4% of 500 = 20 8% of 250 = 20 Which statement best describes this pattern? A When the percent is doubled and the other number is doubled, the answer is 20. B When the percent is doubled and the other number is halved, the answer is 20. C When the percent is increased by 2 and the other number remains the same, the answer is 20. D When the percent remains the same and the other number is increased by 2, the answer is 20. 10. Tonya wants to order a roast beef sandwich, a medium order of fries, and a medium drink. How much money will she save by ordering a Daily Special #2?

11. Before the last game of the season, Amy had scored a total of 58 goals. She scored 4 goals in the final game, making her season average 3.1 goals per game. To find the total number of games that Amy played, first find the sum of 58 and 4, and then— A add the sum to 3.1. B subtract 3.1 from 58. C multiply the sum by 3.1. D divide the sum by 3.1.

Question 12 When answering open-ended items on standardized tests, follow these steps: 1. Read the item carefully. 2. Show all of your work. You may receive points for items that are only partially correct. 3. Check your work.

Pre-AP Record your answers on a sheet of paper. Show your work. 12. On graph paper, graph the points A(4, 2), B(-3, 7), and C(-3, 2). Connect the points to form a triangle. a. Add 6 to the x-coordinate of each coordinate pair. Graph and connect the new points to form a new figure. Is the new figure the same size and shape as the original triangle? Describe how the size, shape, and position of the new triangle relate to the size, shape, and position of the original triangle. b. If you add -6 to each original x-coordinate, and graph and connect the new points to create a new figure, how will the position of the new figure relate to that of the original one?

F $1.22 G $0.84 H $0.78 J $0.38


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4

5

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Go to Lesson...

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2-3

2-2

1-1

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2-3

2-2

1-1

1-1

1-1

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121

3

Equations

•


•

Use graphs, tables, and algebraic representations to make predictions and solve problems.

Key Vocabulary area (p. 163) formula (p. 162) like terms (p. 129) sequence (p. 158) simplest form (p. 130)

Real-World Link Skyscrapers Rising 630 feet to the top, the Gateway Arch in St. Louis, Missouri, is 130 feet higher than Mount Rushmore in Black Hills, South Dakota, 75 feet higher than the Washington Monument, and 25 higher than the Seattle Space Needle.

quations Make this Foldable to help you organize information about expressions and equations. Begin ith five sheets of 812’’ × 11’’ paper.

1 Stack 5 sheets of paper 3 4 inch apart.

2 Roll up the bottom edges. All tabs should be the same size.

3 Crease and staple

4 Label the tabs with

along the fold.

topics from the chapter.

122 Chapter 3 Equations Gibson Stock Photography

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Option 1


Find each product. (Lesson 2-4) 1. 2(-3) 2. -4(3) 3. -5(-2)

4. -4 · 6

5. -11 · -8

6. 9 · (-4)

7. STOCK MARKET The price of a stock decreased $2.05 each day for five consecutive days. What was the total change in value of the stock over the five-day period? (Lesson 2-4)

Write each subtraction expression as an addition expression. (Lessons 2-3) 8. 5 - 7 9. 6 - 10 10. -13 - 9

11. 11 - 10

12. 15 - 6

13. -19 - 10

Example 1 Find 7(-2).

7(-2) = -14

The factors have different signs, so the product is negative.

Example 2 Find -5 · -9.

-5 · -9 = 45 The factors have the same sign, so the product is positive.

Example 3 Write 8 - 12 as an addition expression.

8 - 12 = 8 + (-12) To subtract 12, add -12. = -4

Simplify.

14. MONEY Student Council spent $178 on decorations and $110 on snacks for the dance. Write an addition expression for the amount remaining in the dance budget if Student Council initially had $593. (Lessons 2-3)

Find each sum. (Lesson 2-2) 15. 6 + (-9) 16. -8 + 4 17. 4 + (-4)

18. 7 + (-10)

19. -13 + (-8)

20. -11 + 12

Example 4 Find -5 + 7.

-5 + 7 = 2 Subtract | -5 | from | 7 |. The sum is positive because | 7 | > | -5 |.

21. CAVERNS A tour group began 26 feet underground. During their tour, they descended 15 feet more and then ascended 19 feet. Express their current depth as an integer. (Lesson 2-2)

Chapter 3 Get Ready for Chapter 3

123

3-1

The Distributive Property BrainPOP at pre-alg.com

Main Ideas • Use the Distributive Property to write equivalent numerical expressions. • Use the Distributive Property to write equivalent algebraic expressions.

To find the area of a rectangle, multiply the length and width. You can find the total area of the blue and yellow rectangles in two ways. Method 1

Method 2

Put them together. Add the lengths. Then multiply. 4

+

Separate them. Multiply to find each area. Then add. 4

2

2

New Vocabulary equivalent expressions

3

3

3(4 + 2) = 3 · 6 Add. = 18 Multiply.

+

3

3 · 4 + 3 · 2 = 12 + 6 Multiply. = 18 Add.

a. Draw a 2-by-5 and a 2-by-4 rectangle. Find the total area in two ways. b. Draw a 4-by-4 and a 4-by-1 rectangle. Find the total area in two ways. c. Draw any two rectangles that have the same width. Find the total area in two ways. d. What did you notice about the total area in each case?

Distributive Property The expressions 3(4 + 2) and 3 · 4 + 3 · 2 are equivalent expressions because they have the same value, 18. This example shows how the Distributive Property combines addition and multiplication. Vocabulary Link Distribute Everyday Use to deliver to each member of a group Distributive Math Use property that allows you to multiply a number by a sum

Distributive Property Words

To multiply a number by a sum, multiply each number inside the parentheses by the number outside the parentheses.

Symbols

a(b + c) = ab + ac

Examples 3(4 + 2) = 3 · 4 + 3 · 2

(b + c)a = ba + ca (5 + 3)2 = 5 · 2 + 3 · 2

You can use the Distributive Property to evaluate numerical or algebraic expressions. 124 Chapter 3 Equations

EXAMPLE

Use the Distributive Property

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate the expression. a. 2(6 + 4)

b. (8 + 3)5

2(6 + 4) = 2 · 6 + 2 · 4 = 12 + 8 Multiply. = 20 Add.

(8 + 3)5 = 8 · 5 + 3 · 5 = 40 + 15 Multiply. = 55 Add.

1A. (6 + 3)4

1B. 4(2 + 9)

AMUSEMENT PARKS A one-day pass to an amusement park costs $40. A round-trip bus ticket to the park costs $5. a. Write two equivalent expressions to find the total cost of a one-day pass and a bus ticket for 15 students. Method 1 Find the cost for 1 person, then multiply by 15. 15($40 + $5) 15 times the cost for 1 person Method 2 Find the cost of 15 passes and 15 tickets. Then add. 15($40) + 15($5) cost of 15 passes + cost of 15 tickets b. Find the total cost.

Real-World Link Attendance at U.S. amusement parks increased 22% in the 1990s. In 2004, about 328 million people attended these parks.

15($40 + $5) = 15($40) + 15($5) Distributive Property = $600 + $75 Multiply. = $675 Add. The total cost is $675. You can check your results by evaluating 15($45).

2. FOOD A spaghetti dinner costs $10 and a slice of pie costs $2. Write two equivalent expressions to find the total cost of a spaghetti dinner and a slice of pie for each member of a family of 4. Then find the total cost.

Source: International Association of Amusement Parks and Attractions


Algebraic Expressions You can also model the Distributive Property by using algebra tiles and variables. 4HE MODEL SHOWS X 4HERE ARE GROUPS OF X

3EPARATE THE TILES INTO GROUPS OF X AND GROUPS OF

X Î Ó

X X

£ £

X £ £

£ £

Ó

X X

Î

Ó

£ £

£ £

£ £

2(x + 3) = 2x + 2 · 3 = 2x + 6 The expressions 2(x + 3) and 2x + 6 are equivalent expressions because for every value of x, these expressions have the same value. Extra Examples at pre-alg.com AP/Wide World Photos

Lesson 3-1 The Distributive Property

125

EXAMPLE


Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 3(x + 1)

b. (y + 4)5

3(x + 1) = 3x + 3 · 1 = 3x + 3

(y + 4)5 = y · 5 + 4 · 5

3A. 2(a + 5)

EXAMPLE

= 5y + 20

Simplify.

Simplify.

3B. (b + 6)3

Simplify Expressions with Subtraction

Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 2(x - 1) 2(x - 1) = 2[x + (-1)]

Look Back To review subtraction expressions, see Lesson 2–3.

Rewrite x - 1 as x + (-1).

= 2x + 2(-1)

Distributive Property

= 2x + (-2)

Simplify.

= 2x - 2

Definition of subtraction

b. -3(n - 5) -3(n - 5) = -3[n + (-5)]

Rewrite n - 5 as n + (-5).

= -3n + (-3)(-5) Distributive Property = -3n + 15

4A. 4(d - 3)

Example 1 (p. 125)

(p. 125)


4B. -7(e - 4)

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. 1. 5(7 + 8)

Example 2

Simplify.

2. 2(9 + 1)

3. (2 + 4)6

4. (3 + 6)4

MONEY For Exercises 5 and 6, use the following information. Suppose you work in a grocery store 4 hours on Friday and 5 hours on Saturday. You earn $6.25 an hour. 5. Write two different expressions to find your wages. 6. Find the total wages for that weekend. ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 7. 4(x + 3) 11. 8(y - 2)

126 Chapter 3 Equations

8. 8(m + 4) 12. 9(a - 10)

9. (n + 2)3 13. -6(x - 5)

10. (p + 4)5 14. -3(s - 7)

HOMEWORK

HELP

For See Exercises Examples 15–26 1 27, 28 2 29–36 3 37–44 4

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. 15. 2(6 + 1) 19. (9 + 2)4 23. -3(9 - 2)

16. 5(7 + 3) 20. (8 + 8)2 24. -2(8 - 4)

17. (4 + 6)9 21. 7(3 - 2) 25. -5(8 - 4)

18. (4 + 3)3 22. 6(8 - 5) 26. -5(10 - 3)

2 7. MOVIES One movie ticket costs $7, and one small bag of popcorn costs $3. Write two equivalent expressions for the total cost of four movie tickets and four bags of popcorn. Then find the cost. 28. SPORTS A volleyball uniform costs $15 for the shirt, $10 for the pants, and $8 for the socks. Write two equivalent expressions for the total cost of 12 uniforms. Then find the cost. ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 29. 33. 3 7. 41.

2(x + 3) (x + 3)4 3(x - 2) (r - 5)6

30. 34. 38. 42.

3 1. 35. 39. 43.

5(y + 6) (y + 2)10 9(m - 2) (x - 3)12

ANALYZE GRAPHS For Exercises 45–47, use the double bar graph.

7(y + 8) (2 + x)5 15(s - 3) (a - 6)(-5)

Annual Fashion Spending $2200 $1964

$2000

Average Spending per Teen

45. Find the total amount spent on average by two male teenagers and two female teenagers on fashion products in 2003. 46. Find the total amount spent on average by three female teenagers in both 2002 and 2003 on fashion products. 47. Did teen spending on fashion products increase? How do you know? Explain.

32. 36. 40. 44.

3(n + 1) (3 + y)6 8(z - 3) -2(z - 4)

$1800 $1600 $1400

$1342

$1200 $890

$1000

$834

$800 $600 $400 $200 0

2002 Female

2003 Male

Source: www.piperjaffray.com

ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 48. 2(x + y) 5 1. 4(j - k) EXTRA

PRACTICE


49. 3(a + b) 52. 10(r - s)

50. (e + f)(-5) 53. (u - w)(-8)

MENTAL MATH Find each product mentally. Example 15 · 12 = 15(10 + 2) Think: 12 is 10 + 2. = 150 + 30 or 180 Distributive Property 54. 7 · 14

55. 8 · 23

56. 9 · 32

57. 16 · 11

Lesson 3-1 The Distributive Property

127

58. THEME PARKS Admission to an amusement park is $41.99 for adults and $26.99 for children. The Diego family has a coupon for $10 off each ticket. Write an expression for the cost for x adults and y children.

H.O.T. Problems

59. OPEN ENDED Write an equation using three integers that is an example of the Distributive Property. 60. FIND THE ERROR Julia and Catelyn are using the Distributive Property to simplify 3(x + 2). Who is correct? Explain your reasoning. Catelyn 3(x + 2) = 3x + 6

Julia 3(x + 2) = 3x + 2

61. CHALLENGE Is 3 + (x · y) = (3 + x) · (3 + y) a true statement? If so, explain your reasoning. If not, give a counterexample. 62.

Writing in Math Explain how rectangles can be used to show the Distributive Property.

63. A ticket to a baseball game costs t dollars. A soft drink costs s dollars. Which expression represents the total cost of a ticket and soft drink for p people? A pst

C t(p + s)

B p + (ts)

D p(t + s)

64. Which equation is always true? F 5(a + b) = 5a + b G 5(ab) = (5a)(5b) H 5(a + b) = 5(b + a) J 5(a + 0) = 5a + 5

ALGEBRA The table shows several solutions of the equation x + y = 4. (Lesson 2-6) 65. Graph the ordered pairs on a coordinate plane. 66. Describe the graph. -4y

x+y=4 x

y

(x, y)

-1

5

(-1, 5)

1 2

3 2

(1, 3) (2, 2)

67. ALGEBRA Evaluate x if x = 2 and y = -3. (Lesson 2-5) 68. FITNESS Jake ran x miles on Monday, y miles on Tuesday, and z miles on Wednesday. Write an expression for the average number of miles Jake ran. (Lesson 1-2)

PREREQUISITE SKILL Write each subtraction expression as an addition expression. (Lesson 2-3) 69. 5 - 3 70. -8 - 4 71. 10 - 14 72. 8 - (-6) 128 Chapter 3 Equations

3-2

Simplifying Algebraic Expressions

Main Idea • Use the Distributive Property to simplify algebraic expressions.

New Vocabulary term coefficient like terms constant simplest form simplifying an expression

In a set of algebra tiles, X represents the variable x,

represents the

£

integer 1, and £ represents the integer -1. You can use algebra tiles to represent expressions. You can also sort algebra tiles by their shapes and group them. The tiles below represent the expression 2x + 3 + 3x + 1. On the right, the algebra tiles have been sorted and combined. X TILES £

X

X

£

£

X

X

X

X

£

X

X

X

X

£

£

£

£

TILES ÓX

Î

ÎX

xX

£

{

Therefore, 2x + 3 + 3x + 1 = 5x + 4. Model each expression with algebra tiles or a drawing. Then sort them by shape and write an expression represented by the tiles. a. 3x + 2 + 4x + 3

b. 2x + 5 + x

c. 4x + 5 + 3

d. x + 2x + 4x

Simplify Expressions When plus or minus signs separate an algebraic expression into parts, each part is a term. The numerical part of a term that contains a variable is called the coefficient of the variable. Four terms

2x + 8 + x + 8 2 is the coefficient of 2x.

Vocabulary Link Constant Everyday Use unchanging Math Use a fixed value in an expression

1 is the coefficient of x because x = 1x.

Like terms are terms that contain the same variables, such as 2n and 5n or 6xy and 4xy. A term without a variable is called a constant. Constant terms are also like terms. Like terms

5y + 3 + 2y + 8y Constant

Lesson 3-2 Simplifying Algebraic Expressions

129

Rewriting a subtraction expression using addition will help you identify the terms of an expression.

EXAMPLE

Identify Parts of Expressions

Identify the terms, like terms, coefficients, and constants in the expression 3x - 4x + y - 2. 3x - 4x + y - 2 = 3x + (– 4x) + y + (–2)


= 3x + (– 4x) + 1y + (–2) Identity Property The terms are 3x, – 4x, y, and –2. The like terms are 3x and – 4x. The coefficients are 3, –4, and 1. The constant is –2.

1. Identify the terms, like terms, coefficients, and constants in the expression 9a – 2a + 3b – 5.

An algebraic expression is in simplest form if it has no like terms and no parentheses. When you use the Distributive Property to combine like terms, you are simplifying the expression.

EXAMPLE


Simplify each expression. a. 6n + 3 + 2n Equivalent Expressions To check whether 6n + 2n and 8n are equivalent expressions, substitute any value for n and see whether the expressions have the same value.

6n and 2n are like terms. 6n + 3 + 2n = 6n + 2n + 3 Commutative Property = (6 + 2)n + 3 Distributive Property = 8n + 3

Simplify.

b. 3x - 5 - 8x + 6 3x and -8x are like terms. -5 and 6 are also like terms. 3x - 5 - 8x + 6 = 3x + (-5) + (-8x) + 6


= 3x + (-8x) + (-5) + 6

Commutative Property

= [3 + (-8)]x + (-5) + 6


= -5x + 1

Simplify.

c. m + 3(n + 4m) m + 3(n + 4m) = m + 3n + 3(4m) Distributive Property

2A. 4x + 6 - 3x 130 Chapter 3 Equations

= m + 3n + 12m

Associative Property

= 1m + 3n + 12m

Identity Property

= 1m + 12m + 3n

Commutative Property

= (1 + 12)m + 3n


= 13m + 3n

Simplify.

2B. 2m + 3 - 7m - 4

2C. 4(q + 8p) + p Extra Examples at pre-alg.com

BASEBALL CARDS Suppose your brother has 15 more baseball cards in his collection than you have. Write an expression in simplest form that represents the total number of cards in both collections. Words Variables Expression

Source: CMG Worldwide

plus

number of your brother’s cards

Let x = number of cards you have. Let x + 15 = number of cards your brother has. x + (x + 15)

x + (x + 15) = (x + x) + 15 = (1x + 1x) + 15 = (1 + 1)x + 15 = 2 x + 15

Real-World Link Honus Wagner is considered by many to be baseball’s greatest all-around player. In July, 2000, one of his baseball cards sold for $1.1 million.

number of your cards

Associative Property Identity Property Distributive Property Simplify.

The expression 2x + 15 represents the total number of cards, where x is the number of cards you have.

3. STAMPS Matt and Lola both collect stamps. Lola has 25 more stamps in her collection than Matt has. Write an expression in simplest form that represents the total number of stamps in both collections. Personal Tutor at pre-alg.com

Example 1 (p. 130)

Identify the terms, like terms, coefficients, and constants in each expression. 2. 2m - n + 6m

3. 4y - 2x - 7

4. 6a + 4 + 2a

5. x + 9x + 3

6. 9y + 8 - 8

7. 3x + 2y + 4y

8. 6c + 4 + c + 8

9. 2x - 5 - 4x + 8

10. x + 3(x + 4y)

11. 8e - 4(2f + 5e)

1. 4x + 3 + 5x + y Example 2 (p. 130)

12. 5 - 3(y + 7)

(p. 131)

13. MONEY You have saved some money. Your friend has saved $20 more than you. Write an expression in simplest form that represents the total amount of money you and your friend have saved.

HELP

Identify the terms, like terms, coefficients, and constants in each expression.

Example 3

HOMEWORK

Simplify each expression.


14. 3 + 7x + 3x + x

15. y + 3y + 8y + 2

16. 2a + 5c - a + 6a

17. 5c - 2d + 3d - d

18. 6m - 2n + 7

19. 7x - 3y + 3z - 2 Lesson 3-2 Simplifying Algebraic Expressions

Kit Kittle/CORBIS

131

Simplify each expression. 20. 2x + 5x

21. 7b + 2b

22. y + 10y

23. 5y + y

24. 2a + 3 + 5a

25. 4 + 2m + m

26. 2y + 8 + 5y + 1

27. 8x + 5 + 7 + 2x

28. 5x - 3x

29. 10b - 2b

30. 4y - 5y

31. r - 3r

32. 8 + x - 5x

33. 6x + 4 - 7x

34. 2x + 3 - 3x + 9

For Exercises 35–38, write an expression in simplest form that represents the total amount in each situation. 35. SCHOOL SUPPLIES You bought 5 folders that each cost x dollars, a calculator for $45, and a set of pens for $3. 36. SHOPPING Suppose you buy 3 shirts that each cost s dollars, a pair of shoes for $50, and jeans for $30. 37. FASHION Your friend Natasha has y pairs of shoes. Her sister has 5 fewer pairs. 38. BABY-SITTING Alicia earned d dollars baby-sitting. Her friend earned twice as much. You earned $2 less than Alicia’s friend earned. Simplify each expression. Real-World Link In a recent survey, 10% of students in grades 6–12 reported that most of their spending money came from baby-sitting. Source: USA WEEKEND

39. 6m + 2n + 10m

40. -2y + x + 3y

41. c + 2(d - 5c)

42. 3(b + 2) + 2b

43. 5(x + 3) + 8x

44. -3(a + 2) - a

45. -2(x + 3) + 2x

46. 4x - 4(2 + x)

47. 8a - 2(a - 7)

GEOMETRY You can find the perimeter of a geometric figure by adding the measures of its sides. Write an expression in simplest form for the perimeter of each figure. 2x ⫹ 1 48. 49. 3x

EXTRA

PRACTIICE

H.O.T. Problems

x

x 2x ⫹ 1


5x

4x

Simplify to make the calculation as easy as possible. 50. 16 · (-31) + 16 · 32

51. 72(38) + (-72)(18)

52. OPEN ENDED Write an expression in simplest form containing three terms. One of the terms should be a constant. 53. FIND THE ERROR Koko and John are simplifying the expression 5x - 4 + x + 2. Who is correct? Explain your reasoning. Koko 5x - 4 + x + 2 = 6x - 2

John 5x - 4 + x + 2 = 5x - 2

54. Which One Doesn’t Belong? Identify the algebraic expression that does not belong with the other three. Explain your reasoning. -6(x - 2) 132 Chapter 3 Equations Mary Kate Denny/PhotoEdit

x + 12 - 7x

-x - 5x + 12

-6x - 12

CHALLENGE You use deductive reasoning when you base a conclusion on mathematical rules or properties. Indicate the property that justifies each step that was used to simplify 3(x + 4) + 5(x + 1). 55. 3(x + 4) + 5(x + 1) = 3x + 12 + 5x + 5 56.

= 3x + 5x + 12 + 5

57.

= 3x + 5x + 17

58.

= 8x + 17

59.

Writing in Math Explain how algebra tiles can be used to simplify an algebraic expression. Illustrate your reasoning with an example.

60. The perimeter of DEF is 4x + 3y. What is the measure of the third side of the triangle? A -2x + 2y

$ xX ÎY

B 2x + 2y C x-y

X {Y

&

%

D -x + 2y

61. You spend x minutes reading a book on Saturday. On Sunday, you spend 35 more minutes reading than you did on Saturday. Which expression represents the total amount of time spent reading the book on Saturday and Sunday? F 2x + 35

H 2x - 35

G x + 35

J x - 35

ALGEBRA Use the Distributive Property to write each expression as an equivalent expression. (Lesson 3-1) 62. 3(a + 5)

63. -2(y - 8)

64. 7(d - 10)

65. -3(x - 1)

66. Name the quadrant in which P(-5, -6) is located. (Lesson 2-6) 67. CRUISES The table shows the number of people who took a cruise in various years. Make a scatter plot of the data. (Lesson 1-7)

9EAR .UMBER MILLIONS -ÕÀVi\ #RUISE ,INES )NTERNATIONAL !SSOCIATION

3(4a - 3b) b-4

68. ALGEBRA What is the value of _ if a = 6 and b = 7? (Lesson 1-3) 69. DECORATING A wallpaper roll contains a sheet that is 40 feet long and 18 inches wide. What is the minimum number of rolls of wallpaper needed to cover 500 square feet of wall space? (Lesson 1-1)

PREREQUISITE SKILL Find each sum. (Lesson 2-2) 70. -5 + 4

71. -8 + (-3)

72. 10 + (-1)

Lesson 3-2 Simplifying Algebraic Expressions

133

EXPLORE

3-3 AND 3-4

Algebra Lab

Solving Equations Using Algebra Tiles ACTIVITY 1 In a set of algebra tiles, X represents the variable x,

£

represents the

integer 1, and £ represents the integer -1. You can use algebra tiles and an equation mat to model equations.

£

X

£

£

£

X Î

£

£

£

£

X

£

£

£

X Ó

x

£

When you solve an equation, you are trying to find the value of x that makes the equation true. The following example shows how to solve x + 3 = 5 using algebra tiles.

£

X

£

£

X Î

£

£

£

X

x

X

Remove the same number of 1-tiles from each side of the mat until the x-tile is by itself on one side.

£

X

Model the equation.

£

X

£

£

The number of tiles remaining on the right side of the mat represents the value of x.

Ó

Therefore, x = 2. Since 2 + 3 = 5, the solution is correct.

EXERCISES Use algebra tiles or a drawing to model and solve each equation. 1. 3 + x = 7 2. x + 4 = 5 3. 6 = x + 4 4. 5 = 1 + x 134 Chapter 3 Equations

ACTIVITY 2

Animation at pre-alg.com

Some equations are solved by using zero pairs. You may add or subtract a zero pair from either side of an equation mat without changing its value. The following example shows how to solve x + 2 = -1 by using zero pairs.

£

X

£

X Ó

£

£

£

£

£

X Ó Ó®

X

£

X

£

Model the equation. Notice it is not possible to remove the same kind of tile from each side of the mat.

£

£ £

X

£

£

Add 2 negative 1-tiles to the left side of the mat to make zero pairs. Add 2 negative 1-tiles to the right side of the mat.

£ Ó®

£

£

£

£

Remove all of the zero pairs from the left side. There are 3 negative 1-tiles on the right side of the mat.

Î

Therefore, x = -3. Since -3 + 2 = -1, the solution is correct.

EXERCISES Use algebra tiles or a drawing to model and solve each equation. 5. x + 2 = -2 6. x - 3 = 2 7. 0 = x + 3 8. -2 = x + 1

ACTIVITY 3 The equation 2x = -6 is modeled using more than one x-tile. Arrange the tiles into equal groups to match the number of x-tiles. X

X ÓX

£

£

£

£

£

£ È

X

£

£

£

£

£

£

X X

Î

Therefore, x = -3. Since 2(-3) = -6, the solution is correct.

EXERCISES Use algebra tiles or a drawing to model and solve each equation. 9. 3x = 3 10. 2x = -8 11. 6 = 3x 12. -4 = 2x Explore 3-3 and 3-4 Algebra Lab: Solving Equations Using Algebra Tiles

135

3-3 Main Ideas • Solve equations by using the Subtraction Property of Equality. • Solve equations by using the Addition Property of Equality.

New Vocabulary inverse operation equivalent equations

Solving Equations by Adding or Subtracting On the balance at the right, the paper bag contains a certain number of blocks. (Assume that the paper bag weighs nothing.) a. Without looking in the bag, how can you determine the number of blocks in the bag? b. Explain why your method works.

Solve Equations by Subtracting The equation x + 4 = 7 is a model of the situation above. You can use inverse operations to solve the equation. Inverse operations “undo” each other. For example, to undo the addition of 4 in the expression x + 4, you would subtract 4. To solve the equation x + 4 = 7, subtract 4 from each side. x+4=7 x+4-4=7-4 Subtract 4 from the left side of the equation to isolate the variable.

x+0=3 x=3

Subtract 4 from the right side of the equation to keep it balanced.

The solution is 3. You can use the Subtraction Property of Equality to solve any equation like x + 4 = 7.

Subtraction Property of Equality Words

If you subtract the same number from each side of an equation, the two sides remain equal.

Symbols

For any numbers a, b, and c, if a = b, then a - c = b - c.

Examples

5=5 5-3=5-3 2=2

x+2=3 x+2-2=3-2 x=1



The equations x + 4 = 7 and x = 3 are equivalent equations because they have the same solution, 3. When you solve an equation, you should always check to be sure that the first and last equations are equivalent.

EXAMPLE

Solve Equations by Subtracting

Solve x + 8 = -5. Check your solution and graph it on a number line. x + 8 = -5 x + 8 - 8 = -5 - 8 x + 0 = -13 x = -13 Checking Equations It is always wise to check your solution. You can often use arithmetic facts to check the solutions of simple equations.

Write the equation. Subtract 8 from each side. 8 - 8 = 0, -5 - 8 = -13 Identity Property; x + 0 = x

To check your solution, replace x with -13 in the original equation. CHECK

x + 8 = -5 -13 + 8 -5 -5 = -5

Write the equation. Check to see whether this sentence is true. The sentence is true.

The solution is -13. To graph it, draw a dot at -13 on a number line. £{ £Î £Ó ££ £ä n

Ç

Solve each equation. Check your solution and graph it on a number line. 1A. 4 = x + 10 1B. 16 + z = 14

Solve Equations by Adding Some equations can be solved by adding the same number to each side. This uses the Addition Property of Equality. Addition Property of Equality Words

If you add the same number to each side of an equation, the two sides remain equal.

Symbols

For any numbers a, b, and c, if a = b, then a + c = b + c.

Examples

6=6 6+3=6+3 9=9

x-2=5 x-2+2=5+2 x=7

If an equation has a subtraction expression, first rewrite the expression as an addition expression. Then add the additive inverse to each side.

EXAMPLE

Solve Equations by Adding

Solve y - 7 = -25. y - 7 = -25 y + (-7) = -25 y + (-7) + 7 = -25 + 7 y + 0 = -25 + 7 y = -18 The solution is -18.

Write the equation. Rewrite y - 7 as y + (-7). Add 7 to each side. Additive Inverse Property; (-7) + 7 = 0. Identity Property; y + 0 = y Check your solution.

Solve each equation. 2A. -20 = y - 13 Extra Examples at pre-alg.com

2B. -115 + b = -84 Lesson 3-3 Solving Equations by Adding or Subtracting

137

Jessica downloaded 54 songs onto her digital music player. This is 17 less than the number of songs Kaela downloaded earlier. Which equation can be used to find the number of songs Kaela downloaded onto her digital music player? Key Words When translating words to equation, look for key words that indicate operations. The phrase “less than” can indicate subtraction or an inequality.

A x - 17 = 54 B x + 17 = 54

C 17 - x = 54 D -54 = 17 + x

Read the Test Item Translate the verbal sentence into an equation. Solve the Test Item Words

Jessica downloaded 17 less songs than

Variable

Let = the number of songs Kaela downloaded.

Equation

54 = x - 17

So, the equation 54 = x - 17 or x - 17 = 54 can be used to find the number of songs Kaela downloaded. This is choice A.

3. During the night, the temperature dropped 14° to -9°F. Which equation can be used to find the temperature at the beginning of the night? F -9 + x = -14

H -9 - x = 14

G 14 - x = -9

J x - 14 = -9


SLEDDING Use the information at the left. Write and solve an equation to find the distance of the Northern Route of the Iditarod Trail Sled Dog Race. The Southern Route is 49 miles longer than the Northern Route.

Real-World Link There are two different routes for the Iditarod Trail Sled Dog Race. During the odd years, the race takes place on the 1161-mile Southern Route. This is 49 miles longer than the Northern Route that takes place during the even years. Source: iditarod.com

138 Chapter 3 Equations AP/Wide World Photos

Let d = the distance of the Northern Route. 1161= d + 49 Write the equation. 1161 - 49 = d + 49 - 49 Subtract 49 from each side. 1112 = d Simplify. CHECK

1161 = d + 49 1161 1112 + 49 1161 = 1161

Write the equation. Check to see whether this statement is true. The statement is true.

The Northern Route is 1112 miles long.

4. BUILDINGS The Jefferson Memorial in Washington, D.C., is 129 feet tall. This is 30 feet taller than the Lincoln Memorial. Write and solve an equation to find the height of the Lincoln Memorial.

2. w + 4 = -10

3. 16 = y + 20

4. n - 8 = 5

5. k - 25 = 30

6. r - 4 = -18

7. MULTIPLE CHOICE A video store sells a DVD for $12 more than it pays for it. If the selling price of the DVD is $19, which equation can be used to find how much the store paid for the DVD? A x + 19 = -12

For See Exercises Examples 9–26 1, 2 27, 28 4 43, 44 3

ALGEBRA Solve each equation. Check your solution and graph it on a number line. 9. y + 7 = 21

10. x + 5 = 18

1 1. m + 10 = -2

12. x + 5 = -3

13. a + 10 = -4

14. t + 6 = -9

15. y + 8 = 3

16. 9 = 10 + b

17. k - 6 = 13

18. r - 5 = 10

19. 8 = r - 5

20. 19 = g - 5

21. x - 6 = -2

22. y - 49 = -13

23. -15 = x - 16

24. -8 = t - 4

25. 23 + y = 14

26. 59 = s + 90

27. ELECTIONS In the 2004 presidential election, Georgia had 15 electoral votes. That was 19 votes fewer than the number of electoral votes in Texas. Write and solve an equation to find the number of electoral votes in Texas. 28. WEATHER The difference between the record high and low temperatures in Charlotte, North Carolina, is 109˚F. The record low temperature was -5˚F. Write and solve an equation to find the record high temperature. 29. RESEARCH Use the Internet or another source to find record temperatures in your state. Use the data to write a problem. ANALYZE GRAPHS For Exercises 30 and 31, use the graph and the following information. Tokyo’s population is 10 million greater than New York City’s population. Los Angeles’ population is 2 million less than New York City’s population. 30. Write two different equations to find New York City’s population. 31. Solve the equations to find the population of New York City.

Most Populous Urban Areas 28

?

18

18

18 16 14

Shanghai

HELP

8. FUND-RAISING Jim sold 43 candles to raise money for a class trip. This is 15 less than the number Diana sold. Write and solve an equation to find the number of candles Diana sold.

Los Angeles

HOMEWORK

D 12 - x = 19

Sao Paulo

(p. 138)

C x + 12 = 19

Bombay

Example 4

B 19 + x = 12

Mexico City

(p. 138)

1. x + 14 = 25

New York City

Example 3

ALGEBRA Solve each equation. Check your solution and graph it on a number line.

Tokyo

(p. 137)

Population (millions)

Examples 1, 2

Cities Source: infoplease.com

Lesson 3-3 Solving Equations by Adding or Subtracting

139

ALGEBRA Solve each equation. Check your solution.

PRACTICE

32. a - 6.1 = 3.4

33. 14.8 + x = - 20.1

34. 17.6 = y + 11.5

See pages 766, 796.

35. p - (-13.35) = -19.72

36. -52.23 + b = 40.04

37. z - 37.98 = 65.21


38. ALGEBRA If a number x satisfies x + 4 = -2, find the numerical value of -3x - 2.

EXTRA

H.O.T. Problems

39. OPEN ENDED Write two equations that are equivalent. Then write two equations that are not equivalent. Justify your reasoning. 40. SELECT A TECHNIQUE Jaime’s golf score was -9 today. She decreased her score by 5 strokes from yesterday. Which of the following techniques might you use to determine what her golf score was yesterday? Justify your selection(s). Then use the technique(s) to solve the problem. computer

draw a model

real objects

41. CHALLENGE Write two equations in which the solution is -5. 42.

Writing in Math Formulate a problem situation for the equation x + 7 = 20.

The table shows the five nearest stars to Earth, excluding the Sun. Star

Distance (light-years)

Proxima Centauri

4.22

Alpha Centauri A

4.40

Alpha Centauri B

4.40

Barnard’s Star

5.94

Wolf 359

7.79

43. Which equation will best help you find how much closer Proxima Centauri is to Earth than Barnard’s Star? A x - 5.94 = 4.22 C 5.94 + x = 4.22 B x + 4.22 = 5.94 D 5.94 + 4.22 = x 44. GRIDDABLE How many light years closer is Alpha Centauri B to Earth than Wolf 359?

ALGEBRA Simplify each expression. (Lessons 3-1 and 3-2) 45. -2(x + 5)

46. (t + 4)3

47. -4(x - 2)

48. 6z - 3 - 10z + 7

49. 2(x + 6) + 4x

50. 3 - 4(m + 1)

51. GEOLOGY The width of a beach is changing at a rate of -9 inches per year. How long will it take for the width of the beach to change -4.5 feet? (Lesson 2-5) 52. MONEY Xavier opened a checking account with a deposit of $200. During the next week, he wrote checks for $65, $83, and $28 and made a deposit of $50. Write an addition expression for this situation and find the balance in his account. (Lesson 2-2)

PREREQUISITE SKILL Divide. (Lesson 2–5) 53. -100 ÷ 10 54. 50 ÷ (-2) 140 Chapter 3 Equations

55. -49 ÷ (-7)

72 56. _ -8

3-4

Solving Equations by Multiplying or Dividing

Main Ideas • Solve equations by using the Division Property of Equality. • Solve equations by using the Multiplication Property of Equality.

An exchange rate allows people to exchange one currency for another. In Mexico, about 11 pesos can be exchanged for $1 of U.S. currency, as shown in the table.

U.S. Value ($)

Number of Pesos

1

11(1) ⴝ 11

2

11(2) ⴝ 22

3

11(3) ⴝ 33

4

11(4) ⴝ 44

In general, if we let d represent the number of U.S. dollars and p represent the number of pesos, then 11d = p.

a. Suppose lunch in Mexico costs 77 pesos. Write an equation to find the cost in U.S. dollars. b. How can you find the cost in U.S. dollars?

Solve Equations by Dividing The equation 11x = 77 is a model of the relationship described above. To undo the multiplication operation in 11x, you would divide by 11. To solve the equation 11x = 77, divide each side by 11. 11x = 77 Divide the left side of the equation by 11 to undo the multiplication 11 · x.

11x 77 _ = _ 11 11

1x = 7 x = 7

Divide the right side of the equation by 11 to keep it balanced.

The solution is 7. You can use the Division Property of Equality to solve any equation like 11x = 77.

Division Property of Equality Words

When you divide each side of an equation by the same nonzero number, the two sides remain equal.

b _ For any numbers a, b, and c, where c ≠ 0, if a = b then _ c = c. Examples 14 = 14 3x = -12 a

Symbols

14 _ 14 _ =

3x _ -12 _ =

2=2

x = -4

7

7

3

3

Lesson 3-4 Solving Equations by Multiplying or Dividing

141

EXAMPLE

Solve Equations by Dividing

Solve 5x = -30. Check your solution and graph it on a number line. 5x = -30

Write the equation.

5x -30 = 5 5

Divide each side by 5 to undo the multiplication in 5 · x.

1x = -6

5 ÷ 5 = 1, -30 ÷ 5 = -6

x = -6

Identity Property; 1x = x

To check your solution, replace x with -6 in the original equation. CHECK

5x = -30 5(-6) -30 -30 = -30


The solution is -6. To graph it, draw a dot at -6 on a number line. ⫺7 ⫺6

⫺5

⫺4

⫺3

⫺2 ⫺1

0

1. Solve -48 = 6x. Check your solution and graph it on a number line.

PARKS It costs $3 per car to use the hiking trails along the Columbia River Highway. If income from the hiking trails totaled $1275 in one day, how many cars entered the park? The cost per car

Words

$3

Equation

The Columbia River Highway, built in 1913, is a historic route in Oregon that curves around twenty waterfalls through the Cascade Mountains. Source: columbiariverhighway.com

Write the equation.

3x 1275 _ =_

Divide each side by 3.

3

x = 425 CHECK

equals

the total

x

=

$1275

·

3x = 1275 3

the number of cars

Let = the number of cars.

Variable

Real-World Link

times

Simplify.

3x = 1275 3(425) 1275 1275 = 1275


Therefore, 425 cars entered the park.

2. PARKS In-state camping permits for New Mexico State Parks cost $180 per year. If income from the camping permits totaled $8280 during the first day of sales, how many people bought permits? Personal Tutor at pre-alg.com

142 Chapter 3 Equations Stuart Westmorland/CORBIS

Solve Equations by Multiplying Some equations can be solved by multiplying each side by the same number. This property is called the Multiplication Property of Equality. Multiplication Property of Equality Interactive Lab pre-alg.com

Words

When you multiply each side of an equation by the same number, the two sides remain equal.

Symbols

For any numbers a, b, and c, if a = b, then ac = bc.

_x = 7

8=8

Examples

6

_6x 6 = (7)6

8(-2) = 8(-2)

x = 42

-16 = -16

EXAMPLE Reading Math Division Expressions y Remember, _ means -4 y divided by -4.

Solve Equations by Multiplying

y -4

Solve _ = -9. Check your solution and graph it on a number line. y _ = -9 -4 y _ (-4) = -9(-4) -4

y = 36

Write the equation. y -4

Multiply each side by −4 to undo the division in _. Simplify.

y CHECK _ = -9 -4

Write the equation.

36 _ -9

Check to see whether this statement is true.

-9 = -9

The statement is true.

-4

The solution is 36. To graph it, draw a dot at 36 on a number line. ÎÓ

ÎÎ

Î{

Îx

ÎÈ

ÎÇ

În

x 3. Solve 7 = _ . Check your solution and graph it on a number line. -2

Example 1 (p. 142)

Example 2 (p. 142)

Example 3 (p. 143)

ALGEBRA Solve each equation. Check your solution. 1. 4x = 24

2. -2a = 10

3. -7t = -42

4. TOYS A spiral toy that can bounce down a flight of stairs is made from 80 feet of wire. Write and solve an equation to find how many of these toys can be made from a spool of wire that contains 4000 feet. ALGEBRA Solve each equation. Check your solution. k =9 5. _ 3


y 6. _ = -8 5

n 7. -11 = _ -6


143

HOMEWORK

HELP

For See Exercises Examples 8–31 1, 3 32–35 2

ALGEBRA Solve each equation. Check your solution. 8. 3t = 21

h 10. _ =6

9. 8x = 72

4

11. _c = 4

g 12. _ = -7 -2

x 13. -42 = _

14. -32 = 4y

15. 5n = -95

16. -56 = -7p

17. -8j = -64

b 18. 11 = _

h 19. _ = 20

20. 45 = 5x

21. 3u = 51

m = -3 22. _

24. 86 = -2v

25. -8a = 144

v 27. _ = -132

k 28. -21 = _

30. -116 = -4w

31. -68 = -4m

9

d 23. _ = -3 3 f 26. _ = -10 -13 29. -56 = _t 9

-2

-3

-7 45

-11

8

32. BOATING A forest preserve rents canoes for $12 per hour. Corey has $36. Write and solve an equation to find how many hours he can rent a canoe.

Indian Ocean

Pacific Ocean Outback

AUSTRALIA Southern Ocean

Real-World Link Some students living in the Outback are so far from schools that they get their education by special radio programming. They mail in their homework and sometimes talk to teachers by two-way radio. Source: Kids Discover Australia

33. FRUIT Jenny picked a total of 960 strawberries in 1 hour. Write and solve an equation to find how many strawberries Jenny picked per minute. 34. RANCHING The largest ranch in the world is in the Australian Outback. It is about 12,000 square miles, which is five times the size of the largest United States ranch. Write and solve an equation to find the size of the largest United States ranch. 35. RANCHING In the driest part of an Outback ranch, each cow needs about 40 acres for grazing. Write and solve an equation to find how many cows can graze on 720 acres of land. ALGEBRA Graph the solution of each equation on a number line. 36. -6r = -18

37. -42 = -7x

n 38. _ =3 12

MEASUREMENT The chart shows several conversions in the customary system. Write and solve an equation to find each quantity. 40. the number of feet in 132 inches 41. the number of yards in 15 feet 42. the number of miles in 10,560 feet

y 39. _ = -1 -4

Customary System (length) 1 mile = 5280 feet 1 mile = 1760 yards 1 yard = 3 feet 1 foot = 12 inches 1 yard = 36 inches

EXTRA

PRACTICE

43. PAINTING A person-day is a unit of measure that represents one person working for one day. A painting contractor estimates that it will take 24 person-days to paint a house. Write and solve an equation to find how many painters the contractor will need to hire to paint the house in 6 days.

See pages 766, 796 Self-Check Quiz at pre-alg.com

44. FIND THE DATA Refer to the United States Data File on pages 18–21 of your book. Choose some data and write a real-world problem in which you would solve an equation by multiplying or dividing.


H.O.T. Problems

45. OPEN ENDED Write an equation of the form ax = c where a and c are integers and the solution is 4. 46. NUMBER SENSE Find an equation that is equivalent to -9t = 18. x = 3, what is the value of 7x + 13? 47. CHALLENGE If 10

48.

Writing in Math Explain how equations are used to find the U.S. value of foreign currency. Illustrate your reasoning by finding the cost in U.S. dollars of a 12-pound bus trip in Egypt, if 6 pounds can be exchanged for one U.S. dollar.

49. Suppose that one pyramid balances two cubes and one cylinder balances three cubes as shown below. Which statement is NOT true?

50. The solution of which equation is NOT graphed on the number line below? { Î

A One pyramid and one cube balance three cubes. B One pyramid and one cube balance one cylinder.

Ó

£

ä

£

Ó

Î

F 12 = -6x

H -14 = 7x

G 8x = -16

J -18x = -36

51. During a vacation, the Mulligan family drove 63 miles in 1 hour. If they averaged the same speed during their trip, which equation can be used to find how far the Mulligan family drove in 6 hours? 63 A _ x =6

C One cylinder and one pyramid balance four cubes. D One cylinder and one cube balance two pyramids.

x = 63 B _ 6

C 6x = 63 D 63x = 6

ALGEBRA Solve each equation. Check your solution. (Lesson 3-3) 52. 3 + y = 16

53. 29 = n + 4

54. k - 12 = -40

ALGEBRA Simplify each expression. (Lesson 3-2) 55. 4x + 7x

56. 2y + 6 + 5y

57. 3 - 2(y + 4)

58. AGE Patricia is 12 years old, and her younger sister Renee is 2 years old. How old will each of them be when Patricia is twice as old as Renee? (Lesson 1-1)

PREREQUISITE SKILL Find each difference. (Lesson 2-3) 59. 8 - (-2)

60. -5 - 5

61. -10 - (-8)

62. -18 - 4

63. -45 - (-9)

64. 33 - (-19)


145

CH

APTER

3


1. MULTIPLE CHOICE Lucita works at a fitness center and earns $5.50 per hour. She worked 3 hours on Friday and 7 hours on Saturday. Which expression does NOT represent her wages that weekend? (Lesson 3-1) A 5.50(3 + 7) B 10(5.50)

9. AVIATION On December 17, 1903, the Wright brothers made the first flights in a power-driven airplane. Orville’s flight covered 120 feet, which was 732 feet shorter than Wilbur’s. Find the length of Wilbur Wright’s flight. (Lesson 3-3) 10. WEATHER Before a storm, the barometric pressure dropped to 29.2, which was 1.3 lower than the pressure earlier in the day. Write an equation to represent this situation.

C 5.50(3) + 5.50(7) D 7(5.50 + 3) 2. FUND-RAISING Debbie sold 23 teen magazines at $3.25 each, 38 sports magazines at $3.50 each, and 30 computer magazines at $2.95 each. How much money did Debbie raise? (Lesson 3-1) Simplify each expression. (Lessons 3-1 and 3-2) 3. 6(x + 2) 4. 5(x - 7) 5. 6y - 4 + y 6. 2a + 4(a - 9) 7. SCHOOL You spent m minutes studying on Monday. On Tuesday, you studied 15 more minutes than you did on Monday. Write an expression in simplest form that represents the total amount of time spent studying on Monday and Tuesday. (Lesson 3-2)

(Lesson 3-3)

ALGEBRA Solve each equation. (Lessons 3-3 and 3-4) 11. 4h = -52 12. y - 5 = -23 x =4 13. -3

14. n + 16 = 44 15. MULTIPLE CHOICE The table shows the five nearest train stops to Main Street. Which equation will best help you find how much further Peach Court is from Main Street than City Center is from Main Street? (Lesson 3-3)

Train Stop City Center 14th Street Grand Hotel Stadium Peach Court

8. MULTIPLE CHOICE A paving brick is shown. Find the perimeter of 5 bricks. (Lesson 3-2) ÎX ÓX Ó

Distance to Main Street (miles) 4 6 7 12 17

A x - 17 = 4 ÓX Ó

B x + 17 = 4 C x - 4 = 17

xX Î

D x + 4 = 17

F 12x + 1 G 40x + 10 H 60x + 5 J 50x - 10 146 Chapter 3 Equations

16. MONEY Ricardo spends $3.50 for lunch each day. Write and solve an equation to find how long it takes him to spend $21 on lunch. (Lesson 3-4)

3-5

Solving Two-Step Equations

Main Idea • Solve two-step equations.

The equation 2x + 1 = 9, modeled below, can be solved with algebra tiles.

New Vocabulary two-step equation

X

X £

£

£

£

£

£

£

£

£

£

ÓX £

Step 1 Remove 1 tile from each side of the mat.

X

X £

£

£

£

£

£

£

£

£

£

ÓX £ £

£

Step 2 Separate the remaining tiles into two equal groups. X

£

£

£

£

£

£

£

£

X

ÓX

n

a. What property is shown by removing a tile from each side? b. What property is shown by separating the tiles into two groups? c. What is the solution of 2x + 1 = 9?

Solve Two-Step Equations A two-step equation contains two operations. In the equation 2x + 1 = 9, x is multiplied by 2 and then 1 is added. To solve two-step equations, use inverse operations to undo each operation in reverse order. Step 1 First, undo addition. 2x + 1 = 9 2x + 1 - 1 = 9 - 1 Subtract 1 from each side. 2x = 8 Step 2 Then, undo multiplication. 2x = 8 8 2x _ =_ 2

2


x=4 The solution is 4. Lesson 3-5 Solving Two-Step Equations

147

EXAMPLE

Solve Two-Step Equations

a. Solve 5x - 2 = 13. Check your solution. 5x - 2 = 13

Write the equation.

5x - 2 + 2 = 13 + 2 Undo subtraction. Add 2 to each side. 5x = 15

Simplify.

5x 15 _ =_

Undo multiplication. Divide each side by 5.

5

5

x=3 CHECK

Simplify.

5x - 2 = 13

Write the equation.

5(3) - 2 13

Check to see whether this statement is true.

13 = 13

The statement is true.

The solution is 3. n b. Solve 4 = _ + 11.

4 4 - 11 -7 6(-7) -42

6 n = _ + 11 Write the equation. 6 n =_ + 11 - 11 Undo addition. Subtract 11 from each side. 6 n =_ Simplify. 6 n =6 _ Undo division. Multiply each side by 6. 6 =n Check your solution.

( )

Solve each equation. n 1B. _ + 15 = 8

1A. 6x + 1 = 25

3

SALES Liana bought a DVD recorder. If she pays $80 now, her monthly payments will be $32. The total cost will be $400. Solve 80 + 32x = 400 to find how many months she will make payments. 80 + 32x = 400 Checking Your Solution Use estimation to determine whether your solution is reasonable: 80 + 30(10) = 380. Since $380 is close to $400, the solution is reasonable.

Write the equation.

80 - 80 + 32x = 400 - 80 Subtract 80 from each side. 32x = 320

Simplify.

32x 320 _ =_


32

32

x = 10

Simplify.

Therefore, Liana will make payments for 10 months.

2. COMPUTERS Salvatore purchased a computer for $550. He paid $105 initially, and then he will pay $20 per month until the computer is paid off. Solve 105 + 20x = 545 to find how many payments he will make. 148 Chapter 3 Equations

EXAMPLE

Equations with Negative Coefficients

Solve 4 - x = 10. 4 - x = 10

Write the equation.

4 - 1x = 10

Identity Property; x = 1x

4 + (-1x) = 10


-4 + 4 + (-1x) = -4 + 10 -1x = 6

Simplify.

6 -1x _ =_

Divide each side by -1.

-1

-1

x = -6

EXAMPLE

Check your solution.

Solve each equation. 3B. 35 - k = 21

3A. 19 = 9 - y

Mental Computation

Add -4 to each side.

Combine Like Terms Before Solving

Solve m - 5m + 3 = 47.

You use the Distributive Property to simplify 1m - 5m. 1m - 5m = (1 - 5)m = -4m You can also simplify the expression mentally.

m - 5m + 3 = 47 1m - 5m + 3 = 47 -4m + 3 = 47 -4m + 3 - 3 = 47 - 3

Write the equation. Identity Property; m = 1m Combine like terms, 1m and –5m. Subtract 3 from each side.

-4m = 44

Simplify.

-4m 44 _ =_

Divide each side by – 4.

-4

-4

m = -11

4A. 4 - 9d + 3d = 58

Simplify.

Solve each equation. 4B. 34 = 4m - 2 + 2m


Example 1 (p. 148)

Example 2 (p. 148)


ALGEBRA Solve each equation. Check your solution. 1. 2x - 7 = 9

2. -16 = 6a - 4

y 3. _ + 2 = 10 3

4. MEDICINE For Jillian’s cough, her doctor says that she should take eight tablets the first day and then four tablets each day until her prescription runs out. There are 36 tablets. Solve 8 + 4d = 36 to find how many more days she will take four tablets. ALGEBRA Solve each equation. Check your solution. 5. -7 - 8d = 17

6. 1 - 2k = -9

8. 2a - 8a = 24

9. -4 = 8y - 9y + 6


-n -5 7. 8 = _ 7

10. -6j + 4 + 3j = -23

Lesson 3-5 Solving Two-Step Equations

149

HOMEWORK

HELP


ALGEBRA Solve each equation. Check your solution. 11. 3x + 1 = 7

12. 5x - 4 = 11

13. 4h + 6 = 22

14. 8n + 3 = -5

15. 37 = 4d + 5

16. 9 = 15 + 2p

17. 2n - 5 = 21

18. 3j - 9 = 12

19. -1 = 2r - 7

20. 12 = 5k - 8 23. 3 + _t = 35 2

w 26. _ - 4 = -7 8

y 21. 10 = 6 + _ 7 p 24. 13 + _ = -4 3 c 27. 8 = _ + 15 -3

n 22. 14 = 6 + _ 5

k 25. _ - 10 = 3 5

b 28. -42 = _ +8 -4

29. POOLS There were 640 gallons of water in a 1600-gallon pool. Water is being pumped into the pool at a rate of 320 gallons per hour. Solve 1600 = 320t + 640 to find how many hours it will take to fill the pool. 30. PHONE CARDS A telephone calling card allows for 25¢ per minute plus a one-time service charge of 75¢. If the total cost of the card is $5, solve 25m + 75 = 500 to find the number of minutes you can use the card. ALGEBRA Solve each equation. Check your solution. 31. 8 - t = -25

32. 3 - y = 13

33. 8 = -5 - b

34. 10 = -9 - x

35. 2w - 4w = -10

36. 3x - 5x = 22

37. x + 4x + 6 = 31

38. 5r + 3r - 6 = 10

39. 1 - 3y + y = 5

40. 16 = w - 2w + 9

41. 23 = 4t - 7 - t

42. -4 = -a + 8 - 2a

ALGEBRA Find each number. 43. Five more than twice a number is 27. Solve 2n + 5 = 27. n 44. Ten less than the quotient of a number and 2 is 5. Solve _ - 10 = 5. 2

45. Three less than four times a number is -7. Solve 4n - 3 = -7. n 46. Six more than the quotient of a number and 6 is -3. Solve _ + 6 = -3. 6

EXTRA

PRACTICE


H.O.T. Problems

47. PERSONAL CARE In nine visits to the styling salon, Andre had spent $169 for haircuts. Of that amount, $16 was in tips. Write and solve an equation to find how much Andre pays for each haircut before the tip. 48. BUSINESS Jarret bought old bikes at an auction for $350. He fixed them and sold them for $50 each. He made a $6200 profit. Write and solve an equation to determine how many bikes he sold. 49. CHALLENGE The model represents the equation 6y + 1 = 3x + 1. What is the value of x? 50. OPEN ENDED Write a two-step equation that could be solved by using the Addition and Multiplication Properties of Equality. 51.


Writing in Math Use the information about solving equations on page 147 to explain how algebra tiles can show the properties of equality. Illustrate your reasoning by showing how to solve 2x + 3 = 7 using algebra tiles.

52. GRIDDABLE The cost to park at an art fair is a flat rate plus a per-hour fee. The graph shows the cost for parking up to 4 hours. If x represents the number of hours and y represents the total cost, what is the cost in dollars for 7 hours? 4

53. A local health club charges an initial fee of $45 for the first month and then a $32 fee each month after that. The table shows the cost to join the health club for up to 6 months. What is the cost to join the health club for 10 months?

y

1 Cost (dollars) 45 Months

3 2 1 O

2 77

3 4 5 6 109 141 173 205

A $215

C $333

B $320

D $450

1 2 3 4 5 6 7 8x

ALGEBRA Solve each equation. Check your solution. (Lessons 3-3 and 3-4) 54. 5y = 60

55. 14 = -2n

x 56. _ = -9

57. x - 4 = -6

58. -13 = y + 5

59. 18 = 20 + x

3

ALGEBRA Simplify each expression. (Lesson 3-1) 60. 4(x + 1)

61. -5(y + 3)

62. 3(k - 10)

63. -9(y - 4)

64. 7(a - 2)

65. -8(r - 5)

66. ANALYZE TABLES The table shows the average game attendance for three football teams in consecutive years. What was the total change in attendance from Year 1 to Year 2 for the Bobcats? (Lesson 2-3)

Team

Year 1

Year 2

Bobcats Cheetahs Wildcats

6234 7008 6873

5890 7162 6516

Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 1-7)

67. age and number of siblings

68. temperature and sales of sunscreen

PREREQUISITE SKILL Write an algebraic expression for each verbal expression. (Lesson 1-3)

69. two times a number less six 70. the quotient of a number and 15 71. the difference between twice a number and 8 72. three times a number increased by 10 73. the sum of 2x, 7x, and 4

Lesson 3-5 Solving Two-Step Equations

151

Translating Verbal Problems into Equations An important skill in algebra is translating verbal problems into equations. Consider the following situation. Jennifer is 6 years older than Akira. The sum of their ages is 20. You can explore this problem situation by asking and answering questions. Who is older? Jennifer How many years older? 6 If Akira is x years old, how old is Jennifer? x + 6 You can summarize this information in an equation.

Words

Jennifer is 6 years older than Akira. The sum of their ages is 20.

Variable

Let x = Akira’s age. Let x + 6 = Jennifer’s age.

Equation

x + ( x + 6) = 20

Exercises For each verbal problem, answer the related questions. 1. Lucas is 5 inches taller than Tamika, and the sum of their heights is 137 inches. a. Who is taller? b. How many inches taller? c . If x represents Tamika’s height, how tall is Lucas? d. What expression represents the sum of their heights? e. What equation represents the sentence the sum of their heights is 137? 2. There are five times as many students as teachers on the field trip, and the sum of students and teachers is 132. a. Are there more students or teachers? b. How many times more? c . If x represents the number of teachers, how many students are there? d. What expression represents the sum of students and teachers? e. What equation represents the sum of students and teachers is 132? 152 Chapter 3 Equations

3-6

Writing Two-Step Equations

Main Ideas Logan collected pledges for the charity walk-a-thon. He is receiving total contributions of $68 plus $20 for every mile that he walks. The table shows how to find the total amount that Logan could raise.

• Write verbal sentences as two-step equations. • Solve verbal problems by writing and solving two-step equations.

Number of Miles

Total Amount Raised

0

20(0) + 68 = $68

4

20(4) + 68 = $148

6

20(6) + 68 = $188

10

20(10) + 68 = $268

16

20(16) + 68 = $388

a. Write an expression that represents the amount Logan can raise when he walks m miles.

b. Suppose Logan raised $308. Write and solve an equation to find the number of miles Logan walked. c. Why is your equation considered to be a two-step equation?

Write Two-Step Equations In Chapter 1, you learned how to write Review Vocabulary Expression any combination of numbers and operations; Example: x - 3 (Lesson 1-2)

verbal phrases as expressions. Phrase

the sum of 20 times some number and 68

Expression

20n

+ 68

An equation is a statement that two expressions are equal. The expressions are joined with an equals sign. You can write verbal sentences as equations. Sentence

The sum of 20 times some number and 68 is 308.

Equation

EXAMPLE

20n + 68

= 308

Translate Sentences into Equations

Translate each sentence into an equation. Sentence a. Six more than twice a number is -20.

Equation 2n + 6 = -20

b. Eighteen is 6 less than four times a number.

18 = 4n - 6

c. The quotient of a number and 5, increased by 8, is equal to 14.

n _ + 8 = 14 5

1A. Four more than three times a number is -26. 1B. Twenty-four is 6 less than twice a number. 1C. The quotient of a number and 7, increased by 6, is equal to 12. Extra Examples at pre-alg.com

Lesson 3-6 Writing Two-Step Equations

153

EXAMPLE

Translate and Solve an Equation

Seven more than three times a number is 31. Find the number. Let n = the number.

Equations Look for the words is, equals, or is equal to when you translate sentences into equations.

3n + 7 = 31

Write the equation.

3n + 7 - 7 = 31 - 7 Subtract 7 from each side. 3n = 24 n=8

Simplify. Mentally divide each side by 3.

Therefore, the number is 8.

2. Translate the following sentence into an equation. Then find the number. Eight less than three times a number is -23.

Two-Step Verbal Problems In some real-world situations you start with a given amount and then increase it at a certain rate. These situations can be represented by two-step equations.

CELL PHONES Suppose you are saving money to buy a cell phone that costs $100. You have already saved $60 and plan to save $5 each week. How many weeks will you need to save? Explore

You have already saved $60. You plan to save $5 each week until you have $100.

Plan

Organize the data for the first few weeks in a table. Notice the pattern. Write an equation to represent the situation. Let x = the number of weeks. 5x + 60 = 100

Solve

5x + 60 = 100 5x + 60 - 60 = 100 - 60 5x = 40 x=8

Week

Amount

0 1 2 3

5(0) + 60 = 60 5(1) + 60 = 65 5(2) + 60 = 70 5(3) + 60 = 75

Write the equation. Subtract 60 from each side. Simplify. Mentally divide each side by 5.

You need to save $5 each week for 8 weeks. Real-World Link About 100 million cell phones in the United States are retired each year. Source: Inform Inc.

Check

If you save $5 each week for 8 weeks, you’ll have an additional $40. The answer appears to be reasonable.

3. SHOPPING Jasmine bought 6 CDs, all at the same price. The tax on her purchase was $7, and the total was $73. What was the price of each CD? Personal Tutor at pre-alg.com

154 Chapter 3 Equations Jim West/The Image Works

OLYMPICS In the 2004 Summer Olympics, the United States won 11 more medals than Russia. Together they won 195 medals. How many medals did the United States win? Let x = number of medals won by Russia. Then x + 11 = number of medals won by the United States.

Alternative Method Let x = number of U.S. medals. Then let x - 11 = number of Russian medals. x + (x - 11) = 195

x + (x + 11) = 195

Write the equation.

(x + x) + 11 = 195

Associative Property

2x + 11 = 195

Combine like terms.

2x + 11 - 11 = 195 - 11

Subtract 11 from each side.

x = 103

2x = 184

Simplify.

In this case, x is the number of U.S. medals, 103.

184 2x _ =_


2

2

x = 92

Simplify.

Since x represents the number of medals won by Russia, Russia won 92 medals. The United States won 92 + 11 or 103 medals.

4. CAR WASH During the spring car wash, the Activities Club washed 14 fewer cars than during the summer car wash. They washed a total of 96 cars during both car washes. How many cars did they wash during the spring?

Examples 1, 2 (pp. 153–154)

Translate each sentence into an equation. Then find each number. 1. Three more than four times a number is 23. 2. Four less than twice a number is -2. 3. The quotient of a number and 3, less 8, is 16. Solve each problem by writing and solving an equation.

Example 3 (p. 154)

Example 4 (p. 155)

HOMEWORK

HELP

For See Exercises Examples 6–11 1, 2 12, 13 3 14, 15 4

4. TEMPERATURE Suppose the current temperature is 17°F. It is expected to rise 3°F each hour for the next several hours. In how many hours will the temperature be 32°F? 5. AGES Lawana is five years older than her brother Cole. The sum of their ages is 37. How old is Lawana?

Translate each sentence into an equation. Then find each number. 6. Seven more than twice a number is 17. 7. Twenty more than three times a number is -4. 8. Four less than three times a number is 20. 9. Eight less than ten times a number is 82. 10. Ten more than the quotient of a number and -2 is three. 11. The quotient of a number and -4, less 8, is -42. Lesson 3-6 Writing Two-Step Equations

155

For Exercises 12–15, solve each problem by writing and solving an equation. 12. WILDLIFE Your friend bought 3 bags of wild birdseed and an $18 bird feeder. Each bag of birdseed costs the same amount. If your friend spent $45, find the cost of one bag of birdseed. 13. TEMPERATURE The temperature is 8°F. It is expected to fall 5° each hour for the next several hours. In how many hours will the temperature be -7°F? 14. POPULATION By 2020, California is expected to have 2 million more senior citizens than Florida, and the sum of the number of senior citizens in the two states is expected to be 12 million. Find the expected senior citizen population of Florida in 2020.

Real-World Career Meteorologist A meteorologist uses math to forecast the weather and analyzes how weather affects air pollution and agriculture.

15. BUILDINGS In New York City, the Chrysler Building is 320 feet taller than the Times Square Tower. The combined height of both buildings is 1772 feet. How tall is the Times Square Tower?

Building

Height (ft)

Citigroup Center

915

Chrysler Building

?

Empire State Building

1250

Times Square Tower

?

Woolworth Building

792

Source: emporis.com

Translate each sentence into an equation. Then find each number. 16. If 5 is decreased by 3 times a number, the result is -4.

For more information, go to pre-alg.com.

17. If 17 is decreased by twice a number, the result is 5. 18. Three times a number plus twice the number plus 1 is - 4. 19. Four times a number plus five more than three times the number is 47.

EXTRA

PRACTICE


H.O.T. Problems

20. POPULATIONS Georgia’s Native-American population is 10,000 greater than Mississippi’s. Mississippi’s Native-American population is 106,000 less than Texas’. If the total population of all three is 149,000, find each state’s Native-American population. 21. CONSTRUCTION Henry is building a front door. The height of the door is 1 foot more than twice its width. If the door is 7 feet high, what is its width? 22. OPEN ENDED Write a two-step equation that has 6 as the solution. Write the equation using both words and symbols. 23. FIND THE ERROR Alicia and Ben are translating the following sentence into an equation: Three less than two times a number is 15. Who is correct? Explain your reasoning. Alicia 3 - 2x = 15

Ben 2x - 3 = 15

24. CHALLENGE If you begin with an even integer and count by two, you are counting consecutive even integers. Write and solve an equation to find two consecutive even integers whose sum is 50. 156 Chapter 3 Equations Dwayne Newton/PhotoEdit

25. NUMBER SENSE The table shows the expected population age 65 or older for certain states in 2030. Use the data to write a problem that can be solved by using a two-step equation. 26.

Population (age 65 or older) Number State (millions) CA 8.3

Writing in Math Explain how two-step equations are used to solve real-world problems. Formulate a problem situation that starts with a given amount and then increases.

FL

7.8

TX

5.2

NY

3.9

Source: U.S. Census Bureau

27. An electrician charges $35 for a house call and $80 per hour for each hour worked. If the total charge was $915 to wire a new house, which equation would you use to find the number of hours n that the electrician worked?

28. You and your friend spent a total of $15 for lunch. Your friend’s lunch cost $3 more than yours did. How much did you spend for lunch? F $6 G $7

A 35n + 2n(80) = 915

H $8

B 80 + 35n = 915

J $9

C 35 + (80 - n) = 915 D 35 + 80n = 915

ALGEBRA Solve each equation. Check your solution. (Lessons 3-3, 3-4, and 3-5) 29. 6 - 2x = 10

30. -4x = -16

31. y - 7 = -3

32. 7y + 3 = -11

33. CONCERTS A concert ticket costs t dollars, a hamburger costs h dollars, and soda costs s dollars. Write an expression that represents the total cost of a ticket, hamburger, and soda for n people. (Lesson 3-1) y

Name the ordered pair for each point graphed on the coordinate plane at the right. (Lesson 2-6) 34. T

35. C

36. R

T

R x

O

37. P

38. FOOD The SubShop had 36, 45, 41, and 38 customers during the lunch hour the last four days. Find the mean of the number of customers per day. (Lesson 2-5)

P

C

ALGEBRA Evaluate each expression if x = -12, y = 4, and z = -1. (Lesson 2-2) 39. ⎪x⎥ - 7

40.

⎪x⎥

+ ⎪y⎥

41. ⎪z⎥ - ⎪x⎥

42. ⎪y⎥ - ⎪x⎥ + ⎪z⎥

PREREQUISITE SKILL Find the next term in the pattern. (Lesson 1-1) 43. 5, 9, 13, 17, … 44. 326, 344, 362, 380, … 45. 20, 22, 26, 32, …

Lesson 3-6 Writing Two-Step Equations

157

3-7

Sequences and Equations

Main Ideas • Describe sequences using words and symbols. • Find terms of arithmetic sequences.

The table shows the distance a car moves during the time it takes to apply the brakes and while braking.

Speed (mph)

New Vocabulary

a. What is the braking distance for a car going 70 mph?

sequence arithmetic sequence term common difference

b. What is the difference in reaction distances for every 10-mph increase in speed?

Reaction Braking Distance (ft) Distance (ft)

20

20

20

30

30

45

40

40

80

50

50

125

60

60

180

c. Describe the braking distance as speed increases.

Describing Sequences A sequence is an ordered list of numbers. An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same. So, you can find the next term in the sequence by adding the same number to the previous term. Each number is called a term of the sequence.

EXAMPLE

20, +10

30,

40, +10

50,

+10

60, … The difference is called the common difference.

+10

Describe an Arithmetic Sequence

Describe the sequence 4, 8, 12, 16, … using words and symbols. +1

Substitute numbers from the table to check whether your equation is true.


+1

Term Number (n)

1

2

3

4

Term (t )

4

8

12

16

+4

Check Your Answers

+1

+4

+4

The difference of the term numbers is 1. The terms have a common difference of 4. Also, a term is 4 times the term number. The equation t = 4n describes the sequence.

Describe each sequence using words and symbols. 1A. 10, 11, 12, 13, … 1B. 5, 10, 15, 20, …

Finding Terms Once you have described a sequence with a rule or equation, you can use the rule to extend the pattern and find other terms.

EXAMPLE

Find a Term in an Arithmetic Sequence

Find the 15th term of 7, 10, 13, 16, … . First write an equation that describes the sequence. +1

+1

+1

Term Number (n)

1

2

3

4

Term (t )

7

10

13

16

+3

+3

The difference of the term numbers is 1. The terms have a common difference of 3.

+3

The common difference is 3 times the difference in the term numbers. This suggests that t = 3n. However, you need to add 4 to get the exact value of t. Thus, t = 3n + 4. CHECK If n = 2, then t = 3(2) + 4 or 10. If n = 4, then t = 3(4) + 4 or 16. To find the 15th term in the sequence, let n = 15 and solve for t. t = 3n + 4 = 3(15) + 4 or 49

Write the equation.

So, the 15th term is 49.

2. Find the 20th term of 5, 8, 11, 14, … . Personal Tutor at pre-alg.com

Real-World Link The restaurant industry employs about 12.2 million people, making it the nation’s largest employer outside of government. Source: restaurant.org

RESTAURANTS The diagram shows the number of square tables needed to seat 4, 6, or 8 people at a restaurant. How many tables are needed to seat 16 people? Make a table to organize your sequence and find a rule. Number of Tables (t)

1

2

3

The difference of the term numbers is 1.

Number of People (p)

4

6

8

The terms have a common difference of 2.

The pattern in the table shows the equation p = 2t + 2. If p = 2t + 2 or 16 = 2t + 2, then t = 7. So, seven tables are needed to seat a party of 16.

3. RESTAURANTS Suppose the tables are shaped like hexagons. Find how many tables are needed for a group of 22 diners.

Extra Examples at pre-alg.com Don Tremain/Getty Images

Lesson 3-7 Sequences and Equations

159

Example 1 (p. 158)

Example 2 (p. 159)

Example 3 (p. 159)

Describe each sequence using words and symbols. 1. 2, 3, 4, 5, …

2. 6, 7, 8, 9, …

3. 3, 6, 9, 12, …

4. 7, 14, 21, 28, …

Write an equation that describes each sequence. Then find the indicated term. 5. 10, 11, 12, 13, …; 10th term

6. 6, 12, 18, 24, …; 11th term

7. 2, 5, 8, 11, …; 20th term

8. 2, 6, 10, 14, …; 14th term

9. GEOMETRY Suppose each side of a square has a length of 1 foot. Determine which figure will have a perimeter of 60 feet. &IGURE

HOMEWORK

HELP

For See Exercises Examples 10–21 1 22–29 2 30, 31 3

&IGURE

&IGURE

Describe each sequence using words and symbols. 10. 3, 4, 5, 6, …

11. 8, 9, 10, 11, …

12. 14, 15, 16, 17, …

13. 15, 16, 17, 18, …

14. 2, 4, 6, 8, …

15. 8, 16, 24, 32, …

16. 12, 24, 36, 48, …

17. 20, 40, 60, 80, …

18. 3, 5, 7, 9, …

19. 4, 6, 8, 10, …

20. 1, 4, 7, 10, …

21. 3, 7, 11, 15, …

Write an equation that describes each sequence. Then find the indicated term. 22. 16, 17, 18, 19, …; 23rd term

23. 14, 15, 16, 17, …; 16th term

24. 4, 8, 12, 16, …; 13th term

25. 11, 22, 33, 44, …; 25th term

26. 7, 10, 13, 16, …; 20th term

27. 7, 9, 11, 13, …; 33rd term

28. 1, 5, 9, 13, …; 89th term

29. 3, 8, 13, 18, …; 70th term

30. GEOMETRY Study the pattern. Which figure will have 40 squares? }ÕÀi £

}ÕÀi Ó

}ÕÀi Î

31. CONSTRUCTION A building frame consists of beams in the form of triangles. The frame of a new office building will use 27 beams. Use the pattern below to find the number of triangles that will be formed for the frame.

Î EXTRA

PRACTIICE


x

Ç

THEATERS One section of a movie theater has 26 seats in the first row, 35 seats in the second row, 44 seats in the third row, and so on. 32. How many seats are in the eighth row? 33. If there are 10 rows of seats, how many seats are in the section?


ANALYZE GRAPHS For Exercises 34 and 35, use the graph. 34. Write an equation for the points (x, y) graphed at the right. (Hint: Make a table of ordered pairs.)

Y

35. Find x when y is 101.

H.O.T. Problems

36. OPEN ENDED Write an arithmetic sequence whose common difference is -8.

X

"

37. CHALLENGE Use an arithmetic sequence to find the number of multiples of 6 between 41 and 523. 38.

Writing in Math

Explain how sequences can be used to make

predictions.

39. The expression 1 + 2n(n + 2) describes a pattern of numbers. If n represents a number’s position in the sequence, which pattern does the expression describe? A 7, 17, 31, 49, 71, . . . B 4, 7, 9, 17, 27, . . . C 7, 17, 27, 31, 49, . . .

40. Use the pattern Side Length in the table to 1 find the 2 equation that 3 shows the 4 relationship between the side 5 length s and perimeter p of a pentagon.

D 7, 9, 17, 27, 31, . . .

Perimeter

F p=5+s

H s = 5p + 5

G p = 5s

J s=5+p

5 10 15 20 25

ALGEBRA Translate each sentence into an equation. (Lesson 3-6) 41. Five more than three times a number is 20. 42. Thirty-six is 8 less than twice a number. 43. The quotient of a number and -10, less 3, is -63 ALGEBRA Solve each equation. Check your solution. (Lesson 3-5) 44. 6 - 3x = 21

45. 4y - 3 = 25

46. -3 + 2 z = -19

47. SOCCER A ticket to a soccer game is $12, a team pennant is $7, and a T-shirt is $15. Write two equivalent expressions for the total cost of a group outing for 10 people if each person buys a ticket, a pennant, and a T-shirt. Then find the cost. (Lesson 3-1) 48. WEATHER On Saturday, the temperature fell 10 degrees in 2 hours. Find the integer that expresses the temperature change per hour. (Lesson 2-5) PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 3-4) 49. 2x = -8

50. 15s = 75

51. 108 = 18x

52. 25z = 175

Lesson 3-7 Sequences and Equations

161

3-8

Using Formulas

Main Ideas • Solve problems by using formulas. • Solve problems involving the perimeters and areas of rectangles.

New Vocabulary formula perimeter area

Reading Math Formulas A formula is a concise way to describe a relationship among quantities.

The top recorded speed of a mallard duck in level flight is 65 miles per hour. You can make a table to record the distances that a mallard could fly at that rate.

Speed (mph)

Time (h)

Distance (mi)

65

1

65

b. What disadvantage is there in showing the data in a table?

65

2

130

65

3

195

c. Describe an easier way to summarize the relationship between the speed, time, and distance.

65

t

?

a. Write an expression for the distance traveled by a duck in t hours.

Formulas A formula is an equation that shows a relationship among certain quantities. A formula usually contains two or more variables. One of the most commonly used formulas is d = rt, which shows the relationship between distance d, rate (or speed) r, and time t.

SCIENCE What is the rate in miles per hour of a dolphin that travels 120 miles in 4 hours? Method 1 Substitute first. d = rt 120 = r · 4 120 = r · 4 4 4

30 = r

Method 2 Solve for r first.

Write the formula.

d = rt

Replace d with 120 and t with 4.

t = t

Divide each side by t.


d =r t

Simplify.

Simplify.

d

rt

120 = r 4

30 = r

Write the formula.

Replace d with 120 and t with 4. Simplify.

The dolphin travels at a rate of 30 miles per hour.

1. SCIENCE How long does it take a zebra to travel 160 miles at a speed of 40 miles per hour? 162 Chapter 3 Equations Getty Images

Perimeter and Area Formulas are commonly used in measurement. The distance around a geometric figure is called the perimeter. One method of finding the perimeter P of a rectangle is to add the measures of the four sides. Perimeter of a Rectangle Words

The perimeter of a rectangle is twice the sum of the length and width.

Symbols P = + + w + w P = 2 + 2w or P = 2( + w)

EXAMPLE Common Misconception Although the length of a rectangle is usually greater than the width, it does not matter which side you choose to be the length.

ᐉ

Model w

Find Perimeters and Lengths of Rectangles

a. Find the perimeter of the rectangle. P = 2( + w)

11 in.

Write the formula.

5 in.

= 2(11 + 5) Replace with 11 and w with 5. = 2(16)

Add 11 and 5.

= 32

Simplify. The perimeter is 32 inches.

b. The perimeter of a rectangle is 28 meters. Its width is 8 meters. Find the length. P = 2 + 2w

Write the formula.

28 = 2 + 2(8)

Replace P with 28 and w with 8.

28 = 2 + 16

Simplify.

28 - 16 = 2 + 16 - 16 12 = 2 6=

Subtract 16 from each side. Simplify. Mentally divide each side by 2.

The length is 6 meters.

2A. Find the perimeter of a rectangle with length 15 meters and width 10 meters. 2B. The perimeter of a rectangle is 26 yards. Its length is 8 yards. Find the width. The measure of the surface enclosed by a figure is its area. Area of a Rectangle Words

The area of a rectangle is the product of the length and width.

Symbols A = w

ᐉ

Model w


Lesson 3-8 Using Formulas

163

EXAMPLE

Find Areas and Lengths of Rectangles

a. Find the area of a rectangle with length 15 meters and width 7 meters. 15 m A = w

Write the formula.

= 15 · 7

Replace with 15 and w with 7.

= 105

Simplify.

7m

The area is 105 square meters. b. The area of a rectangle is 45 square feet. Its length is 9 feet. Find its width. Method 1 Substitute, then solve for the variable.

Method 2 Solve, then substitute.

A = w Write the formula. 45 = 9w

Replace A with 45 and with 9.

5=w

Mentally divide each side by 9.

A = w

Write the formula.

w A _ =_

Divide each side by .

A _ =w

Simplify.

45 _ =w

Replace A with 45 and with 9.

9

5=w

Simplify.

The width is 5 feet.

3A. Find the area of a rectangle with a length of 14 inches and a width of 12 inches. 3B. The area of a rectangle is 198 square meters. Its width is 11 meters. Find its length. Personal Tutor at pre-alg.com

Example 1 (p. 162)

Examples 2, 3 (pp. 163–164)

1. ANIMALS How long would it take a bottlenose dolphin to swim 168 miles at 12 miles per hour? GEOMETRY Find the perimeter and area of each rectangle. 2.

3.

8 ft 3 ft

15 km 2 km

4. a rectangle with length 15 feet and width 6 feet GEOMETRY Find the missing dimension in each rectangle. 5.

6.

12 in.

8m

w Area 96 m Perimeter 32 in.


ᐉ

2

HOMEWORK

HELP

For See Exercises Examples 7–8 1 9–25 2, 3

7. TRAVEL Find the distance traveled by driving at 55 miles per hour for 3 hours. 8. BALLOONING What is the rate, in miles per hour, of a balloon that travels 60 miles in 4 hours? GEOMETRY Find the perimeter and area of each rectangle. 9.

10.

3 mi

11.

9 cm

12 ft 5 ft

2 mi

12.

18 cm

13.

18 in.

14. 12 m 6m

50 in. 12 m

17 m

15. a rectangle that is 38 meters long and 10 meters wide 16. a square that is 5 meters on each side GEOMETRY Find the missing dimension in each rectangle. 17.

18.

15 cm Area 270 cm2

19.

w

Area 176 yd2

11 m 16 yd Perimeter 70 m

ᐉ

20.

Freddy Adu became the youngest professional player in modern American team sports history when he joined D.C. United at 14 years of age. Soccer is played on a rectangular field that is usually 120 yards long and 75 yards wide. Source: sportsillustrated. cnn.com

21.

7m

Real-World Link

ᐉ

w

w Area 154 in2

22.

12 ft

Area 468 ft2 ᐉ 14 in.

Perimeter 24 m

23. SOCCER Find the perimeter and area of the soccer field described at the left. 24. COMMUNITY SERVICE Each participant in a community garden is allotted a rectangular plot that measures 18 feet by 45 feet. How much fencing is needed to enclose each plot? 25. LANDSCAPING Jordan is paving a rectangular patio with bricks. If the patio contains a total of 308 bricks and there are 22 bricks running along the length of the patio, how many bricks run along the width of the patio? For Exercises 26 and 27, translate each sentence into a formula. 26. PROFITS The profit made during a year p is equal to sales s minus costs c. 27. GEOMETRY In a circle, the diameter d is twice the length of the radius r. Lesson 3-8 Using Formulas

Thanassis Stavrakis/AP/Wide World Photos

165

28. RUNNING The stride rate r of a runner is the number of strides n (or long steps) that he or she takes divided by the amount of time

Runner

Number of Strides

Time(s)

A B

20 30

5 10

n . The best runners usually have the t, or r = _ t

greatest stride rate. Which runner has the greater stride rate?

Using a formula can help you find the cost of a vacation. Visit pre-alg.com to continue work on your project.

GEOMETRY Draw and label the dimensions of each rectangle whose perimeter and area are given. 29. P = 14 ft, A = 12 ft2

30. P = 16 m, A = 12 m2

31. P = 16 cm, A = 16 cm2

LANDSCAPING For Exercises 32 and 33, use the figure at the right.

80 ft

32. What is the area of the lawn?

Lawn

33. Suppose your family wants to fertilize the lawn that is shown. If one bag of fertilizer covers 2500 square feet, how many bags of fertilizer should you buy?

House 28 ft x 50 ft

75 ft

Driveway 15 ft x 20 ft

34. TRAVEL An airplane flying at 500 miles per hour leaves Minneapolis. One-half hour later, a second airplane leaves Minneapolis flying in the same direction at a rate of 600 miles per hour. How long will it take the second airplane to overtake the first? GEOMETRY Find the area of each rectangle. 35.

Y

!

X

"

$

36.

"

Y %

X

"

# (

EXTRA

PRACTIICE


H.O.T. Problems

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BICYCLING For Exercises 37 and 38, use the following information. American Lance Armstrong won the 2005 Tour de France, completing the 2102-mile race in 83 hours, 36 minutes, 2 seconds. 37. Estimate Armstrong’s average rate in miles per hour for the race. 38. Armstrong also won the 2003 Tour de France. He completed the 2125-mile race in 80 hours, 2 minutes, 8 seconds. Without calculating, determine which race was completed with a faster average speed. Explain. 39. OPEN ENDED Draw and label a rectangle that has a perimeter of 18 inches. 40. CHALLENGE Is it sometimes, always, or never true that the perimeter of a rectangle is numerically greater than its area? Give an example to justify your answer.


41. REASONING A rectangle has width w. Its length is one less than twice its width. Write an expression in simplest form for its perimeter. 42.

Writing in Math Explain why formulas are important in math and science. Include an example of a formula from math or science that you have used and an explanation of how you used the formula.

43. The formula d = rt can be rewritten as dt = r. How is the rate affected if the time t increases and the distance d remains the same?

44. The area of each square in the figure is 16 square units. Find the perimeter.

A It increases.

F 16 units

B It decreases.

G 32 units

C It remains the same.

H 48 units

D There is not enough information.

J 64 units

Write an equation that describes each sequence. Then find the indicated term. (Lesson 3-7) 45. 11, 12, 13, 14, …; 60th term

46. 4, 11, 18, 25, …; 100th term

47. Eight more than five times a number is 78. Find the number. (Lesson 3-6) LIGHT BULBS The table shows the average life of an incandescent bulb for selected years. (Lesson 1-6)

scen Light Bulbs Incandescent 1200 Hours

49. State the domain and the range of the relation.

1500

1500

48. Write a set of ordered pairs for the data.

1000

900

600

600 300 0

14

1870

1881 1910 Years

2000

Math and Technology I Need a Vacation! It’s time to complete your Internet project. Use the information and data you have

gathered about the costs of lodging, transportation, and entertainment for each of the vacations. Prepare a brochure or Web page to present your project. Be sure to include graphs and/or tables in the presentation. Cross-Curricular Project at pre-alg.com

Lesson 3-8 Using Formulas

167

EXTEND

Spreadsheet Lab

3-8

Perimeter and Area

A spreadsheet allows you to use formulas to investigate problems. When you change a numerical value in a cell, the spreadsheet recalculates the formula and automatically updates the results.

EXAMPLE Suppose a gardener wants to enclose a rectangular garden using part of a wall as one side and 20 feet of fencing for the other three sides. What are the dimensions of the largest garden she can enclose? If s represents the length of each side attached to the wall, 20 - 2s represents the length of the side opposite the wall. These values are listed in column B. The areas are listed in column C.

'ARDEN DIMENSIONSXLS "

!

,ENGTH OF 3IDE /PPOSITE 7ALL

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The greatest possible area is 50 square feet. It occurs when the length of each side attached to the wall is 5 feet, and the length of the side opposite the wall is 10 feet.

ANALYZE THE RESULTS 1. What is the area if the length of the side attached to the wall is 10 feet? 11 feet? 2. Are the answers to Exercise 1 reasonable? Justify your reasoning. 3. Suppose you want to find the greatest area that you can enclose with 30 feet of fencing. Which cell should you modify to solve this problem? 4. Use a spreadsheet to find the dimensions of the greatest area you can enclose with 40 feet, 50 feet, and 60 feet of fencing. 5. MAKE A CONJECTURE Use any pattern you may have observed in your answers to Exercise 4 to find the dimensions of the greatest area you can enclose with 100 feet of fencing. Explain. 168 Chapter 3 Equations

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Key Concepts Distributive Property

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(Lesson 3-1)

• For any numbers a, b, and c, a(b + c) = ab + ac.

Solving Equations

formula (p. 162) inverse operations (p. 136) like terms (p. 129) perimeter (p. 163) sequence (p. 158) simplest form (p. 130) term (p. 129, 158) two-step equation (p. 147)

(Lessons 3-3 through 3-6)

• When you add or subtract the same number from each side of an equation, the two sides remain equal. • When you multiply or divide each side of an equation by the same nonzero number, the two sides remain equal. • To solve a two-step equation, undo operations in reverse order.

Sequences

area (p. 163) arithmetic sequence (p. 158) coefficient (p. 129) common difference (p. 158) constant (p. 129) Distributive Property (p. 124) equivalent equations (p. 136) equivalent expression (p. 124)

Vocabulary Check Complete each sentence with the correct term. Choose from the list above. 1. Terms that contain the same variables are called _________. 2. The ________ of a geometric figure is the measure of the distance around it. 3. In the term 4b, 4 is the ________ of the expression.

(Lesson 3-7)

• An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same.

4. The equations x + 3 = 8 and x = 5 are ________ because they have the same solution.

• A sequence can be described by a rule or equation that can be used to extend the pattern or find other terms in the pattern.

5. Addition and subtraction are ________ because they “undo” each other.

Perimeter and Area Formulas

6. In the expression 10x + 6, 6 is the ________ term.

(Lesson 3-8)

• The formula for the perimeter of a rectangle is P = 2( + w). • The formula for the area of a rectangle is A = w.


7. The measure of the surface enclosed by a geometric figure is its ________. 8. A(n) ________ is an equation that shows a relationship among certain quantities.


169

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The Distributive Property

(pp. 124–128)

Use the Distributive Property to write each expression as an equivalent algebraic expression. 9. 3(h + 6) 10. 7(x + 2) 11. -5(k + 1)

12. -2(a + 8)

13. (b - 4)(-2)

14. (y - 3)(-6)

Example 1 Use the Distributive Property to write 2(t - 3) as an equivalent algebraic expression. 2(t - 3) = 2[t + (-3)] Rewrite t - 3 as t + (-3).

15. BOWLING At the Bowling Palace, shoe rental is $3.00 and each game is $2.50. Write two equivalent expressions for the cost of a group of 3 people to rent shoes and play 2 games.

3–2



= 2t + (-6)

Simplify.

= 2t - 6


(pp. 129–133)

Simplify each expression. 16. 4a + 5a 17. 3y + 7 + y

Example 2 Simplify 9x + 3 - 7x. 9x + 3 - 7x

18. x - 10 - 3x + 9

19. 4(m - 4) + 2

= 9x + 3 + (-7x) Definition of subtraction

20. 6w + 2(w + 9)

21. 8(n - 1) - 10n

= 9x + (-7x) + 3 Commutative Property

22. BASKETBALL Karen made 5 less than 4 times the number of free throws that Kimi made. Write an expression in simplest form that represents the total number of free throws made.

3–3

= 2t + 2(-3)

Solving Equations by Adding or Subtracting Solve each equation. Check your solution. 23. t + 5 = 8 24. 12 = b + 4 25. z - 10 = -6

26. a + 12 = -16

27. k - 1 = 4

28. -7 = n - 6

29. REPORTS Sonia needs to add 13 more pages to complete an assignment that is supposed to be 37 pages long. Write and solve an equation to find how many pages she has already completed. 170 Chapter 3 Equations

= [9 + (-7)]x + 3 Distributive Property = 2x + 3

Simplify.

(pp. 136–140)

Example 3 Solve x + 3 = 7. x+3=7

Write the equation.

x + 3 - 3 = 7 - 3 Subtract 3 from each side. x=4

Simplify.

Example 4 Solve y - 5 = -2. y - 5 = -2

Write the equation.

y - 5 + 5 = -2 + 5 Add 5 to each side. y=3

Simplify.



3–4

Solving Equations by Multiplying or Dividing Solve each equation. Check your solution. 30. 6n = 48 31. -3x = 30 r = -2 32. -5

d 33. 22 =

-3

34. FASHION Rosa is making scarves for her friends. Each scarf requires 48 inches of material. Write and solve an equation to find how many scarves Rosa can make if she has 336 inches of material.

(pp. 141–145)

Example 5 Solve -5x = -30. -5x = -30

Write the equation.

-5x = -30 -5 -5


x=6

Simplify.

a = 3. Example 6 Solve -8 a =3 Write the equation. -8

a = -8(3) Multiply each side by -8. -8 -8

a = -24

3–5

Solving Two-Step Equations

(pp. 147–151)

Solve each equation. Check your solution. 35. 6 + 2y = 8 36. 3n - 5 = -17 37. _t + 4 = 2 3

38. _c - 3 = 2 9

Writing Two-Step Equations

Example 7 Solve 6k - 4 = 14. 6k - 4 = 14

Write the equation.

6k - 4 + 4 = 14 + 4

Add 4 to each side.

6k = 18

39. BOOKS Dion’s favorite book is 35 pages longer than Eva’s favorite book. The number of pages in both books is 271. Solve x + x + 35 = 271 to find the number of pages in Dion’s book.

3–6

Simplify.

k=3

Simplify. Divide each side by 6.

(pp. 154–157)

Translate each sentence into an equation. Then find each number. 40. Three more than twice a number is 53. 41. Six less than the quotient of a number and 4 is -3. 42. MONEY Suppose you are saving money to buy a digital video camera that costs $340. You have already saved $120 and plan to save $20 each week. How many weeks will you need to save?

Example 8 Translate the sentence into an equation. Then find the number. Seven less than three times a number is -22. 3n - 7 = -22

Write the equation.

3n - 7 + 7 = -22 + 7 Add 7 to each side. 3n = -15 n = -5



171

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3 3–7


Sequences and Equations

(pp. 158–161)

Describe each sequence using words and symbols. 43. 5, 6, 7, 8, … 44. 14, 15, 16, 17, … 45. 6, 12, 18, 24, …

46. 10, 20, 30, 40, …

Write an equation that describes each sequence. Then find the indicated term. 47. 8, 9, 10, 11, …; 19th term 48. 6, 10, 14, 18, …; 47th term 49. 7, 14, 21, 28, …; 70th term 50. GEOMETRY Which figure in the pattern below will have 99 squares?

3–8

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Using Formulas

(pp. 162–167)

Example 9 Find the 35th term of 9, 18, 27, 36, … . Term Number (n)

1

2

3

4

Term (t)

9

18

27

36

The common difference is 9. Each term is 9 times the term number. So, t = 9n. t = 9n

Write the equation.

t = 9(35) Replace n with 35. t = 315

Simplify.

The 35th term of the sequence is 315.

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Find the perimeter and area of each rectangle. 51.

Example 10 Find the perimeter and area of a 14-meter by 6-meter rectangle. P = 2( + w) Formula for perimeter = 2(14 + 6) Replace with 14 and w with 6. = 40 m

°

°

52. a rectangle with length 8 feet and width 9 feet 53. a square with sides that are 2.5 yards 54. SWIMMING Karl’s rectangular pool is 15 feet by 9.5 feet. What are the perimeter and area of the pool? 55. TYPING Toya typed a 1140-word essay in 12 minutes. At what rate is she typing? (Hint: Typing is measured in words per minute.)


A=·w = 14 · 6 = 84

m2

Simplify. Formula for area Replace with 14 and w with 6. Simplify.

Example 11 The speed of light is 299,792,458 meters per second. Use the formula d = rt to calculate how far light travels in one minute. d=r·t

Write the formula.

= 299,792,458 · 60

Substitute.

= 17,987,547,480

Simplify.

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Practice Test

ENTERTAINMENT Suppose you pay $15 per hour to go horseback riding. You ride 2 hours today and plan to ride 4 more hours this weekend. 1. Write two different expressions to find the total cost of horseback riding. 2. Find the total cost. Simplify each expression. 3. x + 3x

4. 9x + 5 - x + 3

5. 10(y + 3) - 4y

6. -7b - 5(b - 4)

22. MULTIPLE CHOICE A carpet store advertises 16 square yards of carpeting for $300, which includes the $60 installation charge. Which equation could be used to determine the cost of one square yard of carpet x? A 16x = 300

C 60x + 16 = 300

B x + 60 = 300

D 16x + 60 = 300

23. MULTIPLE CHOICE In the sequence below, which expression can be used to find the value of the term in the nth position?

7. MUSIC Omar and Deb each have a digital music player. Deb has 37 more songs on her player than Omar has on his player. Write an expression in simplest form that represents the total number of songs on both players.

Position

Value

1

5

2

14

3

23

4

32

n

?

Solve each equation. Check your solution. 8. 19 = f + 5

9. -15 + z = 3

10. x - 7 = 16

11. g - 9 = -10

12. -8y = 72

n = -6 13. -30

14. 25 = 2d - 9

15. 4w - 18 = -34

16. 6v + 10 = -62

d +1 17. -7 =

18. x + 7 - 2x = 18

19. b - 7b + 6 = -30

F 5n

H 9n

G 5n + 4

J 9n - 4

24. Find the perimeter and area of the rectangle. 48 m 20 m

-5

20. TRAVEL Ms. Carter is renting a car from an agency that charges $20 per day plus $0.15 per mile. She has a budget of $80 per day. Use the equation 80 = 20 + 0.15m to find the maximum number of miles she can drive each day. 21. GENETICS Approximately one-seventh of the people in the world are left-handed. Write and solve an equation to estimate how many people in the United States are left-handed if the population of the United States is about 300 million.


25. MULTIPLE CHOICE The rectangle below has a length of 20 centimeters and a perimeter of P centimeters. Which equation could be used to find the width of the rectangle?

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w A P = 40 + 2

B P = 40 + 2w C P = 20 + w D P = 20 + 2w


173

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Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. Which expression can be used to find the nth term of the following sequence, where n represents a number’s position in the sequence?

4. Let n represent the position of a number in the following sequence. 3

5 3

1 , 1 , , 1, , , 7 , 2, . . . 4 2 4 4 2 4

Which expression can be used to find any term in the sequence? n F 4

Position in Sequence

1

2

3

4

n

n G 2

Term

4

6

8

10

?

H 2n J 3n

A 2n B 2n + 1

5. A movie theater sells large boxes of popcorn for $7.50, medium boxes of popcorn for $5.75, and small boxes of popcorn for $3.75. Suppose a group of friends orders 2 large popcorns, 1 medium popcorn, and 2 small popcorns. Which equation can be used to find the total cost of the popcorn?

C 2n + 2 D 3n + 1

2. Mrs. Kelly’s deck has an area of 660 square feet.

A 2(7.5) + 5.75 + 2(3.75) B (2 + 1 + 2)(7.5 + 5.75 + 3.75)

FT

FT

7.5 + 5.75 + 3.75 C (2 + 1 + 2) 3

D 2(7.5) + 5.75 + 3.75

What is the length of the deck if the width is 22 feet? F 25 ft

H 32 ft

G 30 ft

J 35 ft

3. On Saturday the low temperature in Detroit, Michigan, was -7°F, and the high temperature was 23°F. How much warmer was the high temperature than the low temperature? A -30°F B -16°F

6. GRIDDABLE The ordered pairs (-7, -2), (-3, 5), and (-3, -2) are coordinates of three vertices of a rectangle. What is the y-coordinate of the ordered pair that represents the fourth vertex? 7. Todd is 5 inches taller than his brother. The sum of their heights is 139 inches. Find Todd’s height. F 67 in.

H 77 in.

G 72 in.

J 82 in.

8. Which expression represents the greatest integer?

C 16°F

A ⎪4⎥

C ⎪-8⎥

D 30°F

B ⎪-3⎥

D -9




9. The scatter plot shows how many cups of hot chocolate are sold on average at a ski resort based on the outside temperature.

12. GRIDDABLE Six tables positioned in a row will be used to display science projects. Each table is 8 feet long. How many yards of fabric are needed to make a banner that will extend from one end of the row of tables to the other?

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13. Pedro is buying a DVD player that is on sale for $49.89 plus tax. If he pays with a $100 bill, what other information is needed to determine how much change he should receive?

X

A The brand of the DVD player

Which description best represents the relationship of the data? F Negative trend

B The amount of money Pedro has in his wallet

G No trend

C The amount of Pedro’s weekly income

H Positive trend

D The sales tax rate

J Cannot be determined Pre-AP 10. Henry paid $32 to rent a table saw for a two-day period. The maximum length of time a customer can rent the saw is 10 days. Which equation can be used to find c, the cost of renting the table saw for the maximum number of days? A c = 32 10

C c = 32 5

B c = 2 32

D c = 32 15

Record your answers on a sheet of paper. Show your work. 14. A basketball team can score 3-point baskets, 2-point baskets, and 1-point free throws. Josh heard the Springdale Stars scored a total of 63 points in their last game. Soledad says that they made a total of two 3-point baskets and 12 free throws in that game. a. Write an equation to represent the total points scored p. Use f for the number of free throws, g for the number of 2-point baskets, and h for the number of 3-point baskets. b. Can both Josh and Soledad be correct? Explain your reasoning.

11. Find - 19 - (-10). F -29

G -9

H9

J 29

Question 11 When a question involves integers, check to be sure that you have chosen an answer with the correct sign.


1

2

3

4

5

6

7

8

9

10

11

12

13

14

Go to Lesson...

3-7

3-8

2-3

3-7

1-1

2-6

3-3

2-1

1-7

3-4

2-3

1-1

1-1

3-6


175

Algebra and Rational Numbers Focus Understand rational numbers, ways of representing numbers, relationships among numbers, and number systems. Find measures of central tendency in data sets.

CHAPTER 4 Factors and Fractions Understand that different forms of numbers are appropriate for different situations. Select and use appropriate operations to solve problems and justify solutions.

CHAPTER 5 Rational Numbers Understand that different forms of numbers are appropriate for different situations. Use statistical procedures to describe data sets. Evaluate predictions and conclusions based on statistical data.

176 Unit 2 Algebra and Rational Numbers Envision/Corbis

Algebra and Nutrition You Are What You Eat Did you know that Wisconsin produces more cheese and grows more cranberries for processing than any other state? Calcium, found in cheese and cranberries, helps to keep your bones and teeth strong over your lifetime. In this project, you will be exploring how rational numbers are related to nutrition. Log on to pre-alg.com to begin.

Unit 2 Algebra and Rational Numbers

177

4

Factors and Fractions

•

Understand that different forms of numbers are appropriate for different situations

•

Select and use appropriate operations to solve problems and justify solutions

Key Vocabulary exponent (p. 180) power (p. 180) factor (p. 180) scientific notation (p. 214)

Real-World Link Technology The number of transistors on a computer chip (or the processing power of a computer) doubles about every 2 years.

actors and Fractions Make this Foldable to help you organize your notes about factors and fractions. egin with four sheets of notebook paper.

1 Fold four sheets of notebook paper in half from top to bottom.

3 Cut tabs into the margin. Make the top tab 2 lines wide, the next tab 4 lines wide, and so on.

178 Chapter 4 Factors and Fractions Mike Agliolo/Photo Researchers

2 Cut along the fold. Staple eight halfsheets together to form a booklet.

4 Label each of the tabs with the lesson number and title.

&ACTORS AN D &RACTIONS W BEL



Option 1


Evaluate each expression if x = 2, y = 5, and z = -1. (Lesson 1-3) 1. x + 12 2. z + (-5) 3. 4y + 8

4. 10 + 3z

5. (2 + y)9

6. 6(x - 4)

7. 3xy

8. 2z + y

9. POPULATION The freshman class has 15 more than twice the number of students as the previous year. If the freshman class had 140 students the previous year, how many freshman are there this year?

Example 1

Evaluate the expression 5x - 2y + 0z if x = -4, y = 9, and z = 6. 5x - 2y + 0z

Write the expression.

= 5(-4) - 2(9) + 0(6) Replace x with -4, y with 9, and z with 6.

= -20 - 18 + 0

Multiply.

= -38

Simplify.

(Lesson 1-3)

Example 2

Simplify. (Lesson 3-1) 10. 2(x + 1)

11. 3(n - 1)

Simplify -(x - 2).

12. -2(k + 8)

13. -4(x - 5)

-(x - 2) = -1(x - 2)

14. 6(2c + 4)

15. 5(-3s + t)

16. 7(a + b)

17. 9(b - 2c)

18. FOOD The concession stand offers a slice of pizza for $3.50 and a bottle of water for $1.25. Write two equivalent expressions for the total cost of p people each buying a slice of pizza and a bottle of water.

A negative sign before a parenthesis implies -1.

= (-1)(x) + (-1)(-2) Distribute -1 to each term inside the parentheses.

= -x + 2

Simplify.

(Lesson 3-1)

Find each product. (Prerequisite Skills, pp. 747–748) 19. 4.5 · 10 20. 3.26 · 100 21. 0.1 · 780

22. 15 · 0.01

23. 0.01 · 0.5

24. 301.8 · 0.001

25. HOTELS A hotel costs $159 plus 10% in taxes and fees for each night. The amount of taxes and fees is found by multiplying the cost of the hotel by 10% or 0.1. What is the cost of taxes and fees for a one-night stay? (Prerequisite Skills, pp. 747–748)

Example 3

Find the product of 0.1 × 2.78. 2.78 ×

← 2 decimal places

0.1 ← 1 decimal place

0.278

← 3 decimal places

The product is 0.278.


179

4-1

Powers and Exponents

Main Ideas • Write expressions using exponents. • Evaluate expressions containing exponents.

Computer memory is measured in small units called bytes. These units are based on products and factors of 2.

New Vocabulary factor base exponent power

a. Write 16 as a product of factors of 2. How many factors are there?

500MHZ

128MG CD-ROM

Memor y + Speed

Year

Amount of Memory in a Personal Computer

1980 1983 1992 1998 1999 2002 2004

16 kilobytes 1 megabyte 16 megabytes 32 megabytes 128 megabytes 512 megabytes 1 gigabyte

PC Sale 12/16

750MHZ 20 0GB

32MB + 32MB

40X Max.Var CDROM

1 Call 2 Write

b. How many factors of 2 form the product 128? c. One megabyte is Source: islandnet.com 1024 kilobytes. How many factors of 2 form the product 1024?

Exponents Two or more numbers that are multiplied to form a product are called factors. An expression like 2 × 2 × 2 × 2 with equal factors can be written as a power. A power has two parts, a base and an exponent. The expression 2 × 2 × 2 × 2 can be written as 24. The base is the number that is multiplied.

The exponent tells how many times the base is used as a factor.

24

The number that can be expressed using an exponent is called a power.

The table below shows how to write and read powers with positive exponents. Powers

Reading Math First Power When a number is raised to the first power, the exponent is usually omitted. So 21 is written as 2.

21

Words

Repeated Factors

23 24

2 to the first power 2 to the second power or 2 squared 2 to the third power or 2 cubed 2 to the fourth power or 2 to the fourth

2n

2 to the nth power or 2 to the nth

22

2 2·2 2·2·2 2·2·2·2 2·2·2·…·2 n factors

Any number, except 0, raised to the zero power is defined to be 1. 10 = 1 180 Chapter 4 Factors and Fractions

20 = 1

30 = 1

40 = 1

50 = 1

x0 = 1, x ≠ 0

EXAMPLE

Write Expressions Using Exponents

Write each expression using exponents. a. 3 · 3 · 3 · 3 · 3

b. t · t · t · t

The base is 3. It is a factor 5 times, so the exponent is 5. 3 · 3 · 3 · 3 · 3 = 35 Common Misconception (-9)2

is not the same as -92. (-9)2 = (-9)(-9) = 81 -92 = -1 · 92 = -81

The base is t. It is a factor 4 times, so the exponent is 4. t · t · t · t = t4

c. (-9)(-9)

d. (x + 1)(x + 1)(x + 1)

The base is -9. It is a factor 2 times, so the exponent is 2. (-9)(-9) = (-9)2

The base is x + 1. It is a factor 3 times, so the exponent is 3. (x + 1)(x + 1)(x + 1) = (x + 1)3

e. 7 · a · a · a · b · b First, group the factors with like bases. Then, write using exponents. 7 · a · a · a · b · b = 7 · (a · a · a) · (b · b) = 7a3b2 a · a · a = a3 and b · b = b2

Write each expression using exponents. 1A. 6 · 6 · 6 · 6 1B. x · x · x · x · x 1D. (c - d)(c - d) 1E. 9 · f · f · f · f · g

1C. (-2)(-2)(-2) 1F. (m + 1)

Evaluate Expressions Since powers are forms of multiplication, they need to be included in the rules for order of operations. Order of Operations Words

Example

Step 1

Simplify the expressions inside grouping symbols first.

Step 2

Evaluate all powers.

Step 3

Do all multiplications or divisions in order from left to right.

Step 4

Do all additions or subtractions in order from left to right.

(3 + 4)2 + 5 · 2 = 72 + 5 · 2 = 49 + 5 · 2 = 49 + 10 = 59

Reading Math Exponents An exponent goes with the number, variable, or quantity in parentheses immediately preceding it. • In 5 · 32, 3 is squared. 5 · 32 = 5 · 3 · 3 • In (5 · 3)2, (5 · 3) is squared. (5 · 3)2 = (5 · 3)(5 · 3)

Follow the order of operations to evaluate algebraic expressions.

EXAMPLE

Evaluate Numeric Expressions

Evaluate each expression. a. 23

b. 4 · 32

23 = 2 · 2 · 2 =8

2A. 54 Extra Examples at pre-alg.com

4 · 32 = 4 · 3 · 3 3 is a factor 2 times.

2 is a factor 3 times.

= 36

Multiply.

Multiply.

2B. 5 · 24 Lesson 4-1 Powers and Exponents

181

EXAMPLE


Evaluate each expression. a. y2 + 5 if y = -3 Replace y with -3. y2 + 5 = (-3)2 + 5 = (-3)(-3) + 5 -3 is a factor two times. = 9 + 5 or 14 Multiply. Then add.

Powers of Negatives Be sure to follow the order of operations when evaluating powers. (-3)2 = -3 · (-3) or 9 and -32 = -(3 · 3) or -9.

b. 3(x + y)3 if x = -2 and y = 1 3(x + y)3 = 3(-2 + 1)3 Replace x with -2 and y with 1. = 3(-1)3 Simplify the expression inside the parentheses. = 3(-1) or -3 Evaluate (-1)3. Then simplify.

Evaluate each expression if a = 5 and b = -2. 3A. 10 + b2 3B. (a + b)3 Personal Tutor at pre-alg.com

Example 1 (p. 181)

Example 2 (p. 181)

Example 3 (p.182)

Write each expression using exponents. 1. n · n · n 4. 3 · 3 · x · x · x · x

2. 7 · 7 5. (y - 3)(y - 3)(y - 3)

3. (-4)(-4)(-4) 6. (a + 1)(a + 1)

8. 63

9. 2 · 52

Evaluate each expression. 7. 24

ALGEBRA Evaluate each expression if x = -2 and y = 4. 10. x3 - 4

11. 5(y - 1)2

12. x2 + y2

13. SOUND Fireworks can easily reach a sound of 169 decibels, which can be dangerous if prolonged. Write this number using a power greater than 1 and a lesser base.

HOMEWORK

HELP


Write each expression using exponents. 14. 17. 20. 23.

4·4·4·4·4·4 (-8)(-8)(-8)(-8) r·r·r·r 2·x·x·y·y

15. 18. 21. 24.

6 k·k m·m·m·m (n - 5)(n - 5)(n - 5)

16. 19. 22. 25.

(-5)(-5)(-5) (-t)(-t)(-t) a·a·b·b·b·b 9 · (p + 1) · (p + 1)

Evaluate each expression. 26. 72 29. (-2)5 32. 63 · 4 182 Chapter 4 Factors and Fractions

27. 103 30. 3 · 42 33. 35 · 10

28. (-9)3 31. 2 · 43 34. 20 · 10

ALGEBRA Evaluate each expression if a = 2, b = 4, and c = -3. 35. b4

36. c4

37. 4a4

38. ac3

39. b0 - 10

40. c2 + a2

41. 3a + b3

42. a2 + 3a - 1

43. b2 - 2b + 6

44. 3(b - 1)4

45. 2(3c + 7)2

46. 5(a3 + 6)

47. BIOLOGY A man burns approximately 121 Calories by standing for an hour. A woman burns approximately 100 Calories per hour when standing. Write each of these numbers as a power with an exponent greater than 1. 48. MILEAGE Which numbers in the table can be expressed as a power greater than 1? Name the cities and express the numbers as powers.

Miles to Kentucky Dam City

49. Write 7 cubed times x squared as repeated multiplication.

Miles

Bowling Green

120

Chicago

400

Evansville

100

Lexington

250

50. Write negative eight cubed using exponents and as a product of repeated factors.

Louisville

200

Nashville

125

51. Without using a calculator, order 96, 962, 9610, 965, and 960 from least to greatest. Explain.

Paducah

25

St. Louis

225

Source: kentuckylake.com

52. NUMBER THEORY Explain whether the square of any nonzero number is sometimes, always, or never a positive number.

Real-World Link The so noodles are about a yard long and as thin as a piece of yarn. Very few chefs still know how to make these noodles. Source: The Mathematics Teacher

FOOD For Exercises 53–55, use the following information. In an ancient Chinese tradition, a chef stretches and folds dough to make long, thin noodles called so. After the first fold, he makes 2 noodles. He stretches and folds it a second time to make 4 noodles. Each time he repeats this process, the number of noodles doubles. 53. Use exponents to express the number of noodles after each of the first five folds. 54. Legendary chefs have completed as many as thirteen folds. How many noodles is this? 55. If the noodles are laid end to end and each noodle is 5 feet long, after how many of these folds will the length be more than a mile? Replace each 56. 37

73

with , or = to make a true statement. 57. 24

42

58. 63

GEOMETRY For Exercises 59–61, use the cube at the right. 59. The surface area of a cube is the sum of the areas of the faces. Use exponents to write an expression for the surface area of the cube. EXTRA

I E PRACTIC


60. The volume of a cube, or the amount of space that it occupies, is the product of the length, width, and height. Use exponents to write an expression for the volume of the cube.

44

3 cm

3 cm

61. If you double the length of each edge of the cube, are the surface area and volume also doubled? Explain. Lesson 4-1 Powers and Exponents

Christophe Loviny/CORBIS

3 cm

183

H.O.T. Problems

62. OPEN ENDED Use exponents to write a numerical expression and an algebraic expression in which the base is a factor 5 times. CHALLENGE Suppose the length of a side of a square is n units and the length of an edge of a cube is n units. 63. If all the side lengths of a square are doubled, are the perimeter and the area of the square doubled? Explain. 64. If all the side lengths of a square are tripled, show that the area of the new square is 9 times the area of the original square. Explain. 65. If all the edge lengths of a cube are tripled, show that the volume of the new cube is 27 times the volume of the original cube. Explain.

N

N

66. SELECT A TOOL Mercury has a mean distance from the Sun of 60002 miles. Which of the following tools might you use to determine the mean distance from the Sun to Mars if it is four times Mercury’s distance to the Sun? Justify your selection(s). Then use the tool(s) to solve the problem. draw a model 67.

paper/pencil

calculator

Writing in Math Use the information about exponents on page 180 to explain how they are used to describe computer memory. Include an advantage of using exponents.

68. Which expression represents the number of cells after half an hour? A 210

C 220

B 215

D 230

Time (min)

Number of Bacteria

0 3 6 9 12

20 21 22 23 24

69. GRIDDABLE Suppose a certain forest fire doubles in size every 8 hours. If the initial size of the fire was 1 acre, how many acres will the fire cover in 3 days?

70. TORNADOES A tornado travels 300 miles in 2 hours. Use the formula d = rt to find the tornado’s speed in miles per hour. (Lesson 3-8) 71. PATTERNS Study the pattern. Find the equation that represents a relationship between the number of columns c and the number of rows r in the pattern. (Lesson 3-7) ALGEBRA Solve each equation. Check your solution. (Lesson 3-5) 72. 2x + 1 = 7

n 74. _ +8=6

73. 16 = 5k - 4

3

PREREQUISITE SKILL List all the factors for each number. (Pages 740–741) 75. 11

76. 10

184 Chapter 4 Factors and Fractions

77. 16

78. 50

EXTEND

4-1

Algebra Lab

Base 2

A computer contains a large number of tiny electronic switches that can be turned ON or OFF. The digits 0 and 1, also called bits, are the alphabet of computer language. This binary language uses a base two system of numbers.

The digit 1 represents the ON switch.

24

23

22

21

20

Place values are powers of 2.

1

0

1

1

0

The digit 0 represents the OFF switch.

101102 = (1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (0 × 20) = 16

+

0

+

4

+

2

+

0

or 22

So, 101102 = 2210 or 22. You can also reverse the process and express base ten numbers as equivalent numbers in base two.

ACTIVITY Express the decimal number 13 as a number in base two. Step 1 Make a base 2 place-value chart. Find the greatest power of 2 that is less than 13. Place a 1 in that place value.

16

1 8

4

2

1

Step 2 Subtract 13 - 8 = 5. Now find the greatest power of 2 that is less than 5. Place a 1 in that place value.

16

1 8

1 4

2

1

16

1 8

1 4

0 2

1 1

Step 3 Subtract 5 - 4 = 1. Place a 1 in that place value. Step 4 There are no powers of 2 left, so place a 0 in any unfilled spaces.

So, 13 in the base 10 system is equal to 1101 in the base 2 system. Or, 13 = 11012.

ANALYZE THE RESULTS 1. Express 10112 as an equivalent number in base 10. Express each base 10 number as an equivalent number in base 2. 2. 6

3. 9

4. 15

5. 21

6. The first five place values for base 5 are shown. Any digit from 0 to 4 can be used to write a base 5 number. Write 179 in base 5.

625 125 25

5

1

7. OPEN ENDED Write 314 as an equivalent number in a base other than 2, 5, or 10. Include a place-value chart. 8. OPEN ENDED Choose a base 10 number and write it as an equivalent number in base 8. Include a place-value chart. Extend 4-1 Algebra Lab: Base 2

185

4-2

Prime Factorization

Main Ideas • Write the prime factorizations of composite numbers.

There are two ways that 10 can be expressed as the product of whole numbers. This can be shown by using 10 squares to form rectangles. 10

• Factor monomials.

1

2

1 10 10

New Vocabulary prime number composite number prime factorization factor tree monomial factor

5

2 5 10

a. Use grid paper to draw as many different rectangular arrangements of 2, 3, 4, 5, 6, 7, 8, and 9 squares as possible. b. Which numbers of squares can be arranged in more than one way? c . Which numbers of squares can only be arranged one way? d. What do the rectangles in part c have in common? Explain.

Prime Numbers and Composite Numbers A prime number is a whole number that has exactly two factors, 1 and itself. A composite number is a whole number that has more than two factors. Zero and 1 are neither prime nor composite. Whole Numbers

⎧ Prime ⎨ Numbers ⎩ ⎧ Composite ⎨ Numbers ⎩ Neither Prime ⎧ ⎨ nor Composite ⎩

Vocabulary Link Composite Everyday Use materials that are made up of many substances Math Use numbers having many factors

EXAMPLE

Factors

Number of Factors

2 3 5 7

1, 2 1, 3 1, 5 1, 7

2 2 2 2

4 6 8 9

1, 2, 4 1, 2, 3, 6 1, 2, 4, 8 1, 3, 9

3 4 4 3

0 1

all numbers 1

infinite 1

Identify Numbers as Prime or Composite

a. Determine whether 19 is prime or composite. Find factors of 19 by listing the whole number pairs whose product is 19. 19 = 1 × 19 The number 19 has only two factors. Therefore, 19 is a prime number. 186 Chapter 4 Factors and Fractions


b. Determine whether 28 is prime or composite. Find factors of 28 by listing the whole number pairs whose product is 28.

Mental Math

28 = 1 × 28 28 = 2 × 14 28 = 4 × 7 The factors of 28 are 1, 2, 4, 7, 14, and 28. Since the number has more than two factors, it is composite.

To determine whether a number is prime or composite, you can mentally use the rules for divisibility rather than listing factors. You can review divisibility rules on pages 740–741.

Determine whether each number is prime or composite. 1A. 21

1B. 37

When a composite number is expressed as the product of prime factors, it is called the prime factorization of the number. One way to find the prime factorization of a number is to use a factor tree. Write the number that you are factoring at the top.

24

24

Choose any pair of whole number factors of 24.

· 12

2

2·3

·

4

2·3·2

·

Continue to factor any number that is not prime.

2

2

8

·

4

·

2·3

·

2·2·3

3

The factor tree is complete when you have a row of prime numbers.

Both trees give the same prime factors, except in different orders. There is exactly one prime factorization of 24. The prime factorization of 24 is 2 · 2 · 2 · 3 or 23 · 3.

EXAMPLE

Write Prime Factorization

Write the prime factorization of 36. 36


6

·

6

36 = 6 · 6

2 · 3 · 2 · 3 6=2·3 The factorization is complete because 2 and 3 are prime numbers. The prime factorization of 36 is 2 · 2 · 3 · 3 or 22 · 32.

Write the prime factorization of each number. Use exponents for repeated factors. 2A. 16 2B. 27 Personal Tutor at pre-alg.com Lesson 4-2 Prime Factorization

187

Factor Monomials A number such as 80 or an expression such as 8x is called a monomial. A monomial is a number, a variable, or a product of numbers and/or variables. 8 · 10 = 80

8 · x = 8x

8 and 10 are factors of 80.

8 and x are factors of 8x.

To factor a number means to write it as a product of its factors. A monomial can also be factored as a product of prime numbers and variables with no exponent greater than 1. Negative coefficients can be factored using -1 as a factor.

EXAMPLE

Factor Monomials

Factor each monomial. a. 8ab2 8ab2 = 2 · 2 · 2 · a · b2 =2·2·2·a·b·b

8=2·2·2 a · b2 = a · b · b

b. -30x3y -30x3y = -1 · 2 · 3 · 5 · x3 · y -30 = -1 · 2 · 3 · 5 = -1 · 2 · 3 · 5 · x · x · x · y x3 · y = x · x · x · y c. -16e2f 3 -16e2f 2 = -1 · 2 · 2 · 2 · 2 = -1 · 2 · 2 · 2 · 2 · e · e · f · f · f

3A. 10x2y

Example 1 (pp. 186–187)

-16 = -1 · 2 · 2 · 2 · 2 e2 · f 3 = e · e · f · f · f

3B. -18mn4

Determine whether each number is prime or composite. 1. 7

2. 23

3. 15

4. NUMBER THEORY One mathematical conjecture that is unproved states that there are infinitely many twin primes. Twin primes are prime numbers that differ by 2, such as 3 and 5. List all the twin primes that are less than 50. Example 2 (p. 187)

Write the prime factorization of each number. Use exponents for repeated factors. 5. 18

Example 3 (p. 188)

6. 39

7. 50

ALGEBRA Factor each monomial. 8. 4c2


9. 5a2b

10. -70xyz

HOMEWORK

HELP



12. 33

13. 23

14. 70

15. 17

16. 51

17. 43

18. 31


20. 81

21. 66

22. 63

23. 104

24. 100

25. 392

26. 110

ALGEBRA Factor each monomial. 27. 14w

28. 9t2

29. -7c2

30. -25z3

31. 20st

32. -38mnp

33. 28x2y

34. 21gh3

35. 13q2r2

36. 64n3

37. -75ab2

38. -120r2st3

TECHNOLOGY Mersenne primes are prime numbers in the form 2n - 1. In 2004, Josh Findlay used special software to discover the largest prime number so far, 224,036,583 - 1. Write the prime factorization of each number, or write prime if the number is a Mersenne prime. 39. 25 - 1

40. 26 - 1

41. 27 - 1

42. 28 - 1

43. CALENDARS February 3 is a prime day because the month and day (2/3) are represented by prime numbers. How many prime days are there in a non-leap year? EXTRA

PRACTIICE


H.O.T. Problems

44. Is the value of n2 - n + 41 prime or composite if n = 3? 45. PACKAGING A beverage company is developing the packaging for a supercase of soda that contains 36 cans. List the arrangement of the cans that could be used for the package. (Hint: The cans can be stacked as well as arranged in a rectangular pattern one-can high.) 46. OPEN ENDED Write a monomial whose factors include -1, 5, and x. 47. FIND THE ERROR Cassidy and Francisca each factored 88. Who is correct? Explain your reasoning. Francisca 88

Cassidy 88 4 ·

22

4 · 2 · 11 88 = 4 · 2 · 11

2

8

11

·

4 · 11

2 · 2 · 2 · 11 88 = 2 · 2 · 2 · 11

48. CHALLENGE Find the prime factors of these numbers that are divisible by 12: 12, 60, 84, 132, and 180. Then, write a rule to determine when a number is divisible by 12. 49. NUMBER SENSE Find the least number that gives you a remainder of 1 when you divide it by 2, 3, 5, or 7. Lesson 4-2 Prime Factorization

189

50.

Writing in Math Use the information about prime numbers on page 186 to explain how models can be used to determine whether numbers are prime. Include the number of rectangles that can be drawn to represent a prime or a composite number and an explanation of how one model can show that a number is not prime.

51. Prime numbers are used to help keep messages sent over the Internet private. One step in the process involves multiplying two prime numbers to produce a key N. Which number could be N?

52. How many rectangles with different wholenumber dimensions can be drawn if each rectangle has an area of 30 square centimeters?

A 27

C 31

F 2

H 4

B 29

D 33

G3

J

Ài> Îä VÓ

5

53. Write (-5) · (-5) · (-5) · h · h · k using exponents. (Lesson 4-1) 54. FENCING Luis has 48 feet of fencing and is planning to make a rectangular pen for his dog. The length of the fence is 3 times as long as the width. If he uses all of the fencing, what are the dimensions of the pen? (Lesson 3-8) TIME ZONES The table shows a relationship between times in the Pacific Standard Time Zone (PST), where Seattle is located, and the Eastern Standard Time Zone (EST), where New York is located. (Lesson 3-7) 55. What time is it in New York if it is 6 P.M. in Seattle? 56. What time is it in Seattle if it is 11 P.M. in New York? 57. Write in words the relationship between PST and EST. ALGEBRA Solve each equation. Check your solution. (Lesson 3-4) n 58. _ = -4 8

59. 2x = -18

60. 30 = 6n

y 4

61. -7 = _

62. ALGEBRA Evaluate 9 + t if t = -1. (Lesson 2-2) Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 1-7) 63. outside temperature and amount of heating bill 64. size of a television screen and the number of channels it receives

PREREQUISITE SKILL Use the Distributive Property to write each expression as an equivalent expression. (Lesson 3-1) 65. 2(n + 4)

66. 5(x - 7)

67. -3(t + 4)

68. (a + 6)10

69. (b - 3)(-2)

70. 8(9 - y)


*-/

-/

£ *°°

{ *°°

Ó *°°

x *°°

Î *°°

È *°°

{ *°°

Ç *°°

4-3

Greatest Common Factor

Main Ideas • Find the greatest common factor of two or more numbers or monomials. • Use the Distributive Property to factor algebraic expressions.

New Vocabulary Venn diagram greatest common factor

A Venn diagram shows the relationships among sets of numbers or objects by using overlapping circles in a rectangle.

Prime Factors of 12

3

The Venn diagram at the right shows the prime factors of 12 and 20. The common prime factors are in both circles.

12 2 · 2 · 3

Prime Factors of 20 2 2

5

20 2 2 5

a. Which numbers are in both circles? b. Find the product of the numbers that are in both circles. c. Is the product also a factor of 12 and 20? d. Make a Venn diagram showing the prime factors of 16 and 28. Then use it to find the common factors of the numbers.

Greatest Common Factor Often, numbers have some of the same factors. The greatest number that is a factor of two or more numbers is called the greatest common factor (GCF). Example 1 shows several ways to find the GCF.

EXAMPLE Choosing a Method

Find the GCF

a. Find the GCF of 12 and 20.

To find the GCF of two or more numbers, it is easier to

Method 1 List the factors.

• list the factors if the numbers are small, or • use prime factorization if the numbers are large.

factors of 20: 1, 2, 4, 5, 10, 20

factors of 12: 1, 2, 3, 4, 6, 12

Common factors of 12 and 20: 1, 2, 4

The greatest common factor of 12 and 20 is 4. Method 2 Use prime factorization. 12 = 2 · 2 · 3 20 = 2 · 2 · 5

Common prime factors of 12 and 20: 2, 2

The GCF is the product of the common prime factors. 2·2=4 Again, the GCF of 12 and 20 is 4. (continued on the next page) Lesson 4-3 Greatest Common Factor

191

b. Find the GCF of 30 and 24. First, factor each number completely. Then circle the common factors. 30

24

2 · 15

2 · 12

2· 3 · 5

2·2 · 6 2·2·2 · 3

30 = 2 ·

3 ·5

24 = 2 · 2 · 2 · 3 Writing Prime Factors Try to line up the common prime factors so that it is easier to circle them.

The common prime factors are 2 and 3.

The GCF of 30 and 24 is 2 · 3 or 6. c. Find the GCF of 54, 36, and 45. 54 = 2 ·

3 · 3 ·3

36 = 2 · 2 · 3 · 3 45 =

The common prime factors are 3 and 3.

Prime Factors of 54 3

3 · 3 ·5

The GCF is 3 · 3 or 9. Prime Factors of 45

Prime Factors of 36 2 33

2

5

Find the GCF of each set of numbers. 1A. 6, 24 1B. 16, 60

1C. 10, 25, 30

TRACK AND FIELD There are 208 boys and 240 girls participating in a field day competition. a. What is the greatest number of teams that can be formed if all the teams have the same number of girls and the same number of boys? Find the GCF of 208 and 240. 208 = 2 · 2 · 2 · 2 · 13 240 = 2 · 2 · 2 · 2 · 3 · 5

The common prime factors are 2, 2, 2, and 2.

The GCF is 2 · 2 · 2 · 2 or 16. So, 16 teams can be formed. b. How many boys and girls will be on each team? Real-World Link In some events such as sprints and the long jump, if the wind speed is greater than 2 meters per second (or 4.5 miles per hour) then the time or mark cannot be considered for record purposes. Source: encarta.msn.com

boys: 208 ÷ 16 = 13

girls: 240 ÷ 16 = 15

So, each team will have 13 boys and 15 girls.

FOOD Marta is cutting a 16-inch and a 28-inch submarine sandwich for a party. 2A. How long is the longest possible piece if she cuts them all to be the same length? 2B. How many total pieces are there? Personal Tutor at pre-alg.com

192 Chapter 4 Factors and Fractions Andy Lyons/Getty Images

Factor Algebraic Expressions You can also find the GCF of two or more monomials by finding the product of their common prime factors.

EXAMPLE

Find the GCF of Monomials

Find the GCF of 16xy2 and 30xy. Completely factor each expression. 16xy2 = 2 · 2 · 2 · 2 · x · y · y 30xy = 2 · 3 · 5 ·

Circle the common factors.

x · y

The GCF of 16xy2 and 30xy is 2 · x · y or 2xy.

3. Find the GCF of 8ab and 18b2.

You can use a GCF to factor an algebraic expression such as 2x + 6.

EXAMPLE

Factor Expressions

Factor 2x + 6. First, find the GCF of 2x and 6. 2x = 2 · x 6 = 2 · 3 The GCF is 2. Now write each term as a product of the GCF and its remaining factors.

Look Back To review the Distributive Property, see Lesson 3-1.

2x + 6 = 2(x) + 2(3) = 2(x + 3) Distributive Property

4A. 4d + 8

Example 1 (pp. 191–192)

Example 2 (p. 192)

Example 3 (p. 193)

Example 4 (p. 193)

Factor each expression. 4B. 3x + 9

Find the GCF of each set of numbers. 1. 6, 8

2. 21, 45

3. 16, 56

4. 28, 42

5. 12, 24, 36

6. 6, 15, 24

7. PARADES In the parade, 36 members of the color guard are to march in front of 120 members of the high school marching band. Both groups are to have the same number of students in each row. Find the greatest number of students that could be in each row. Find the GCF of each set of monomials. 8. 2y, 10y2

9. 14n, 42n3

10. 36a3b, 56ab2

Factor each expression. 11. 3n + 9


12. t2 + 4t

13. 15 + 20x Lesson 4-3 Greatest Common Factor

193

HOMEWORK

HELP


Find the GCF of each set of numbers or monomials. 14. 12, 8

15. 3, 9

16. 24, 40

17. 21, 14

18. 20, 30

19. 12, 18

20. 42, 56

21. 30, 35

22. 9, 15, 24

23. 20, 21, 25

24. 20, 28, 36

25. 66, 90, 150

26. QUILTING Suki is making a quilt from two different kinds of fabrics. One is 48 inches wide and the other is 54 inches wide. What are the dimensions of the largest squares she can cut from both fabrics so there is no wasted fabric? 27. DESIGN Lauren is covering the surface of an end table with equal-sized ceramic tiles. The table is 30 inches long and 24 inches wide. What is the largest square tile that Lauren can use and not have to cut any tiles? How many tiles will Lauren need? Find the GCF of each set of monomials. 28. 18, 45mn

29. 24t2, 32

30. 12x, 40x2

31. 4st, 10s

32. 5ab, 6b2

33. 7x2, 15xy

34. 14b, 56b2

35. 25k, 35j

36. 21x2y, 63xy2

37. 2x + 8

38. 3r + 12

39. 8 + 40a

40. 6 + 3y

41. 15f + 18

42. 14 + 21c

Factor each expression.

PATTERNS For Exercises 43 and 44, consider the pattern 7, 14, 21, 28, 35, . . . 43. Find the GCF of the terms in the pattern. Explain how you know. 44. Write the next two terms in the pattern. Find the GCF of each set of monomials. 45. 30a3b2, 24a2b

46. 32mn2, 16n, 12n3

NUMBER THEORY Two numbers are relatively prime if their only common factor is 1. Determine whether the numbers in each pair are relatively prime. Write yes or no. 47. 7 and 8

48. 13 and 11

49. 27 and 18

50. 20 and 25

Factor each expression. 51. k3 + k2 + 5k

52. 2x + 4y - 16

53. 5n - 10m + 25

Find possible dimensions of each rectangle, given the area. EXTRA

PRACTICE

55.

54.

56. Ài>

See pages 768, 797.

Ài> ÎX È Self-Check Quiz at pre-alg.com


XÓ

ÓX

Ài> Y Ó Y

H.O.T. Problems

57. OPEN ENDED Name two different numbers whose GCF is 12. 58. FIND THE ERROR Christine and Jack both found the GCF of 2 · 32 · 11 and 23 · 5 · 11. Who is correct? Explain your reasoning. Jack 2 · 32 · 11 2 · 2 · 2 · 5 · 11 GCF = 2 · 11 or 22

Christine 2 · 32 · 11 23 · 5 · 11 GCF = 11

59. CHALLENGE Can the GCF of a set of numbers be equal to one of the numbers? Give an example or a counterexample to support your answer. 60.

Writing in Math Use the information about GCF on page 191 to explain how a Venn diagram can be used to show the greatest common factor.

61. The Venn diagram shows the factors of 10x and 18x2. What is the greatest common factor of the two monomials? £äÝ

A x

B 2x

£nÝÓ X X

C x2

62. Terrell is cutting paper streamers to decorate for a party. He has a blue roll of paper 144 inches long, a red roll 192 inches long, and a yellow roll 360 inches long. If he wants to have all colors the same length, what is the longest length that he can cut?

D 2

F 24 in.

H 16 in.

G 18 in.

J 12 in.

ALGEBRA Factor each monomial. (Lesson 4-2) 63. 9n

64. 15x2

66. 22ab3

65. -5jk

67. ALGEBRA Evaluate 7x2 + y3 if x = -2 and y = 4. (Lesson 4-1) SALES An online bookstore charges a delivery fee for every book order placed on its Web site. The table shows the relationship between the book order amount and the total amount due. (Lesson 3-7) 68. What is the total if a $7 book order was placed? 69. What was the amount of the order placed if the total amount was $16?

Book Order (dollars)

Total (dollars)

1

4

2

5

3

6

4

7

Find each quotient. (Lesson 2-5) 70. -69 ÷ 23

71. 48 ÷ (-8)

72. -24 ÷ (-12)

73. -50 ÷ 5

PREREQUISITE SKILL Find each equivalent measure. (Pages 753–756) 74. 1 ft = ? in. 75. 1 yd = ? in. 76. 1 day = ? h 77. 1 m = ? cm Lesson 4-3 Greatest Common Factor

195

4-4

Simplifying Algebraic Fractions

Main Ideas • Simplify fractions using the GCF.

You can use a fraction to compare a part of something to a whole. The figures below show what part 15 minutes is of 1 hour.

• Simplify algebraic fractions.

11

12

1 2

10

New Vocabulary

11

simplest form algebraic fraction

8 6

2

15 60

2 3

8

4 7

5

6

1

9

4 7

12

10

3

8

5

15 of 60 parts are shaded.

11

9

4 7

1

10

3

9

12

3 of 12 parts are shaded.

3 12

6

5

1 of 4 parts is shaded.

1 4

a. Are the three fractions equivalent? Explain your reasoning. b. Which figure is divided into the least number of parts? c. Which fraction would you say is written in simplest form? Why?

Simplify Numerical Fractions A fraction is in simplest form when the GCF of the numerator and the denominator is 1. Fractions in Fractions not in Simplest Form Simplest Form 17 _1 , _1 , _3 , _

3 _ 6 _ _ , 15 , _ , 5

4 3 4 50

12 60 8 20

One way to write a fraction in simplest form is to write the prime factorizations of the numerator and the denominator. Then divide the numerator and denominator by the GCF.

EXAMPLE

Simplify Fractions

Prime Factors of 9

9 Write _ in simplest form. 12

9=3·

3

12 = 2 · 2 · 3

Factor the numerator. Factor the denominator.

Use a Venn Diagram

The GCF of 9 and 12 is 3.

To simplify fractions, let one circle in a Venn diagram represent the factors of the numerator and the other circle represent the factors of the denominator. The product of factors in the intersection is the GCF.

12

9÷3 9 _ =_ 12 ÷ 3 3 =_ 4

3 3 2 2

Divide the numerator and the denominator by the GCF. Simplest form

Prime Factors of 12

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 16 8 1A. _ 1B. _


20

9

The division in Example 1 can be represented in another way. 1

9 3·3 _ =_

The slashes mean that the numerator and the denominator are both divided by the GCF, 3.

2·2·3

12

1

3 3 =_ or _ Simplify. 2·2

EXAMPLE

4

Simplify Fractions

Write each fraction in simplest form. 12 a. _ 48

1

1

1

2·2·3 12 _ = __

Divide the numerator and denominator 1 by the GCF, 2 · 2 · 3.

2·2·2·2·3

48


1

1 =_

1

Simplify.

4

17 b. _

30 17 _ is in simplest form because the GCF of 17 and 30 is 1. 30

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 16 2A. _

24 2B. _

45

40

Eighty-eight feet is what part of 1 mile? 1 C _

1 A _

15

60 _ B 1 30

1 D _ 5

Read the Test Item Estimation You can solve some problems without much calculating by estimating your answer. 100 88 1 _ = _ or _ 5280 5000 50

The phrase what part indicates a relationship that can be written as a fraction. You need to write a fraction comparing 88 feet to the number of feet in 1 mile. Solve the Test Item 88 There are 5280 feet in 1 mile. Write the fraction _ in simplest form. 5280

1

1

1

1

88 2 · 2 · 2 · 11 _ = __ 2 · 2 · 2 · 2 · 2 · 3 · 5 · 11

5280

1

1

1

Divide the numerator and denominator by the GCF, 2 · 2 ?· 2 · 11.

1

1 =_ 60

1 So, 88 feet is _ of a mile. The answer is A. 60


(continued on the next page) Lesson 4-4 Simplifying Algebraic Fractions

197

CHECK You can check whether your answer is correct by solving the problem in a different way. Divide the numerator and denominator by common factors until the fraction is in simplest form. 88 44 _ =_ 5280

2640 22 =_ 1320 11 1 =_ or _ 660 60

3. Six hundred sixteen yards is what part of 1 mile? (Hint: 1 mi = 1760 yd) 6 F_

1 G_

25

3 H_

4

7 J_ 20

10


Simplify Algebraic Fractions A fraction with variables in the numerator or denominator is called an algebraic fraction. Algebraic fractions can also be written in simplest form.

EXAMPLE Check Reasonableness of Results You can check whether your answers are reasonable. In Example 4a, you can see that the variable y does not appear in the final answer since y can be divided into both the numerator and denominator. 21x2y

_

Simplify Algebraic Fractions

Simplify each fraction. If the fraction is already in simplest form, write simplified. 2

21x y a. _

3

abc b. _ 2

35xy

ab

1

2

1

1

21x y 3·7·x·x·y _ = __ 5·7·x·y

35xy

1

1

(pp. 196–197)

(p.198)

Multiply.

3

xyz

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 9 2. _

25 4. _

5 3. _

15

40

11

64 5. _ 68

6. MULTIPLE CHOICE Nine inches is what part of 1 yard? 1 A_

1 B _

8

Example 4

Factor.

1

xy 4B. _2

28ab 4A. _ 2

14

(pp.197–198)

a·a·b

1

c =_ a

Simplify.

5

2 1. _

Example 3

a2b

1

42ab

Examples 1, 2

abc3 a·b·c·c·c _ = __ 3

3x =_

35xy

1 1

Factor out the GCF, 7 · x ?· y.

1 C_

5

1 D_ 2

4

ALGEBRA Simplify each fraction.If the fraction is already in simplest form, write simplified. x 7. _ 3 x


8a2 8. _ 16a

12c 9. _ 15d

24 10. _ 5k

25mn 11. _ 65n

HOMEWORK

HELP

For See Exercises Examples 12–21 1, 2 22–29 4 42–44 3


18 18 17. _ 44

10 13. _

15 14. _

12 16 18. _ 64

8 15. _

17 16. _

36 34 20. _ 38

21 30 19. _ 37

20 17 21. _ 51

ALGEBRA Simplify each fraction. If the fraction is already in simplest form, write simplified. a 22. _4

a 4k _ 26. 19m

3

y 23. _ y

12m 24. _

15m _ 28. 16n2 18n p

4t 27. _2 64t

40d 25. _ 42d

28z3 29. _ 16z

30. MEASUREMENT Twelve ounces is what part of a pound? (Hint: 1 lb = 16 oz) 31. ANALYZE TABLES The table shows the number of Nebraska tornadoes that occurred in May and the total for selected years. What fraction of tornadoes occurred in May for each year? Write each fraction in simplest form.

9EAR

-AY 4OTAL

9EAR 4OTAL

3OURCE .EBRASKA 3EVERE 7EATHER

Real-World Career Musician A musician uses math to increase or decrease the tension in the strings of his or her instrument. The pitch is the frequency at which an instrument’s string vibrates when it is struck.

32. AIRCRAFT A model of Lindbergh’s Spirit of St. Louis has a wingspan of 18 inches. The wingspan of the actual airplane is 46 feet. Write a fraction in simplest form comparing the wingspan of the model and the wingspan of the actual airplane. (Hint: Convert 46 feet to inches.) MUSIC For Exercises 33–35, use the following information. Musical notes C and A sound harmonious together because of their frequencies, or vibrations. The fraction that is formed by the two frequencies can be simplified, as shown below. C 264 3 _ =_ or _


A

440

5

When a fraction formed by two frequencies cannot be simplified, the notes sound like noise. Determine whether each pair of notes would sound harmonious together. Explain why or why not. 33. E and A EXTRA

PRACTICE

34. D and F

Note

Frequency (hz)

C D E F G A B C

264 294 330 349 392 440 494 528

35. first C and last C

36. TIME Fifteen hours is what part of one day?


37. FIND THE DATA Refer to the United States Data File on pages 18–21 of your book. Choose some data and write a real-world problem in which you would simplify fractions. Lesson 4-4 Simplifying Algebraic Fractions

Geoff Butler

199

H.O.T. Problems

38. OPEN ENDED Write a numerical fraction and an algebraic fraction in simplest form and a numerical fraction and an algebraic fraction not in simplest form. 39. Which One Doesn’t Belong? Identify the fraction that does not belong with the other three. Explain your reasoning. 6y _

_4

5

_1

10 _ 12

x2

7

1

23 23 2 40. CHALLENGE Is it true that _ =_ or _ ? Explain why or why not. 53

Writing in Math

41.

53 1

5

Explain how simplified fractions are useful in representing measurements. Include an explanation of how measurements represent parts of a whole and examples of fractions that represent measurements.

42. Which store offers the best buy? A A B B

Store

Potatoes

A

18 for $12

B

30 for $24

C C

C

36 for $30

DD

D

42 for $30

43. In a pen factory, an average of 5 pens out of every p pens tested are rejected. What fraction of those p pens is NOT rejected? 5 F _ p

p G _ 5

p-5 H _ 5

44. Ninety-six centimeters is what part of a meter? 3 A _ 5

4 B _ 5

23 C _ 25

Find the GCF of each set of numbers or monomials. (Lesson 4-3) 45. 9, 15

46. 4, 12, 10

47. 40x2, 16x

48. 25a, 30b

Determine whether each number is prime or composite. (Lesson 4-2) 49. 13

50. 34

51. 99

52. 79

ALGEBRA Write and solve an equation to find each number. (Lesson 3-3) 53. The sum of a number and 9 is -2.

54. The sum of -5 a number and is -15.

55. GEOMETRY The area of a trapezoid is the product of one half the height and the sum of both bases. If h is the height, b1 is one base, and b2 is the second base, write an expression for the area of the trapezoid. (Lesson 1-3)

PREREQUISITE SKILL For each expression, use parentheses to group the numbers together and to group the powers with like bases together. (Lesson 1-4) Example: a · 4 · a3 · 2 = (4 · 2)(a · a3) 56. 6 · 7 · k3

57. s · t2 · s · t

58. b · 5 · 10 · b4

59. 3 · x4 · (-5) · x2

60. 5 · n3 · p · 2 · n · p

61. 12 · 15 · a · 9 · a5 · c3


p-5 J _ p

24 D_ 25

Powers The phrase the quantity is used to indicate parentheses when reading expressions. Recall that an exponent indicates the number of times that the base is used as a factor. Suppose you are to write each of the following in symbols. Words

Symbols

Examples (Let x = 2.)

3x2 =

3 · 22 = 3 · 4 Evaluate 22. = 12 Multiply 3 · 4.

3x2

three times x squared

three times x the quantity squared

(3x)2 = (3 · 2)2 = 62 Evaluate 3 · 2. = 36 Square 6.

(3x)2

In the expression (3x)2, parentheses are used to show that 3x is used as a factor twice. (3x)2 = (3x)(3x) The quantity can also be used to describe division of monomials. Words

Symbols

Examples (Let x = 2.)

8 8 _ =_ x2

8 _

eight divided by x squared

x2

22 8 = _ Evaluate 22. 4 = 2 Divide 8 ÷ 4.

8x = 82 2

eight divided by x the quantity squared

8x

2

2

= 42 Evaluate 8 ÷ 2. = 16 Square 4.

Exercises State how you would read each expression. 5 _

1. 4a2

2. (10x)5

3.

6. (a - b)4

7. a - b4

a 8. _4

n3 b

4 2 4. _r

5. (m + n)3

c

8 10. _2

9. (4c2)3

3

Determine whether each pair of expressions is equivalent. Write yes or no. 11. 4ab5 and 4(ab)5

12. (2x)3 and 8x3

13. (mn)4 and m4 · n4

14. c3d3 and cd3

2 x 15. _2 and x y

2 n2 16. _ and n 2 r

y

r

Reading Math Powers

201

CH

APTER

4


1. ALGEBRA Evaluate b2 - 4ac if a = -1, b = 5, and c = 3. (Lesson 4-1)

Factor each monomial. (Lesson 4-2)

2. MULTIPLE CHOICE The number of acres consumed by a forest fire triples every two hours. Which expression represents the number of acres consumed after 1 day? (Lesson 4-1)

11. -23n3

Hours

2

4

6

8

10

Acres Consumed

31

32

33

34

35

A 310 acres

C 318 acres

B 312 acres

D 324 acres

3. Write his reward on each of the first three days as a power of 2. 4. Write his reward on the 8th day as a power of 2. Then evaluate. Write the prime factorization of each number. Use exponents for repeated factors. (Lesson 4-2) 6. 99

10. 18st 12. 30cd2

13. BAKE SALE Joanna baked 81 cookies and 54 cupcakes for the bake sale. She wants to place the same number of cookies and the same number of cupcakes in a plastic bag. What is the maximum number of bags that she can make if she uses all of the cookies and cupcakes? (Lesson 4-3) Find the GCF of each set of numbers or monomials. (Lesson 4-3)

LITERATURE In a story, a knight received a reward for slaying a dragon. He received 1 cent on the first day, 2 cents on the second day, 4 cents on the third day, and so on, continuing to double the amount for 30 days. (Lesson 4-1)

5. 42

9. 77x

7. 64

8. MULTIPLE CHOICE A kitchen floor with the dimensions shown is to be tiled. If the tiles are only available in dimensions that are prime numbers, which set of tile dimensions could NOT be used to tile the floor? (Lesson 4-2)

14. 18, 45

15. 22, 21

16. 16, 40

17. 10, 12m

18. 3x, 18x2

19. 15g, 35h

Factor each expression. (Lesson 4-3) 20. 9s + 18

21. 6y + 21

22. 60 + 15h

23. 9z - 99

24. 21x - 63

25. 18 - 12a

26. MULTIPLE CHOICE Three hundred thirty yards is what part of 1 mile? (Lesson 4-4) 1 A 16

2 27. x3

x

G 2 ft by 3 ft

J 3 ft by 3 ft


20d

30. ANIMALS The table shows the average amount of food each animal can eat in a day and its average weight. What fraction of its weight can each animal eat per day? (Lesson 4-4)

elephant

H 2 ft by 5 ft

5c3d 29. 2

2 28. 27n 15

Animal

F 2 ft by 2 ft

D 1 4

ALGEBRA Simplify each fraction. If the fraction is already in simplest form, write simplified. (Lesson 4-4)

£Ó vÌ

Óä vÌ

3 C 16

B 1 8

hummingbird polar bear tiger

Daily Weight of Amount Animal of Food 450 lb 9000 lb 2g

3g

25 lb 20 lb

1500 lb 500 lb

Source: Animals as Our Companions, Wildlife Fact File

4-5

Multiplying and Dividing Monomials

Main Ideas • Multiply monomials. • Divide monomials.

For each increase of 1 on the Richter scale, an earthquake’s vibrations, or seismic waves, are 10 times greater. So, an earthquake of magnitude 4 has seismic waves that are 10 times greater than that of a magnitude 3 earthquake.

Richter Scale

Times Greater than Magnitude 3 Earthquake

Written Using Powers

4

10

101

5

10 ⴛ 10 ⴝ 100

101 ⴛ 101 ⴝ 102

6

10 ⴛ 100 ⴝ 1000

101 ⴛ 102 ⴝ 103

7

10 ⴛ 1000 ⴝ 10,000

101 ⴛ 103 ⴝ 104

8

10 ⴛ 10,000 ⴝ 100,000

101 ⴛ 104 ⴝ 105

a. Examine the exponents of the factors and the exponents of the products in the last column. What do you observe? b. MAKE A CONJECTURE Write a rule for determining the exponent of the product when you multiply powers with the same base. Test your rule by multiplying 23 · 24 using a calculator.

Multiply Monomials Recall that exponents are used to show repeated multiplication. You can use the definition of exponent to help find a rule for multiplying powers with the same base. 3 factors

4 factors

23 · 24 = (2 · 2 · 2) · (2 · 2 · 2 · 2) = 27

7 factors

Notice the sum of the original exponents and the exponent in the final product. This relationship is stated in the following rule. Common Misconception

Product of Powers Multiply powers with the same base by adding their exponents.

When multiplying powers, do not multiply the bases.

Words

32 · 34 = 36, not 96

Example 32 · 34 = 32 + 4 or 36

Symbols a m · an = a m + n

Lesson 4-5 Multiplying and Dividing Monomials

203

EXAMPLE

Multiply Powers

Find 73 · 7. 73 · 7 = 73 · 71 7 = 71 = 73 + 1

The common base is 7.

= 74

Add the exponents.

Find each product. 1A. 52 · 53

1B. 24 · 26

Monomials can also be multiplied using the rule for the product of powers.

EXAMPLE

Multiply Monomials

Find each product. a. x5 · x2 x5 · x2 = x 5 + 2 = x7

The common base is x. Add the exponents.

b. (-4n3)(2n6) Look Back

(-4n3)(2n6) = (-4 · 2)(n3 · n6)

To review the Commutative and Associative Properties of Multiplication, see Lesson 1-4.

Group the coefficients and variables.

= (-8)(n3 + 6)

The common base is n.

= -8n9

Add the exponents.

2A. y6 · y3

2B. (5a2)(-3a4)

Divide Monomials You can also find a rule for quotients of powers. 26 = 2 · 2 · 2 · 2 · 2 · 2 1 2 2

6 factors 1 factor

1

2·2·2·2·2·2 = 2

1

= 25

5 factors

Divide the numerator and the denominator by the GCF, 2. Simplify.

Compare the difference between the original exponents and the exponent in the final quotient. This relationship is stated in the following rule. Quotient of Powers

BrainPOP® pre-alg.com

Words

Divide powers with the same base by subtracting their exponents.

Symbols

am _ = am - n, where a ≠ 0

an 45 Example _ = 45 - 2 or 43 42


EXAMPLE

Divide Powers

Find each quotient. 5

y b. _3

57 a. _ 4

5 57 _ = 57 - 4 54

=

53

The common base is 5.

= y5 - 3

The common base is y.

y2

Subtract the exponents.

=

b7 3B. _ 6

3

How Many/ How Much How many times faster indicates that division is to be used to solve the problem. If the question had said how much faster, then subtraction (1010 - 109) would have been used to solve the problem.

y3

Subtract the exponents.

39 3A. _ 2

Reading Math

y y5 _

b

COMPUTERS The table compares the processing speeds of a specific type of computer in 1999 and in 2004. Find how many times faster the computer was in 2004 than in 1999.

Year

(instructions per second)

Write a division expression to compare the speeds. 1010 _ = 1010 - 9 109

= 101 or 10

Processing Speed

1999

109

2004

1010

Subtract the exponents. Simplify.

So, the computer was 10 times faster in 2004 than in 1999.

Source: The Intel Microprocessor Quick Reference Guide

4. TRAVEL The table compares the number of people who drove to work versus the number of people who walked to work in Wyoming in 2004. How many times more people drove than walked to work?

Mode of Transportation

Number of People

Drove

105

Walked

103

Source: U.S. Bureau of the Census


Examples 1–3 (pp. 204–205)

Find each product or quotient. Express using exponents. 1. 93 · 92

2. 114 · 116

3. 6 · 66

4. a · a5

5. (n4)(n4)

6. -3x2(4x3)

38

7. _5 3

Example 4 (p. 205)

105

8. _3 10

a10 9. _ 6 a

10. EARTHQUAKES In 2005, an earthquake measuring 5 on the Richter scale struck the Philippines. Four days later, an earthquake of magnitude 3 struck Southern Alaska. How many times greater were the seismic waves in the Philippines than in Alaska? (Hint: Let 105 and 103 represent the strength of the earthquakes, respectively.)



205

HOMEWORK

HELP

For See Exercises Examples 11–14 1 15–22, 31 2 23–30, 32 3 33–37 4


12. 6 · 67

13. 94 · 95

14. 104 · 103

15. d4 · d6

16. n8 · n

17. t2 · t4

18. a6 · a6

19. 2y · 9y4

20. (5r 3)(4r4)

21. (10x)(4x7)

22. 6p7 · 9p7

25. (-2)6 ÷ (-2)5

26. 1010 ÷ 102

55

23. _2 5

27. b6 ÷ b3

84

24. _3 8 a8 28. _ a8

5

(-x) 29. _

30. m20 ÷ m8

(-x)

31. the product of nine to the fourth power and nine cubed 32. the quotient of k to the fifth power and k squared PHYSICAL SCIENCE For Exercises 33–35, use the information at the left. The pH of a solution describes its acidity. Each one-unit decrease in the pH means that the solution is 10 times more acidic. For example, a pH of 4 is 10 times more acidic than a pH of 5. 33. How much more acidic is vinegar than baking soda? 34. Suppose the pH of a lake is 5 due to acid rain. How much more acidic is the lake than water? 35. Cola is 104 times more acidic than water. What is the pH value of cola?

Real-World Link The pH values of different kitchen items are shown below. Item pH lemon Juice 2 vinegar 3 tomatoes 4 water 7 baking soda 9 Source: Biology, Raven

EXTRA

PRACTIICE


H.O.T. Problems

GEOMETRY For Exercises 36 and 37, use the information in the figures. 36. How many times greater is the length of the edge of the larger cube than the smaller one? 37. How many times greater is the volume of the larger cube than the smaller one?

Volume ⴝ 23 cubic units

Find each missing exponent. • 39. t2 = t14 38. (4•)(43) = 411

Volume ⴝ 26 cubic units

135 40. =1 13•

t

41. What is the product of 73, 75, and 7? 42. Find a4 · a6 ÷ a2. ARTS AND CRAFTS For Exercises 43 and 44, use the information below. When a piece of paper is cut in half, the result is two smaller pieces of paper. When the two smaller pieces are stacked and then cut, four pieces of paper are made. The number of resulting sheets of paper after c cuts is 2c. 43. How many more pieces of paper are there if a piece of paper is cut and stacked 8 times than when a piece of paper is cut and stacked 5 times? 44. Notebook paper usually stacks about 500 sheets to the inch. How thick would your stack be if you were able to make 10 cuts? Find each product or quotient. Express using exponents. 45. ab5 · 8a2b5

46. 10x3y · (-2xy2)

3

5

n (n ) 47. _ 2 n

s7 48. _ 2 s·s

49. OPEN ENDED Write a multiplication expression whose product is 53. 50. CHALLENGE Use the laws of exponents to show why the value of any nonzero number raised to the zero power equals 1.

206 Chapter 4 Factors and Fractions Laura Sifferlin

51. REASONING Determine whether the statement is true or false. If true, explain your reasoning. If false, give a counterexample. For any integer a, (-a)2 = -a2. 52.

Writing in Math Use the data about earthquakes on page 203 to explain how powers of monomials are useful in comparing earthquake magnitudes. Include a comparison of two earthquakes of different magnitudes by using the Quotient of Powers rule.

53. How many times as intense is a rock band as a noisy office?

Sound intensity is measured in decibels. The decibel scale is based on powers of ten as shown in the table. Sound

Decibels

Intensity

rock band noisy office normal conversation whispering

120 60

1012 106

50

105

20

102

A 102

C 1012

B 106

D 1072

54. How many times as intense is a noisy office as a person whispering? F 10,000

H 100

G 1000

J 10

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 4-4) 12 55. _ 40

20 56. _ 53

8n2 57. _

6x3 58. _ 4

32n

4x y

Find the greatest common factor of each set of numbers or monomials. (Lesson 4-3) 59. 36, 4

60. 18, 28

62. 9a, 10a3

61. 42, 54

63. ALGEBRA A number is divided by -6, and the result is 24. What is the original number? (Lesson 3-4) 64. Evaluate |a| - |b| · |c| if a = -16, b = 2, and c = 3. (Lesson 2-1)

PREREQUISITE SKILL Evaluate each expression if x = 10, y = -5, and z = 4. Write as a fraction simplest form. (Lesson 1-3) 1 66. _ x _ 68. 1 zy

z 67. _ 100 1 69. _

Energy Used 160 Electricity Used (megawatts)

65. ENERGY The graph shows the high temperature and the amount of electricity used during each of fifteen summer days. Do the data show a positive, negative, or no relationship? Explain. (Lesson 1-7)

150 140 130 120 110 100 0

85

90

95

100

High Temperature (˚F)

(z)(z)(z)


207

EXTEND

4-5

Algebra Lab

A Half-Life Simulation

A radioactive material such as uranium decomposes or decays in a regular manner best described as a half-life. A half-life is the time it takes for half of the atoms in the sample to decay.

ACTIVITY

COLLECT THE DATA

Number of Half-Lives

Step 1 Place 50 pennies heads up in a shoebox. Put the lid on the box and shake it up and down one time. This simulates one half-life.

Number of Pennies That Remain

1 2 3

Step 2 Open the lid of the box and remove all the pennies that are now tails up. In a table like the one at the right, record the number of pennies that remain.

4 5

Step 3 Put the lid back on the box and again shake the box up and down one time. This represents another half-life. Step 4 Open the lid. Remove all the tails up pennies. Count the pennies that remain. Step 5 Repeat the half-life of decay simulation until fewer than five pennies remain in the shoebox.

ANALYZE THE RESULTS 1. On grid paper, draw a coordinate grid in which the x-axis represents the number of half-lives and the y-axis represents the number of pennies that remain. Plot the points (number of half-lives, number of remaining pennies) from your table. 2. Describe the graph of the data. After each half-life, you expect to remove about one-half of the pennies. So, you expect about one-half to remain. The expressions at the right represent the average number of pennies that remain if you start with 50, after one, two, and three half-lives.

one half-life: two half-lives: three half-lives:

50 1 1 = 50 1 2 2 2 1 1 1 50 = 50 1 2 2 2 2 1 50 1 = 50 1 2 2

3. MAKE A CONJECTURE Use the expressions to predict how many pennies remain after three half-lives. Compare this number to the number in the table above. Explain any differences. 4. MAKE A CONJECTURE Suppose you started with 1000 pennies. Predict how many pennies would remain after three half-lives. 208 Chapter 4 Factors and Fractions Latent Image

2

3

4-6

Negative Exponents

Main Ideas • Write expressions using negative exponents.

Power

Value

26

64

25

32

24

16

23

8

22

4

21

2

?

?

?

?

Copy the table at the right. a. Describe the pattern of the powers in the first column. Continue the pattern by writing the next two powers in the table.

• Evaluate numerical expressions containing negative exponents.

b. Describe the pattern of values in the second column. Then complete the second column. c. Verify that the powers you wrote in part a are equal to the values that you found in part b. d. Determine how 3-1 should be defined.

Negative Exponents Extending the pattern at the right 1 . shows that 2-1 can be defined as _

22 = 4

You can apply the Quotient of Powers rule and the x3 and write a general rule definition of a power to _ x5 about negative powers.

20 = 1

÷2

2-1 = 1 2

÷2

21 = 2

2

Method 1 Quotient of Powers

Method 2 Definition of Power 1

1

1

x3 x·x·x _ = __

x3 _ = x3 - 5

x· x · x · x · x

x5

x5

=

÷2

1 1

1

1 1 =_ or _

x-2

x·x

x3

x2

1 . Since _5 cannot have two different values, you can conclude that x-2 = _ x x2 This suggests the following definition.

Negative Exponents Symbols Example

a-n = _ n , for a ≠ 0 and any whole number n.

1 a 1 5-4 = _ 54

EXAMPLE

Use Positive Exponents

Write each expression using a positive exponent. a. 6-2

b. x -5

1 6-2 = _ Definition of negative 2 6

1A. 3-5 Extra Examples at pre-alg.com

1 x-5 = _ Definition of negative 5 x

exponent

exponent

1B. y-3 Lesson 4-6 Negative Exponents

209

EXAMPLE

Use Negative Exponents

1 Write _ as an expression using a negative exponent. 9

1 _1 = _ 9

Find the prime factorization of 9.

3·3 _ = 12 3

Definition of exponent

= 3-2

Definition of negative exponent

1 2. Write _ as an expression using a negative exponent. 25

Negative exponents are often used in science when dealing with very small numbers. Usually the number is a power of ten.

ANIMALS Geckos have tiny hairs on the bottom of their feet called setae. These setae are about 0.000001 meter long. Write the decimal as a fraction and as a power of ten. The digit 1 is in the millionths place. 1 0.000001 = _ Real-World Link Most geckos lack movable eyelids and the largest species of geckos measure 14 inches.

Write the decimal as a fraction.

1,000,000 1 =_ 106

1,000,000 = 106

= 10-6

Definition of negative exponent

1 Therefore, 0.000001 is _ as a fraction and 10-6 as a power of 10. 1,000,000

Source: encyclopedia.com

3. CHEMISTRY A hydrogen atom is only 0.00000001 centimeter in diameter. Write the decimal as a fraction and as a power of 10. Personal Tutor at pre-alg.com

Evaluate Expressions Algebraic expressions containing negative exponents can be written using positive exponents and then evaluated.

EXAMPLE

Algebraic Expressions with Negative Exponents

Evaluate n-3 if n = 2. n-3 = 2-3 1 =_ 23 1 =_ 8

Replace n with 2. Definition of negative exponent Find 23.

4. Evaluate x-4 if x = 3. 210 Chapter 4 Factors and Fractions Kim Taylor/DK Limited/CORBIS

Example 1 (p. 209)

Example 2 (p. 210)

Write each expression using a positive exponent. 1. 5-2

2. (-7)-1

1 6. _ 2

Example 4

HOMEWORK

1 7. _

1 8. _ 8

49

9

3

(p. 210)

4. n-2

Write each fraction as an expression using a negative exponent other than -1. 1 5. _ 4

Example 3

3. t-6

9. MEASUREMENT A unit of measure called a micron equals 0.001 millimeter. Write this number using a negative exponent. ALGEBRA Evaluate each expression if a = 2 and b = -3.

(p. 210)

10. a-5

HELP

Write each expression using a positive exponent.


11. b-3

12. (ab)-2

13. 2b

14. 4-1

15. 5-3

16. (-6)-2

17. (-3)-3

18. 3-5

19. 10-4

20. p-1

21. a-10

22. d-3

23. q-4

24. b-15

25. r-20


1 27. _ 5

1 28. _ 3

5 1 31. _ 81

9

1 30. _ 100

1 29. _ 2

8 1 32. _ 27

13 1 33. _ 16

34. BIRDS A mockingbird uses about 5-4 Joules of energy to sing a song. Write the amount of energy the bird uses as an expression using a positive exponent and as a decimal. PHYSICAL SCIENCE A nanometer is equal to a billionth of a meter. The visible range of light waves ranges from 400 nanometers (violet) to 740 nanometers (red).

400 nm

Real-World Link The wavelengths of X rays are between 1 and 10 nanometers. Source: Biology, Raven

430 nm

500 nm

560 nm

600 nm

650 nm

740 nm

35. Write one billionth of a meter as a fraction and as an expression with a negative exponent. 36. Use the information at the left to express the greatest wavelength of an X ray in meters. Write the expression using a negative exponent. ALGEBRA Evaluate each expression if w = -2, x = 3, and y = -1. 37. x-4

38. w-7

39. 8w

40. (xy)-6

Lesson 4-6 Negative Exponents Getty Images

211

41. ANALYZE TABLES Consider the pattern in the table in which the exponents are integers. If the pattern continues, what is the value of the eighth term in the pattern? Expression

23

22

21

20

Value

8

4

2

1

Write each decimal using a negative exponent. 42. 0.1

43. 0.01

44. 0.0001

45. 0.00001

1 -inch long can jump about 8 inches high. Write 46. ANIMALS A common flea _ 16 each number as an exponential expression with a base of 2. Then find how many times its body size a flea can jump.

47. MEDICINE Which type of molecule in the table has a greater mass? How many times greater is it than the other type?

EXTRA

PRACTIICE

iVÕi

>ÃÃ }®

PENICILLIN

INSULIN


Use the Product of Power and Quotient of Power rules to simplify each expression. 48. x-2 · x-3 6

y 51. _ -10 y

H.O.T. Problems

4 50. x7

49. r-5 · r9

x 36s3t5 53. 12s6t-3

a4b-4 52. _ -2 ab

54. OPEN ENDED Write a convincing argument that 30 = 1 using the fact that 34 = 81, 33 = 27, 32 = 9, and 31 = 3. 1 . Does it increase or decrease as the 55. REASONING Investigate the fraction _ 2n value of n increases? Explain.

56. CHALLENGE Using what you learned about exponents, is (x3)-2 = (x-2)3? Why or why not? NUMBER SENSE Numbers can also be expressed in expanded form. Example 1: 13,548 = 10,000 + 3000 + 500 + 40 + 8 = (1 × 104) + (3 × 103) + (5 × 102) + (4 × 101) + (8 × 100) Example 2: 0.568 = 0.5 + 0.06 + 0.008 = (5 × 10-1) + (6 × 10-2) + (8 × 10-3) Write each number in expanded form. 57. 5931 61.

58. 29,607

59. 0.173

60. 0.5875

Writing in Math Use the information about negative exponents on page 209 to explain how they represent repeated division. Illustrate your reasoning with an example of a power containing a negative exponent written in fraction form.


62. How many square centimeters does one square millimeter equal? (Hint: 1 cm = 10 mm) A 10-1 B 10-2

£ V

C 10-3 D

63. A nurse draws a sample of blood. A cubic millimeter of the blood contains 223 white blood cells and 225 red blood cells. Compare the number of white blood cells to the number of red blood cells as a fraction.

£ V

103

10,648 F _

1 H_

484 G_

1 J _

1

484

1

10,648

ALGEBRA Find each product or quotient. Express using exponents. (Lesson 4-5) 64. 36 · 3

65. x2 · x4 3

9

y 67. _2

55 66. _ 2

16n 68. ALGEBRA Write _ in simplest form. (Lesson 4-4)

y

5

8n

69. CARPENTRY Danielle is helping her father make shelves to store her sports equipment in the garage. How many shelves measuring 12 inches by 16 inches can be cut from a 48-inch by 72-inch piece of plywood so that there is no waste? (Lesson 4-3) 70. KEYBOARDING Keyboarding speed can be determined by using the formula w - 10r where s represents the speed of words typed per minute, w s=_ m represents the number of words typed, r represents the number of errors, and m represents the total number of minutes typed. If Esteban received a keyboard speed of 80 words per minute and typed 530 words in 6 minutes, how many errors did he make? (Lesson 3-8) ALGEBRA Use the Distributive Property to rewrite each expression. (Lesson 3-1) 71. 8(y + 6)

72. -5(a - 10)

73. (9 + k)(-2)

SCHOOL For Exercises 75 and 76, use the table that shows the heights and grade point averages of the students in Mrs. Stanley’s class. (Lesson 1-7) 75. Make a scatter plot of the data. 76. Does there appear to be a relationship between the scores and the heights? Explain.

74. (n - 3)5 Name

Height (in.)

GPA

Regina

66

3.6

Michael

61

3.2

Latisha

59

3.9

Simon

64

2.8

Maurice

61

3.8

Timothy

65

3.1

Ivan

70

2.6

Helen

64

2.2

Eduardo

65

4.0

PREREQUISITE SKILL Find each product. (Pages 747–748) 77. 7.2 × 100

78. 1.6 × 1000

79. 4.05 × 10

80. 0.05 × 1000

81. 3.8 × 0.01

82. 5.0 × 0.0001

83. 9.24 × 0.1

84. 11.64 × 0.001

Lesson 4-6 Negative Exponents

213

4-7

Scientific Notation

Main Ideas • Express numbers in standard form and in scientific notation. • Compare and order numbers written in scientific notation.

New Vocabulary standard form scientific notation

A compact disc or CD has a single spiral track that stores data. It circles from the inside of the disc to the outside. If the track were stretched out in a straight line, it would be 0.5 micron wide and over 5000 meters long. a. Write the track length in millimeters.

Track Length

Track Width

5000 meters

0.5 micron

b. Write the track width in millimeters. (1 micron = 0.001 millimeter)

Scientific Notation Numbers like 5,000,000 and 0.0005 are in standard form because they do not contain exponents. However, when you deal with very large numbers like 5,000,000 or very small numbers like 0.0005, it is difficult to keep track of the place value. Numbers such as these can be written in scientific notation. Scientific Notation Words

A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10.

Symbols

a × 10n, where 1 ≤ a < 10 and n is an integer

Examples 5,000,000 = 5.0 × 106

Powers of Ten To multiply by a power of 10, • move the decimal point to the right if the exponent is positive, and • move the decimal point to the left if the exponent is negative. In each case, the exponent tells you how many places to move the decimal point.

EXAMPLE

Express Numbers in Standard Form

Express each number in standard form. a. 3.78 × 106 3.78 × 106 = 3.78 × 1,000,000 106 = 1,000,000 = 3,780,000 Move the decimal point 6 places to the right. b. 5.1 × 10-5 5.1 × 10-5 = 5.1 × 0.00001 10-5 = 0.00001 = 0.000051 Move the decimal point 5 places to the left. 1A. 5.94 × 107

214 Chapter 4 Factors and Fractions Getty Images

0.0005 = 5.0 × 10-4

1B. 1.3 × 10-3 Extra Examples at pre-alg.com

EXAMPLE

Express Numbers in Scientific Notation

Express each number in scientific notation. Positive and Negative Exponents When the number is 1 or greater, the exponent is positive. When the number is between 0 and 1, the exponent is negative.

a. 60,000,000 60,000,000 = 6.0 × 10,000,000 The decimal point moves 7 places. = 6.0 × 107

The exponent is positive.

b. 0.0049 0.0049 = 4.9 × 0.001 The decimal point moves 3 places. = 4.9 × 10-3

The exponent is negative.

2A. 32,800

2B. 0.000064

SPACE The table shows the objects in space and their distances from the Sun. Light travels 300,000 kilometers per second. Estimate how long it takes light to travel fromthe Sun to Pluto. 109

kilometers Explore It is 5.90 × from the Sun to Pluto, and the speed of light is 300,000 kilometers per second.

Plan

Use the equation d = rt. To estimate, round 5.90 × 109 to 6.0 × 109. Write 300,000 as 3.0 × 105.

Solve 6.0 ×

Calculator To enter a number in scientific notation on a calculator, enter the decimal portion, press [EE] then enter the exponent. A calculator in Sci mode will display answers in scientific notation. For example, the number 1.0 × 1010 is displayed as 1E10 on the calculator.

d = rt 109 ≈

Earth

1.55 ⴛ 108

Jupiter

7.78 ⴛ 108

Mars

2.28 ⴛ 108

Mercury

5.80 ⴛ 107

Neptune

4.50 ⴛ 109

Pluto

5.90 ⴛ 109

Saturn

1.43 ⴛ 109

Uranus

2.87 ⴛ 109

Venus

1.03 ⴛ 108


Write the formula.

(3.0 ×

105)t 105)t

Replace d with 6.0 × 109 and r with 3.0 × 105.

(3.0 × 6.0 × 109 _ ≈_ 5 5

Divide each side by 3.0 × 105.

6.0 × 109 _ ≈t 5

Simplify.

3.0 × 10

Distance from the Sun (km)

Object

3.0 × 10

3.0 × 10

You can use a calculator to find the quotient. 6.0

[EE] 9 ⫼ 3.0

[EE] 5 ENTER 20000

So, it would take 20,000 (or 2.0 × 104) seconds, or about 51 hours. 2

Check

You can divide each part of the number in scientific notation. 6.0 × 109 6.0 109 _ =_ ×_ 3.0 × 105

3.0

105

= 2.0 × 104 The answer is reasonable.

3. SPACE Estimate how long it takes light to travel from the Sun to Mercury. Personal Tutor at pre-alg.com Lesson 4-7 Scientific Notation

215

Compare and Order Numbers To compare and order numbers in scientific notation, first compare the exponents. With positive numbers, any number with a greater exponent is greater. If the exponents are the same, compare the factors.

SPACE Refer to the table in Example 3. Order Mars, Jupiter, Mercury, and Saturn from least to greatest distance from the Sun. First, order the numbers according to their exponents. Then, order the numbers with the same exponent by comparing the factors. Jupiter and Mars

Step 2

Saturn

1.43 × 109 ⎭

7.78 × 108 2.28 × 108
°Ã°V


HOMEWORK

HELP

For See Exercises Examples 9–17 1 18–26 2 27, 28 3 29, 30 4

Express each number in standard form. 9. 4.24 × 102

10. 5.72 × 104

11. 3.347 × 10-1

12. 5.689 × 10-3

13. 1.5 × 10-4

14. 9.01 × 10-2

15. 1.399 × 105

16. 2.505 × 103

17. 6.1 × 104

Express each number in scientific notation. 18. 2,000,000

19. 499,000

20. 0.006

21. 0.0125

22. 50,000,000

23. 39,560

24. 0.000078

25. 0.000425

26. 5,894,000

27. SPACE Refer to the table in Example 3 on page 215. To the nearest second, about how long does it take light to travel from the Sun to Venus? 28. TRAFFIC In a recent year, route U.S. 59 in the Houston metropolitan area averaged approximately 338,510 vehicles per day. About how many vehicles was this during the entire year? Write the number in scientific notation. 29. OCEANS Rank the oceans in the table at the right by area from least to greatest.

Real-World Link In 2000, the International Hydrographic Organization named a fifth world ocean near Antarctica, called the Southern Ocean. It is larger than the Arctic Ocean and smaller than the Indian Ocean. Source: geography.about.com

Ocean

Area (sq mi)

Arctic

5.44 × 106

Atlantic

3.18 × 107

30. MEASUREMENT The table below shows the Indian 2.89 × 107 values of different prefixes that are used Pacific 6.40 × 107 in the metric system. Write the units attometer, gigameter, kilometer, nanometer, petameter, and picometer in order from greatest to least measure. Prefix Meaning

atto

giga

kilo

nano

peta

pico

10-18

109

103

10-9

1015

10-12

Order each set of numbers from least to greatest. 31. -3.14 × 102, -3.14 × 10-2, 3.14 × 102, 3.14 × 10-2 32. 2.81 × 104, 2805, 2.08 × 105, 3.2 × 104, 3024 33. 9,562,301, 9.05 × 10-6, 9.5 × 106, 905,000 ANALYZE GRAPHS For Exercises 34–36, use the graph. The graph shows the weights of the five heaviest marine and land mammals on Earth in pounds.

i>ÛiÃÌ >>Ã vÀV> i«>Ì Õi 7>i

£°{{ £ä{ L Ó°nÇ £äx L {

EXTRA

PRACTICE


7>i °Ó £ä L 34. Rank the animals in order ,}Ì 7>i n°nÓ £ä{ L from heaviest to lightest. 7Ìi ,ViÀÃ Ç°{ £äÎ L 35. About how many times heavier is the Blue Whale -ÕÀVi\ "OOK OF 7ORLD 2ECORDS than the African Elephant? 36. Estimate the combined weight of the Fin Whale, Right Whale, and White Rhinoceros. Write the weight in scientific notation and in standard form.

Lesson 4-7 Scientific Notation Robert Fried

217

Convert the numbers in each expression to scientific notation. Then evaluate the expression. Express in scientific notation and in standard notation. (420,000)(0.015) 38. __

20,000 37. _

0.025

0.01

H.O.T. Problems

(0.078)(8.5) 39. __ (0.16)(250,000)

40. OPEN ENDED Write a number in standard form and then write the number in scientific notation, explaining each step that you used. NUMBER SENSE Los Angeles is the second largest city in the United States. 41. Which number better describes the population of Los Angeles, 3.8 × 104 or 3.8 × 106? 42. What are some other ways to express Los Angeles’ population? 43. Which form of the number is best to use when describing population? Explain. 44. CHALLENGE In standard form, 3.14 × 10-4 = 0.000314, and 3.14 × 104 = 31,400. What is 3.14 × 100 in standard form? 45.

Writing in Math Explain how scientific notation is an important tool in comparing real-world data. Illustrate your answer with an example of real-world data that is written in scientific notation, and the advantages of using scientific notation to compare data.

46. If you wrote the areas of the bodies of water in the table from least to greatest, which would be third in the list? Body of Water

Area (km2)

Lake Huron Lake Victoria Red Sea Great Salt Lake

5.7 × 104 6.9 × 104 4.4 × 105 4.7 × 103

47. GRIDDABLE The weight of a fruit fly is about 1.3 × 10-4 pound. How many pounds would one million fruit flies weigh? 48. The distance from Earth to the Sun is about 9.6 × 107 miles. Which of the following represents this distance in standard notation?


A Lake Huron

C Red Sea

B Lake Victoria

D Great Salt Lake

F 9,600,000 mi

H 960,000,000 mi

G 96,000,000 mi

J

9,600,000,000 mi

ALGEBRA Evaluate each expression if s = -2 and t = 3. (Lesson 4-6) 49. t-4

50. s-5

51. 7s

52. st

ALGEBRA Find each product or quotient. Express using exponents. (Lesson 4-5) 53. 44 · 47

54. 3a2 · 5a2

55. c5 ÷ c2

57. Write ten million as a power of ten. (Lesson 4-1)

36d6 56. 4 12d

58. BUSINESS An online bookstore adds a $2.50 shipping and handling charge to the total price of every order. If the cost of books in an order is c, write an expression for the total cost. (Lesson 1-3) 218 Chapter 4 Factors and Fractions

CH

APTER

4




&ACTORS AN D &RACTIONS W BEL

Key Concepts Exponents

(Lesson 4-1)

• An exponent is a shorthand way of writing repeated multiplication. • Follow the order of operations to evaluate algebraic expressions containing exponents.

Prime and Composite Numbers

(Lesson 4-2)

algebraic fraction (p. 198) base (p. 180) composite number (p. 186) exponent (p. 180) factor (pp. 180 and 188) greatest common factor (GCF) (p. 191) monomial (p. 188) power (p. 180) prime factorization (p. 187) prime number (p. 186) scientific notation (p. 214) standard form (p. 214) Venn diagram (p. 191)

• A prime number is a whole number that has exactly two factors, 1 and itself. • A composite number is a whole number that has more than two factors.

Factors and Factoring

(Lesson 4-3)

• The greatest number or monomial that is a factor of two or more numbers or monomials is their greatest common factor, or GCF. • The Distributive Property can be used to factor algebraic expressions. • Algebraic fractions can be written in simplest form by dividing the numerator and the denominator by the GCF. • Powers with the same base can be multiplied by adding their exponents. Powers with the same base can be divided by subtracting their exponents.

Negative Exponents and Scientific Notation (Lessons 4-6 and 4-7)

Determine whether each statement is true or false. If false, replace the underlined word or number to make a true statement. 1. The exponent of a number raised to the first power can be omitted. 2. Numbers expressed using exponents are called powers. 3. The number 7 is a factor of 49 because it can divide into 49 with a remainder of zero. 4. A monomial is a number, a variable, or a sum of numbers and/or variables. 5. The number 64 is a composite number. 6. A number is in scientific notation when it does not contain exponents.

1 • For a ≠ 0 and any whole number n, a-n n. a

• A number in scientific notation is the product of a number between 1 and 10 and a power of 10.


Vocabulary Check

7. A fraction is in simplest form when the GCF of the numerator and the denominator is 2.


219

CH

A PT ER

4



Powers and Exponents

(pp. 180–184)

Write each expression using exponents. 8. 6 · 6 · 6 · 6 · 6 10. x · x · x

9. 4 11. f · f · g · g · g · g

Evaluate each expression if x = -3, y = 4, and z = -2. 12. 33

13. (-5)2

14. 2(3z + 4)5

15. x2z4

Example 1 Write a · a · b · b · b · b · b using exponents. Group the factors with like bases. Then, write using exponents. a · a · b · b · b · b · b = (a · a) · (b · b · b · b · b) = a2b5

Example 2 Evaluate 4(a + 2)3 if a = -5. 16. E-MAIL Suppose Theo sends an e-mail to three of his friends. Each of his three friends forwards the e-mail to three of their friends. Each of those friends forwards it to three friends, and so on. Find the number of e-mails sent during the fifth stage as a power. Then find the value of the power.

4–2

Prime Factorization

4(a + 2)3 = 4(-5 + 2)3

Replace a with -5.

= 4(-3)3

Simplify the expression inside the parentheses.

= 4(-27)

Evaluate (-3)3.

= -108

Simplify.

(pp. 186–190)


18. 55

19. 68

20. 200

Example 3 Write the prime factorization of 40. Use exponents for repeated factors. 40 4

Factor each monomial. 21. 18x

22. 10e2

23. 32pq

24. -25ab2

25. PHOTOGRAPHY Jacy picked out 24 photographs to put into a frame in a rectangular arrangement. How many different numbers of rows and columns can she display them in if each row has the same number of photographs? Name each arrangement.


· 10

40 = 4 · 10

2 · 2 · 2 · 5 4 = 2 · 2 and 10 = 2 · 5

The prime factorization of 40 is 2 · 2 · 2 · 5 or 23 · 5.

Example 4 Factor 9s3t2. 9s3t2 = 3 · 3 · s3 · t2

9=3·3

= 3 · 3 · s · s · s · t · t s3 · t2 = s · s · s · t · t



4–3

Greatest Common Factor (GCF)

(pp. 191–195)

Find the GCF of each set of numbers or monomials.

Example 5 Find the GCF of 12a2 and 15ab.

26. 6, 48

12a2 = 2 · 2 · 3 ·

28. 4n,

5n2

27. 16, 24 29.

20c3d,

12cd

31. 3x + 24

32. 30 - 4n

33. 45s + 25

34. 14r - 30

35. 64 - 60k

Example 6 Factor 4n + 8. Step 1 Find the GCF of 4n and 8. 4n = 2 · 2 · n 8 = 2 · 2 · 2 The GCF is 2 · 2 or 4.

36. DESIGN An architect is designing two seating sections for an auditorium. One section will contain 860 seats, and the other will contain 1000 seats. Both sections will have the same number of seats per row. What is the greatest number of seats in each row?

4–4


38. 24 40

39. 15 16 28w 41. 38w2 9mn 43. 18n2

40. 21 30 23x 42. 32y

Step 2 Write the product of the GCF and its remaining factors. 4n + 8 = 4(n) + 4(2) = 4(n + 2)

Rewrite using the GCF. Distributive Property

(pp. 196–200)

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 6 37. 21

3 ·5· a ·b

The GCF of 12a2 and 15ab is 3 · a or 3a.

Factor each expression. 30. 2t + 20

15ab =

a ·a

2 44. 15ac 24ab

45. TRAVEL Of the 267 students in the freshman class, 89 of them take the bus to school. What fraction of the freshman class takes the bus to school?

. Example 7 Simplify 36 60 1 1 1

36 = 2 · 2 · 3 · 3 Divide the numerator and the 60 2 · 2 · 3 · 5 denominator by the GCF, 2 · 2 · 3 or 12. 1

1

1

= 35

Simplify.

17q2 34qr

Example 8 Simplify . 1

1

17 · q · q 17q2 = Divide the numerator and the 17 · 2 · q · r denominator by the GCF, 17 · q. 34qr 1

q = 2r

1

Simplify.


221

CH

A PT ER

4


4–5

Multiplying and Dividing Monomials

(pp. 203–207)


7 47. 32

48. 7x · 2x6

49. k6 ÷ k5

3

50. LIFE SCIENCE Starting from a single bacterium, the number of bacteria after t cycles of reproduction is 2t. A bacteria reproduces every 30 minutes. If there are 1000 bacteria in a dish now, how many will there be in 1 hour?

4–6

Negative Exponents

5

Example 10 Find 43.

4 45 = 45 - 3 The common base is 4. 3 4 = 42 Subtract the exponents.

(pp. 209–213)


Example 11 Write 3-4 as an expression using a positive exponent.

51. 7-2

1 Definition of negative exponent 3-4 = 4

52. b-4

53. (-4)-3

Write each fraction as an expression using a negative exponent other than -1. 1 54. 3 6

1 55. 64

1 56. 125

57. DISTANCE If 1 millimeter is equal to 10-3 meter and 1 nanometer is equal to 10-9 meter, how many nanometers are in 1 millimeter?

4–7

Example 9 Find x3 · x2. x3 · x2 = x3 + 2 The common base is x. = x5 Add the exponents.

Scientific Notation

3

1 as an expression Example 12 Write 32 using a negative exponent other than -1. 1 1 = Find the prime factorization of 32. 32 2 · 2 · 2 · 2 · 2 1 = 5

Definition of exponent.

= 2-5

Definition of negative exponent.

2

(pp. 214–218)

58. 6.1 × 102

59. 2.9 × 10-3

Example 13 Express 3.5 × 10-2 in standard form.

60. 1.85 × 10-2

61. 7.045 × 104

3.5 × 10-2 = 3.5 × 0.01 10-2 = 0.01

Express each number in standard form.

= 0 035 Express each number in scientific notation.

Move the decimal point 2 places to the left.

62. 1200

63. 0.008

Example 14 Express 269,000 in scientific notation.

64. 0.000319

65. 45,710,000

269,000 = 2.69 × 100,000 The decimal point moves 5 places.

66. SPACE The mass of the Sun is 1.98892 × 1015 exagrams. Express this number in standard form. 222 Chapter 4 Factors and Fractions

= 2.69 ×

105

The exponent is positive.

CH

A PT ER

4

Practice Test

Write each expression using exponents.


1. 3 · 3 · 3 · 3 2. b · b · b · b · b 3. -2 · -2 · -2 · a · a · a · a 4. ALGEBRA Is the value of 2n - 1 prime or composite if n = 5? 5. MULTIPLE CHOICE When the United States had 48 states, the stars on the flag were in a 6 × 8 rectangular arrangement. Which rectangular arrangement of the 48 stars would NOT be possible?

16. 4-2

17. 10-10

18. t-6

19. (yz)-3

20. MEASUREMENT How many square centimeters is equivalent to one square millimeter? Write as an expression with a positive exponent. Write each number in standard form.

A 2 × 24

22. 5.206 × 10-3 24. 7.29 × 103

21. 3.71 × 104 23. 3.4 × 10-5

B 3 × 16 C 4 × 12 D 5 × 10

Write each number in scientific notation. 25. 0.09 27. 50,300

Factor each expression. 6. 12r2 8. 7 + 21p

7. 50xy2 9. 24c - 10

10. MULTIPLE CHOICE Eighty fluid ounces is what part of 1 gallon? (Hint: There are 128 fluid ounces in 1 gallon.) 3 F 10

H 5 8

G 2 5

J 3 4

11.

·

56

12.

(4x7)(-6x3)

9 14. w 5

13. k · k5

w

15. MULTIPLE CHOICE Which expression represents the area of the square? A 3ab2 B

3a2b2

C 9ab2 D 9a2b2


ANALYZE GRAPHS For Exercises 29 and 30, use the graph. The graph shows the maximum amounts of lava in cubic meters per second that erupted from six volcanoes in the last century. Eruption Rates

Find each product or quotient. Express using exponents. 53

26. 1,068,300 28. 0.008

ÎAB

Mount St. Helens, 1980 Ngauruhoe, 1975 Hekla, 1970 Agung, 1963 Bezymianny, 1956 Hekla, 1947 Santa Maria, 1902

2.0 104 2.0 103 4.0 103 3.0 104 2.0 105 2.0 104 4.0 104

Source: University of Alaska

ÎAB

29. Rank the volcanoes in order from greatest to least eruption rate. 30. How many times greater was the Santa Maria eruption than the Mount St. Helens eruptions?


223

CH

A PT ER


4

Cumulative, Chapters 1–4


6. In the sequence below, which expression can be used to find the value of the term in the nth position?

1. Which coordinates are most likely to be the coordinates of point P? 20

P

y

10

⫺20 ⫺10 O

10

20 x

⫺10 ⫺20

A (-13, 7)

C ( 13, 7)

B (7, -13)

D (7, 13)

2. A certain bacterium has a diameter of 0.0000000025 centimeter. How is this length expressed in scientific notation? × 10-8 cm

F 2.5

× 109 cm

H 2.5

G 2.5

× 108 cm

J 2.5 × 10-9 cm

3. The shipping charge from an Internet bookstore includes a base fee of $5 plus $3 for each item purchased. Which equation represents the shipping charge for n items? A S = 5(3n)

3 C S=5+ n

B S = 3n - 5

D S = 5 + 3n

4. Suppose you paid for a DVD with a $20 bill. You received 3 dollars, 3 dimes, and 2 pennies in change. How much did you pay for the DVD? F $16.68

H $17.68

G $16.88

J $17.88

5. Express 0.0000000102 in scientific notation. A 1.02 × 10-9

C 1.02 × 10-7

B 1.02 × 10-8

D 1.02 × 10-6


Position

Term

1

0.5

2

1.5

3

2.5

4

3.5

5

4.5

n

?

F n - 0.5

H 2n

n G 2

n J 4

7. GRIDDABLE The low temperatures during the past five days are given in the table. Find the average (mean) of the temperatures. Day Temperature (˚F)

1

2

3

4

5

-2

0

4

5

4

8. Which table of values represents the following rule? Add the input number to the square of the input number. A

C Input (x)

Output (y)

Input (x)

Output (y)

0

1

1

2

2

3

2

6

4

5

4

20

B

D Input (x)

Output (y)

Input (x)

Output (y)

1

1

1

2

2

6

2

4

4

8

4

8

9. GRIDDABLE A bus traveled 185 miles at an average speed of 60 miles per hour. About how many hours did it take for the bus to reach its destination? Round your answer to the nearest tenth. Standardized Test Practice at pre-alg.com


10. The rectangle below is 8 feet long and 5 feet wide.

13. Suppose that the weight of a certain breed of puppy at 3 months is 1.3 times its weight at 2 months. Given x, the weight of a puppy at 2 months, which equation can be used to find y, the weight of the puppy at 3 months? A y = 1.3 + x C y = 1.3x B y = x - 1.3 D y = 1.3 ÷ x

x vÌ n vÌ

If the dimensions of the figure are multiplied by 3, by what factor will the area increase? F 3 H 15 G9 J 40

Pre-AP Record your answers on a sheet of paper. Show your work. 14. Chandra plans to order CDs from an Internet shopping site. She finds that the CD prices are the same at three different sites, but that the shipping costs vary. The shipping costs include a fee per order, plus an additional fee for each item in the order, as shown in the table below.

Question 10 Most standardized tests include any necessary formulas in the test booklet. It helps to be familiar with formulas such as the area of a rectangle, but use any formulas that are given to you.

Shipping Cost

11. Sydney spends 20 minutes traveling to and from work everyday. What fraction of the day is this? 1 A

Company

72 1 B 12 2 C 3 5 D 6

Per Order

Per Item

CDBargains

$4.00

$1.00

WebShopper

$6.00

$3.00

EverythingStore

$2.50

$1.50

a. For each company, write an equation that represents the shipping cost. In each of your three equations, use S to represent shipping cost and n to represent the number of items purchased.

12. The distance from Earth to Mars averages about 2.28 108 kilometers. Which of the following represents this number in standard notation? F 228,000,000 km G 22,800,000 km H 2,280,000 km J 228,000 km

b. If Chandra orders 2 CDs, which company will charge the least for shipping? Use the equations you wrote and show your work. c. If Chandra orders 10 CDs, which company will charge the least for shipping? Use the equations you wrote and show your work.


1

2

3

4

5

6

7

8

9

10

11

12

13

14

Go to Lesson...

1-6

4-7

3-6

1-1

4-7

3-7

2-5

1-6

3-8

3-8

4-5

4-7

3-4

3-6


225

Rational Numbers

5 •

Understand that different forms of numbers are appropriate for different situations.

•


•

Uses statistical procedures to describe data.

Key Vocabulary common multiples (p. 257) least common denominator (p. 258) measures of central tendency (p. 274)

reciprocals (p. 245)

Real-World Link Hurricanes A hurricane can be measured by winds greater than 74 miles per hour, a storm surge greater than 4 feet, and barometric pressure less than 28.94 inches.

Applying Rational Numbers Make this Foldable to help you record information about rational numbers. 1 Begin with three sheets of 8_” × 11” paper. 2

1 Fold the first two sheets in half from top to bottom. Cut along the fold from edges to margin.

3 Insert the first two sheets through the third sheet and align the folds.

226 Chapter 5 Rational Numbers Getty Images

2 Fold the third sheet in half from top to bottom. Cut along the fold from margin to edge.

4 Label each page with a lesson number and title.

>«ÌiÀ x\ ,>Ì> ÕLiÀÃ



Option 1 Take the Quick Check below. Refer to the Quick Review for help.

Find each quotient. Round to the nearest tenth, if necessary. (Lesson 2-5) 1. 3 ÷ 5 2. -1 ÷ 8 3. 2 ÷ 17 4. -12 ÷ 3 5. -2 ÷ (-9) 6. 4 ÷ (-15) 7. -24 ÷ 14 8. -72 ÷ (-9)

Example 1

Find 3 ÷ 8. Round to the nearest tenth. 3 ÷ 8 = 0.375 ≈ 0.4

Find the quotient. Round to the nearest tenth.

9. LANDSCAPING A hedge of roses is 8.25 meters long. Suppose bricks each 0.25 meter long will make a border along one side. How many bricks are needed to make the border? (Prerequisite Skills, pp. 749–750)

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 4-4) 5 12 14 10. _ 11. _ 12. _ 40 36 13. _ 50

20 27 14. _ 54

39 32 15. _ 85

16. SURVEY Ten of the 25 students in math class have blue eyes. What fraction of the students in math class do not have blue eyes? Write in simplest form. (Lesson 4-4)

Example 2 14 Write _ in simplest form. 35

Find the GCF of 14 and 35. factors of 14: 1, 2, 7, 14 factors of 35: 1, 5, 7, 35 The GCF of 14 and 35 is 7. 14 14 ÷ 7 _ = _ 35 35 ÷ 7 2 = _

Simplest form

5

Find each sum or difference. (Lesson 2-2) 17. 4 + (-9) 18. -10 + 16 19. (-3) + (-8) 20. -1 - (-10) 21. SCUBA DIVING A scuba diver descends 21 feet below the surface of the water. She then ascends 14 feet. Find an integer that represents the scuba diver’s position in relation to the surface of the water.

Divide the numerator and the denominator by the GCF.

Example 3

Find 14 - 18. 14 - 18 = 14 + (-18) = -4

To subtract 18, add -18. Simplify.

(Lesson 2-2) Chapter 5 Get Ready for Chapter 5

227

5-1

Writing Fractions as Decimals

Main Ideas • Write fractions as terminating or repeating decimals. • Compare fractions and decimals.

New Vocabulary terminating decimal mixed number repeating decimal bar notation

In the 18th century, a silver dollar contained $1 worth of silver. The sizes of all other coins were based on this coin.

Coin

Fraction of Silver of $1 Coin

quarter-dollar (quarter)

1 4

10-cent (dime)

1 10

half-dime* (nickel)

1 20

a. A half dollar contained half the silver of a silver dollar. What was it worth? b. Write the decimal value of each coin in the table. c. Order the fractions in the table from least to greatest. (Hint: Use the values of the coins.)

* In 1866, nickels were enlarged for convenience

Write Fractions as Decimals Any fraction _a , where b ≠ 0, can be b

written as a decimal by dividing the numerator by the denominator. So, _a = a ÷ b. If the division ends, or terminates, when the remainder b is zero, the decimal is a terminating decimal.

EXAMPLE

Write a Fraction as a Terminating Decimal

3 Write _ as a decimal. 8

Vocabulary Link Terminating Everyday Use bringing to an end Math Use a decimal whose digits end

Method 1 Use paper and pencil.

Method 2 Use a calculator.

0.375 3.000 8 -2.4 60 -56 40 -40 0

3 ⫼ 8 ENTER 0.375

_3 = 0.375 8

Division ends when the remainder is 0.

0.375 is a terminating decimal.

Write each fraction as a decimal. 4 1A. _ 5

3 1B. _ 16

1 is the sum of a whole number and a fraction. A mixed number such as 3_ 2 Mixed numbers can also be written as decimals.

228 Chapter 5 Rational Numbers

EXAMPLE

Write a Mixed Number as a Decimal

1 Write 3_ as a decimal. 2

Mental Math It will be helpful to memorize the following list of fraction-decimal equivalents. 1 2 = 0.5 1 3 = 0.3 1 4 = 0.25 1 5 = 0.2 2 3 = 0.6

3 4 = 0.75 2 5 = 0.4 3 5 = 0.6

4 5 = 0.8

1 1 3_ =3+_ 2

2

Write as the sum of an integer and a fraction.

= 3 + 0.5 12 = 0.5 = 3.5

Add.

Write each mixed number as a decimal. 1 2A. 2_

3 2B. 4_

4

4

Not all fractions can be written as terminating decimals.

2→ 3

CHECK 2 ⫼ 3

The number 6 repeats.

0.666 3 2.000 -1 8 20 -18 20 -18 2

The remainder after each step is 2.

.6666666667 The last digit is rounded.

So, 2 = 0.6666666666… . This decimal is called a repeating decimal. 3 Repeating decimals have a pattern in their digits that repeats without end. You can use bar notation to indicate that a digit or group of digits repeats. 6 The digit 6 repeats, so place a bar over the 6. 0.6666666666… = 0. The table shows three examples of repeating decimals.

EXAMPLE

Decimal

Bar Notation

0.13131313. . . 6.855555. . .

0. 13 6.85

19.1724724. . .

19.1 724

Write Fractions as Repeating Decimals

Write each fraction as a decimal. Use a bar to show a repeating decimal. 2 b. _

6 a. -_

11 0.5454. . . 6 → - 11 11 6.0000. . . 6 So, -_ = -0. 54. 11

7 3A. _ 9


The digits 54 repeat.

15 2 → 15

0.1333. . . The digit 3 15 2.0000. . . repeats.

2 So, _ = 0.1 3. 15

5 3B. -_ 6

Lesson 5-1 Writing Fractions as Decimals

229

GOLF During the 2005 Masters Tournament, Tiger Woods’ first shot landed on the fairway 32 of 56 times. To the nearest thousandth, what part of the time did his shot land on the fairway? Divide the number of fairways on which he landed, 32, by the total number of fairways, 56. 32 4 _ =_ ≈ 0.571428… or 0. 571428 56

7

Look at the digit to the right of the thousandths place. Round down since 4 < 5. Tiger Woods landed on the fairway 0.571 of the time.

4. SOFTBALL The United States women’s softball team had 73 hits out of a total of 213 at bats in the final round of the 2004 Olympics. To the nearest thousandth, what part of the time did the team have a hit in the final round?

Compare Fractions and Decimals It may be easier to compare numbers when they are written as decimals. Real-World Link Tiger Woods became the youngest golfer to win The Masters golf tournament at the age of 21 years 3 months and 14 days when he won in 1997. Source: masters.org

EXAMPLE

Compare Fractions and Decimals

Replace each a. 35


0.75 3 5

0.75

Write the sentence.

0.6

0.75

Write 35 as a decimal.

0.6 < 0.75

In the tenths place, 6 < 7.

0.6

0.75

0.5 0.55 0.6 0.65 0.7 0.75 0.8

3 On a number line, 0.6 is to the left of 0.75, so _ < 0.75. 5 b. -_ 8

5

6 -_ 9

Write the fractions as decimals and then compare the decimals. 5 6 -_ = -0.625 -_ = -0.6 8

9

On a number line, -0.625 is to the right of -0. 6, so -0.625 is 5 6 6. The inequality is -_ > -_ . greater than -0. 8

7 5A. _ 8

230 Chapter 5 Rational Numbers Mike Blake/REUTERS/Landov

0.87

9

7 5B. -_ 15

5 -_ 12

13 17 BREAKFAST In a survey of students, _ of the boys and _ of the girls 20 25 make their own breakfast. Of those surveyed, do a greater fraction of boys or girls make their own breakfast?

Write the fractions as decimals and then compare the decimals. 13 = 0.65 boys: _

0.65

20 17 girls: _ = 0.68 25

0.68

0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70

13 17 °®

ä°Îä

16 1 48. -5_ 3

2 47. -2.2 -2_ 7

Real-World Link After being caught, a marlin can strip more than 300 feet of line from a fishing reel in less than 5 seconds. Source: Incredible Comparisons

ä°Ón

ä°Ó£

ä°Óä ä°£È

ä°£x

ä°£x

ä°£Î

ä°£ä ä°äx

25 3 -5_ 10

>Þ `

>Þ ÕÀ

À

Ã`

Ã` /

8 -_

>Þ

Þ `>

5 46. -_

40. 6.18

ä

99

-2.09

12

4 1 6_ 5

ä°Óx

1 45. -2_

1 -_

,>v> ÕÀ} x >Þ *iÀ`

Replace each with , or = to make a true sentence. 34 7 43. -0.75 -_ 44. _ 0.3 4 9

2

6 36. -_ 25

i

1 38. 1_ 20

0. 5

1 1 35. -_ -_ 8 10 39. 34 3. 4 9

10 _ 14

_1

32. 0.3

0.4

5

>Þ

_8 9

2 31. _

0.65

Ã`

7 33. _ 8 1 37. _ 5

8 5 34. _ 7

7 i`

5 30. _

1 29. 0.3 _ 4

/Õ i

Replace each

>Þ

49. AUTOMOBILES Of all the cars sold in the United States in 2003, 2 were 5 imported from Japan and 0.26 were imported from Germany. Are more Japanese or German cars sold in the United States? Explain. Order each group of numbers from least to greatest. 7 7 50. _ , 0.8, _ 8

2 3 , -_ 51. -0.29, -

11 7 1 1 53. -1. 1, -1_, -1_ 8 10

9

3 2 , 2.67, 2_ 52. 2_ 5 3

54. ANALYZE GRAPHS Find a fraction or mixed number that might represent each point on the graph at the right.

!" ä°x

# $ £

% £°x

Ó

ANIMALS For Exercises 55 and 56, use the following information. 5 mile in one minute. A marlin can swim _ EXTRA PRACTICE

6

5 55. Write _ as a decimal rounded to the nearest hundredth. 6

See pages 770, 798.

56. Which form of the number is best to use? Explain.


57. OPEN ENDED Give one example each of real-world situations where it is most appropriate to give a response in fractional form and in decimal form.

232 Chapter 5 Rational Numbers Burton McNeely/Getty Images

H.O.T. Problems

58. NUMBER SENSE Find a terminating and a repeating decimal between 1 and 8. Explain how you found them. 6

9

59. CHALLENGE Write the prime factorization of each denominator and the decimal equivalent of each fraction. Then explain how prime factors of denominators and the decimal equivalents of fractions are related. 1 _ 1 _ _1 , _1 , _1 , _1 , _1 , _1 , _1 , _ , 1,_ , 1 2 3 4 5 6 8 9 10 12 15 20

60. SELECT A TECHNIQUE Luke is making lasagna that calls for 4 pound of 5 mozzarella cheese. The store only has packages that contain 0.75- and 0.85-pound of mozzarella cheese. Which of the following techniques might Luke use to determine which package to buy? Justify your selection(s). Then use the technique(s) to solve the problem. mental math

61.

Writing in Math

number sense

estimation

Explain how 0.5 and 0. 5 are different. Which is

greater?

62. Which decimal represents the shaded portion of the figure? A 0.6

C 0.63

B 0. 6

D 0.6 3

7 is found between which 63. The fraction _ 9 pair of fractions on a number line? 3 3 and _ F _

5 4 7 4 G _ and _ 5 10

3 7 H _ and _

J

10 4 3 2 _ and _ 5 3

Write each number in scientific notation. (Lesson 4-7) 64. 854,000,000

65. 0.077

66. 0.00016

67. 925,000

Write each expression using a positive exponent. (Lesson 4-6) 68. 10-5

69. (-2)-7

70. x-4

71. y -3

72. ALGEBRA Write (a · a · a)(a · a) using an exponent. (Lesson 4-1) 73. TRANSPORTATION A car travels an average of 464 miles on one tank of gas. If the tank holds 16 gallons, how many miles per gallon does it get? (Lesson 3-8) 74. SCUBA DIVING A scuba diver descends from the surface of the lake at a rate of 6 meters per minute. Where will the diver be in relation to the lake’s surface after 4 minutes? (Lesson 2-4)

PREREQUISITE SKILL Simplify each fraction. (Lesson 4-4) 4 75. 30

5 76. 65

77. 36 60

78. 12 18

79. 21 24

80. 16 28

81. 32 48

125 82. 1000 Lesson 5-1 Writing Fractions as Decimals

233

5-2

Rational Numbers

Main Ideas • Write rational numbers as fractions. • Identify and classify rational numbers.

New Vocabulary rational number

Animation pre-alg.com

The solution of 2x = 4 is 2. It is a member of the set of natural numbers. N = {1, 2, 3, …} The solution of x + 3 = 3 is 0. It is a member of the set of whole numbers. W = {0, 1, 2, 3, …}

7HOLE .UMBERS .ATURAL .UMBERS .ATURAL .UMBERS

The solution of x + 5 = 2 is -3. It is a member of the set of integers. I = { …, -3, -2, -1, 0, 1, 2, 3, …}

)NTEGERS 7HOLE .UMBERS

The solution of 2x = 3 is 3, which is neither a 2 natural number, a whole number, nor an integer. It is a member of the set of rational numbers.

.ATURAL .UMBERS

Rational numbers include fractions and decimals as well as whole numbers and integers. a. Is 7 a natural number? a whole number? an integer?

,>Ì>

b. How do you know that 7 is also a rational number?

ÕLiÀÃ

)NTEGERS 7HOLE .UMBERS

c. Is every whole number a rational number? Is every rational number a whole number? Give an explanation or a counterexample to support your answers.

.ATURAL .UMBERS

Reading Math

Write Rational Numbers as Fractions A number that can be written as

Ratios Rational comes from the word ratio. A ratio is the comparison of two quantities by division. Recall that _a = a ÷ b, where

a fraction is called a rational number. Some examples of rational numbers are shown below. 3 28 5 1 1 0.75 = _ -0.3 = – _ 28 = _ 1_ =_

b

b ≠ 0.

4

EXAMPLE

3

1

4

Write Mixed Numbers and Integers as Fractions

Write each rational number as a fraction. 2 a. 5 _

3 17 _ 52 =_ 3 3

3 1A. 4_ 4


4

b. -3 Write 5 23 as an improper fraction.

-3 3 -3 = _ or - _ 1

1B. 7

1

Terminating decimals are rational numbers because they can be written as a fraction with a denominator of 10, 100, 1000, and so on.

EXAMPLE

Write Terminating Decimals as Fractions

Simplify. The GCF of 48 and 100 is 4.

1000 3 = 6_ 8

6.375 is 6 and 375 thousandths. Simplify. The GCF of 375 and 1000 is 125.

2A. 0.56

hu nd red ths tho us an dth ten s -th ou sa nd ths

0 4 8

b. 6.375 375 6.375 = 6_

ten ths

0.48 is 48 hundredths.

ten ths

100 12 =_ 25

on es

48 0.48 = _

on es

a. 0.48

hu nd red ths tho us an dth ten s -th ou sa nd ths

Write each decimal as a fraction or mixed number in simplest form. tho us an ds hu nd red s ten s

Decimal Point Use the word and to represent the decimal point. • Read 0.375 as three hundred seventy-five thousandths. • Read 300.075 as three hundred and seventy-five thousandths.

tho us an ds hu nd red s ten s

Reading Math

6 3

5

2B. 5.875

Any repeating decimal can be written as a fraction, so repeating decimals are also rational numbers.

EXAMPLE Repeating Decimals When two digits repeat, multiply each side by 100. Then subtract N from 100N to eliminate the repeating part.

Write Repeating Decimals as Fractions

Write 0. 8 as a fraction in simplest form. N = 0.888…

Let N represent the number.

10N = 10(0.888…) Multiply each side by 10 10N = 8.888…

because one digit repeats.

Subtract N from 10N to eliminate the repeating part, 0.888… . 10N = 8.888… -(N = 0.888…) 9N = 8

10N - N = 10N - 1N or 9N

9N 8 _ =_


9

9 8 N=_ 9

CHECK

Simplify.

8 ⫼ 9 ENTER .8888888889

3. Write 0. 3 as a fraction in simplest form. Personal Tutor at pre-alg.com Lesson 5-2 Rational Numbers

235

READING in the Content Area For strategies in reading this Lesson, visit pre-alg.com.

Identify And Classify Rational Numbers All rational numbers can be written as terminating or repeating decimals. Decimals that are neither terminating nor repeating, such as the numbers below, are called irrational. You will learn more about irrational numbers in Chapter 9.

= 3.141592654…

→ The digits does not repeat.

4.232232223…

→ The same block of digits does not repeat.

Rational Numbers A rational number is any number that can be expressed a as the quotient _ of two b integers, a and b, where b ≠ 0.

Words

Model Rational Numbers 1.8 0.7

EXAMPLE

2

45 Integers ⫺12 ⫺5 Whole 2 Numbers 3 6 0 15 2 ⫺3.2222...

Classify Numbers

Identify all sets to which each number belongs. a. -6 -6 is an integer and a rational number. 4 b. 2 _ 5

4 14 =_ , it is rational. It is neither a whole number nor an integer. Because 2_ 5

5

c. 0.914114111… This is a nonterminating, nonrepeating decimal. So, it is not rational.

6 4A. -_ 9

Example 1 (p. 234)

4B. 1.414213562…

4C. 0

Write each number as a fraction. 1 1. -2 _ 3

5 2. 1 _ 6

3. 10

(p. 235)

4. MEASUREMENT A micron is a unit of measure that is approximately 0.000039 inch. Express this as a fraction.

Examples 2, 3

Write each decimal as a fraction or mixed number in simplest form.

Example 2

(p. 235)

5. 0.8 8. -0. 7

Example 4 (p. 236)

6. 6.35 9. 0.45

7. 3.16 10. 0.06

Identify all sets to which each number belongs. 11. -5


12. 6.05

13. 0. 1 Extra Examples at pre-alg.com

HOMEWORK

HELP


Write each number as a fraction. 2 14. 4_

4 15. -1_

3

16. -21

7

17. 60

Write each decimal as a fraction or mixed number in simplest form. 18. 0.4

19. 0.09

20. 5.22

21. 1.68

22. 3.625

23. 8.004

24. WHITE HOUSE The White House covers an area of 0.028 square mile. What fraction of a square mile is this? 25. RECYCLING Use the information at the left to find the fraction of all recycled newspapers that were used to make tissues in 2004. Write each decimal as a fraction or mixed number in simplest form. 27. -0.333… 28. 4. 5 26. 0.2 29. 5. 6

30. 0. 32

31. 2. 05

Identify all sets to which each number belongs. 32. 4

33. -7

5 34. -2_

6 35. _

36. 15.8

37. 9.0202020…

38. 1.7345…

39. 30.151151115…

8

3

40. TRACK AND FIELD During the women’s 100-meter final in the 2004 Olympics, the eight finalists finished within twenty-five hundredths of a second of each other. Write this number as a fraction in simplest form. Real-World Link The portions of recycled newspapers used for other purposes are shown below. Newsprint: 0.31 Exported for Recycling: 0.28 Paperboard: 0.13 Tissues: 0.08 Other products: 0.18 Source: American Forest and Paper Association, Newspaper Association of America

41. ANALYZE TABLES The city of Heath makes 1 of the population in Rockwall County. up _ 10 Use the table to find the fraction of Rockwall County’s population that lives in other cities. Write each fraction in simplest form.

Replace each 42. -0.23

PRACTIICE

Decimal Part of Rockwall County’s Population

Fate

0.018

McLendonChisholm Rockwall Royse City

0.02 0.42 0.07

with , or = to make a true statement.

-0.3

1 45. -1_ -0.9 11

EXTRA

City

43. 8 9

0.888…

5 4_ 46. 4.63 8

44. 0.714

_5

47. -5. 3

5.333…

7

3 -inch 48. MACHINERY Will a steel peg 2.37 inches in diameter fit in a 2 _ 8 diameter hole? How do you know?


H.O.T. Problems

49. GEOMETRY Pi () to six decimal places has a value of 3.141592. Pi is often . Is the estimate for greater than or less than the actual estimated as 22 7 value of ? Explain. 50. OPEN ENDED Give an example of a number that is not rational. Explain why it is not rational. Lesson 5-2 Rational Numbers

Lester Lefkowitz/CORBIS

237

51. CHALLENGE Show that 0.999… = 1. REASONING Determine whether each statement is sometimes, always, or never true. Explain by giving an example or a counterexample. 52. An integer is a rational number. 53. A rational number is an integer. 54. A whole number is not a rational number. 55.

Writing in Math

Explain how rational numbers are related to other sets of numbers. Illustrate your reasoning with examples of numbers that belong to more than one set and examples of numbers that are only rational.

56. There are infinitely many between S and T on the number line. S

57. Which fraction is between 0.12 and 0.15? 3 F _

T

25

1 G_ 8

⫺2⫺1 0 1 2 3 4 5 6 7 8

A rational numbers

3 H_ 20 1 J _ 5

B integers C whole numbers D negatives

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. (Lesson 5-1) 2 58. _

4 59. -7 _

5

5

13 60. -_

5 61. 2_

64. 7.4 × 10-4

65. 1.681 × 10-2

9

20

Write each number in standard form. (Lesson 4-7)

62. 2 × 103

63. 3.05 × 106 2

12n 66. ALGEBRA Write _ in simplest form. (Lesson 4-4) 3an

Find the perimeter and area of each rectangle. (Lesson 3-8)

67.

68. {°x V

7 in. V

16 in.

69. COIN COLLECTING Jada has 156 coins in her collection. This is 12 more than 8 times the number of nickels in the collection. How many nickels does Jada have in her collection? (Lesson 3-6)

PREREQUISITE SKILL Estimate each product. (page 752) 2 1 70. 1_ · 4_ 3

8

1 4 71. -5_ · 3_


3

5

1 1 72. 2_ · 2_ 4

9

9 7 73. 6_ · 1_ 8

10

5-3

Multiplying Rational Numbers

Main Ideas • Multiply positive and negative fractions. • Use dimensional analysis to solve problems.

New Vocabulary

To find 2 · 3, use an area model

3 4

3 4 2 to find of 3. 3 4

2 3

a. The overlapping green area 2 represents the product of _

3 3 and _. What is the product? 4

dimensional analysis

Draw a rectangle and shade three fourths of it yellow.

Then shade two thirds of the rectangle blue.

Use an area model or another manipulative to model each product. Explain how the model shows the product. 3 1 c. _ · _

1 _ b. _ ·1 2

5

3

3 1 d. _ · _ 4

4

3

e. What is the relationship between the numerators and denominators of the factors and the numerator and denominator of the product?

Multiply Fractions This model suggests a rule for multiplying fractions. Multiplying Fractions To multiply fractions, multiply the numerators and multiply the denominators.

Words Symbols

a _ a·c _ ·c=_ , where b, d ≠ 0 b

d

b·d

1 2 2 1·2 Example _ · _ = _ or _ 3 5 15 3·5

EXAMPLE

Multiply Fractions

2 _ Find _ · 3 . Write the product in simplest form. 3

Review Vocabulary GCF (greatest common factor) the greatest number that is a factor of two or more numbers; Example: for 12 and 20, the GCF is 4. (Lesson 4-4)

4

←Multiply the numerators. ←Multiply the denominators. 3·4 6 1 =_ or _ Simplify. The GCF of 6 and 12 is 6. 2 12

2·3 _2 · _3 = _ 3

4

Find each product. Write in simplest form. 1 _ 1A. _ · 4 2

10

5 _ 1B. _ · 6 12

10

Lesson 5-3 Multiplying Rational Numbers

239

If the fractions have common factors in the numerators and denominators, you can simplify before you multiply.

EXAMPLE

Multiply Negative Fractions

5 _ Find -_ · 3 . Write the product in simplest form. Negative Fractions

12 8 1 5 _ 5 _ -_ · 3 = -_ ·3 12 8 12 8

Divide 3 and 12 by their GCF, 3.

4

5 can be written as - 12

-5 · 1 =_

Multiply the numerators and multiply the denominators.

4·8 5 = -_ 32

-5 5 or as . 12 -12

Simplify.

Find each product. Write in simplest form. 3 _ 6 3 · 9 2B. _ · -_ 2A. _ 4

9

12

EXAMPLE

11

Multiply Mixed Numbers

Find 125 · 21. Write the product in simplest form. Estimate 1 · 3 = 3 2

12 1 7 _ _ · 2_ =_ ·5 5

2

5

Rename 125 as 75 and rename 212 as 52 .

2

1

7 _ =_ ·5

Estimation

5

You can justify your answer by using estimation.

7·1 =_

1·2 7 1 = _ or 3_ 2 2

• 125 is close to 1. • 212 is close to 3. So, 125 · 212 ≈ 1 · 3 or 3.

Divide by the GCF, 5.

2

1

Multiply. Simplify.

Find each product. Write in simplest form. 3 5 1 1 3A. 3_ · 2_ 3B. -1_ · 4_ 8

3

6

8

ROLLER COASTERS The first drop on one roller coaster at a theme park is 255 feet. The first drop on another roller coaster at the park is about 11 _ as high. Find the height of the drop on the second roller coaster. 20

11 . To find the height of the drop on the second roller coaster, multiply 255 by _ 20 255 _ 11 255 255 · _ =_ · 11 Rename 255 as _ . 20

1

1

20

51

255 _ =_ · 11 1

20


4

51 · 11 =_

1·4 561 1 _ = or 140_ 4 4


Multiply. Simplify. The height of the drop is about 140 feet. Extra Examples at pre-alg.com

4. SKYSCRAPERS The Sears Tower in Chicago is about 1450 feet. The Empire State Building in New York City is about 4 as tall. About how tall is the 5 Empire State Building? Algebraic fractions are multiplied in the same manner as numeric fractions.

EXAMPLE

Multiply Algebraic Fractions

b2

2a _ Find _ · . Write the product in simplest form. b

d

1

b2

2a _ _ b·b _ · = 2a · _ b

d

d

b 1

2ab =_

The GCF of b and b2 is b. Simplify.

d

Find each product. Write in simplest form. x2 _ · z 5A. _ 3y

2 7 _ 5B. _ · rs 2

2x

r

10

Dimensional Analysis Dimensional analysis is the process of including units of measurement as factors when you compute. You can use dimensional analysis to check whether your answers are reasonable.

SPACE TRAVEL The landing speed of the space shuttle is about 216 miles per hour. How far does the shuttle travel in 13 hour during landing? Words

equals the

Variable

multiplied by the

Let

Equation

.

·

=

216 miles _ d=_ · 1 hour distance = rate · time 1 hour

3

72

216 miles _ =_ · 1 hour

1 hour

3

Divide out the common factors and units.

1

Simplify. The space shuttle travels 72 miles in _ hour 3 during landing. 1 Multiplying by _ is the same as dividing by 3. 3 1 216 · = 216 ÷ 3 3 1

= 72 miles CHECK

= 72

6. SPEED RECORD The record for the fastest land car speed is about 760 miles 1 hour? per hour. How far would the car travel in _ Personal Tutor at pre-alg.com

4

Lesson 5-3 Multiplying Rational Numbers

241

Examples 1– 3 (pp. 239–240)

Find each product. Use an area model if necessary. 1 _ 1. _ ·3

4 5 1 _ 4. 3 · _ 7 6 1 _ 7. _ -5 2 6 1 2 10. 3 _ · -_ 4 11

Example 4 (pp. 240–241)

Example 5 (p. 241)

3 5 2 _ 5. 5 · _ 10 9 2 _ 8. -_ -1 3 6 3 1 11. -5 _ · -3 _ 3 8

3 _ 3. _ ·1 8

4

4 _ 6. _ ·5 5 8 6 _ 9. -_ ·1

10 8 1 2 12. -2 _ · 5_ 2 3

13. GEOGRAPHY “Midway” is the name of 252 towns in the United States. “Pleasant Hill” occurs 5 as many times. How many towns named “Pleasant 9 Hill” are there in the United States? ALGEBRA Find each product. Write in simplest form. 2 _ · 3x 14. _ x

7

5b 15. _a · _ b

4t _ 16. _ · 18r 2

c

t

9r

(p. 241)

1 17. TRAVEL A car travels 65 miles per hour for 3_ hours. What is the distance 2 traveled? Use the formula d = rt to solve the problem and show how you can divide by the common units.

HELP

Find each product. Use an area model if necessary.

Example 6

HOMEWORK

1 _ 2. _ ·2

For See Exercises Examples 18–27 1–2 28–35 3 36, 37 4 38–43 5 44, 45 6

6 _ 18. -_ ·2

7 7 3 _ ·3 21. -_ 4 5 2 _ ·5 24. _ 5 6

4 _ 19. _ ·2

1 1 20. _ -_

9 3 5 _ 22. _ · 8 9 25 8 _ 25. _ · 27 9 28

5

8

1 2 23. -_ -_ 2

7

3 1 26. _ -_ 4

3

27. -7 · 2 8 5

7 28. 2 ·

6 29. (-3) 15

2 _ 30. 6 _ ·1

5 1 31. -_ · 3_

1 2 32. 2_ · 6_

3 2 5 1 · 2_ 33. 3 _ 3 8

12

9 12 2 1 34. -6_ -1_ 3 2

3 7 3 4 35. 1_ -9 _ 7 5

36. BREAD The average person living in Slovakia consumes about 320 pounds of bread per year. The average person living in the United States consumes 1 as much. How many pounds of bread does the average American about _ 5 consume every year? 37. BRIDGES The Golden Gate Bridge in San Francisco is 4200 feet long. 19 as long. How long is the The Brooklyn Bridge in New York City is _ 50 Brooklyn Bridge? ALGEBRA Find each product. Write in simplest form. 4a _ ·3 38. _

5 a 2 8 _ 41. _ · c c 11


3x _ 9y 39. _ y · x n _ 42. _ · 64 18

n

12 _ 40. _ · 3k

4 jk 2 x _ 43. _ · 2z 2z 3

44. HYBRID CARS Hybrid cars can get up to 52 miles per gallon of gas. How far 3 gallon of gas? can the car travel on _ 4

2 1 45. LAWN CARE Dexter’s lawn is _ of an acre. If 7_ bags of fertilizer are needed 3 2 for 1 acre, how much will he need to fertilize his lawn?

Real-World Link An improvement of 5 miles per gallon in fuel economy saves 55 million metric tons of carbon emissions per day. Source: hybridcars.com

ANALYZE TABLES For Exercises 46–48, use the table that shows statistics from the last election for class &RACTION OF STUDENT BODY THAT VOTED ? president. ? &RACTION OF VOTES FOR (ECTOR 46. What fraction of the student ? body voted for Hector? &RACTION OF VOTES FOR .ORA 47. What fraction of the student body voted for Nora? 48. Was there another candidate for class president? How do you know? Explain your reasoning. If there was another candidate, what fraction of the student body voted for this person? 49. FILMS The table shows the number of sports films created with different themes. Which theme occurs 5 as many times as boxing? 12

Sport Theme Boxing Horse Racing Football Baseball

3 4 50. ALGEBRA Evaluate (xy)2 if x = _ and y = -_ . 4

5

Films 204 139 123 85

Source: Top 10 of Everything

MEASUREMENT Complete. 5 51. ? feet = _ mile

3 52. ? ounces = _ pound

6

8

(Hint: 1 mile = 5280 feet)

(Hint: 1 pound = 16 ounces)

2 hour = ? minutes 53. _

3 54. _ yard = ? inches

3

4

CONVERTING MEASURES Use dimensional analysis and the fractions in the table to find each missing measure. EXTRA

PRACTICE


? cm 55. 5 in. = _____

56. 10 km = _____ ? mi

57. 26.3 cm = _____ ? in.

2 2 58. 8_ ft = _____ ? m2

Customary→ Metric

3 3 60. _ cm = _____ ? in. 4

59. 72 m2 = _____ ? ft2 61. 130.5 mi = _____ ? km

H.O.T. Problems

Conversion Factors Metric→ Customary

2.54 cm 1 in. 1.61 km 1 mi

0.39 in. 1 cm 0.62 mi 1 km

0.09 m2 1 ft2

10.76 ft2 1 m2

62. 130.5 km = _____ ? mi

63. OPEN ENDED Choose two rational numbers whose product is a number between 0 and 1. 5 _ · 18 . Who is correct? 64. FIND THE ERROR Terrence and Marie are finding _ 24 25 Explain your reasoning.

Terrence 1

5

3

18

Marie 3

· 25 = 24 20

4

5

1

9

5

18

4

5

9 · = 24 25 20

Lesson 5-3 Multiplying Rational Numbers Ford Motor Company

243

CHALLENGE Use the digits 3, 4, 5, 6, 8, or 9 no more than once to make true sentences. 6 □ _ 65. _ × □ =_ □

67.

□

5 □ _ 66. _ × □ =_ □

5

□

8

Writing in Math Use the information about fractions on page 239 to explain how multiplying fractions is related to areas of rectangles. Illustrate your reasoning with an area model.

68. What is the equivalent length of a chain that is 52 feet long?

8 3 69. The product of _ and _ is a 8 15 . number

A 4 yd 5 ft

F between 0 and 1

B 4.5 yd

G between 1 and 2

C 17 yd 1 ft

H between 2 and 3

D 17.1 yd

J greater than 3

Write each decimal as a fraction or mixed number in simplest form. (Lesson 5-2)

70. 0.18

71. -0.2

73. 0. 7

72. 3.04

1 74. FOOD In an online survey, about _ of teenagers go to sleep between 9 and 4 13 _ of teenagers go to sleep at 12 A.M. or later. Which group is 10 P.M., while 50 larger? (Lesson 5-1)

Express each situation with a number in scientific notation. (Lesson 4-7) 75. The number of possible ways that a player can play the first four moves in a chess game is 3 billion. 76. A particle of dust floating in the air weighs 0.000000753 gram.

77. ALGEBRA What is the product of x2 and x4? (Lesson 4-5) GEOMETRY Find the perimeter and area of each rectangle. (Lesson 3-8) 78.

79. 5 in.

3.5 m 12 in.

4.9 m

ALGEBRA Solve each equation. Check your solution. (Lesson 3-5) 80. 2x - 1 = 9

81. 14 = 8 + 3n

k 82. 7 + _ = -1 5

83. ALGEBRA Simplify 4(y + 2) - y. (Lesson 3-2)

PREREQUISITE SKILL Find the GCF of each pair of monomials. (Lesson 4-3) 84. 8n, 16n

85. 5ab, 8b


86. 9k, 27

87. 4p2, 6p

5-4

Main Ideas

Dividing Rational Numbers

1 1 The model shows 4 ÷ _ . Each of the 4 circles is divided into _ -sections. 3

• Divide positive and negative fractions using multiplicative inverses. • Use dimensional analysis to solve problems.

New Vocabulary multiplicative inverses reciprocals

3

1 2

4 3

5

7 6

8

10 9

11

12

1 1 = 12. Another way to find the There are twelve _ -sections, so 4 ÷ _ 3 3 number of sections is to multiply 4 × 3 = 12.

Use a circle model or another manipulative to model each quotient. Explain how the model shows the quotient. 1 a. 2 ÷ _

1 b. 4 ÷ _

3

1 c. 3 ÷ _

2

4

d. MAKE A CONJECTURE Write about how dividing by a fraction is related to multiplying.

Reading Math Synonyms Multiplicative inverse and reciprocal are different terms for the same concept. They may be used interchangeably.

Divide Fractions Rational numbers have all of the properties of integers. 3 1 _ · = 1. Two numbers whose product Another property is shown by _ 3

1

is 1 are called multiplicative inverses or reciprocals. Inverse Property of Multiplication Words

The product of a number and its multiplicative inverse is 1.

Symbols

a For every number _ , where a, b ≠ 0, there is exactly one b

a _ b b _ number _ a such that b · a = 1.

EXAMPLE

Find Multiplicative Inverses

Find multiplicative inverse of each number. 1 b. 2 _

3 a. - _

8 8 _ =1 - 3 -_ 8 3

( )

5

The product is 1.

The multiplicative inverse 3 8 or reciprocal of -_ is -_ . 8 3

7 1A. -_ 9

1 _ 2_ = 11

5 5 5 11 _ _ · =1 5 11

Write as an improper fraction. The product is 1.

1 _ The reciprocal of 2_ is 5 . 5

11

1 1B. 6 _ 3

Lesson 5-4 Dividing Rational Numbers

245

1 Dividing by 2 is the same as multiplying by _ , 2 its multiplicative inverse. This is true for any rational number.

reciprocals

1 6·_ =3

6÷2=3

2

same result

Dividing Fractions Words

To divide by a fraction, multiply by its multiplicative inverse.

Symbols

a a d c _ ÷_ =_·_ , where b, c, d ≠ 0

Example

1 7 5 1 _ _ ÷_ =_ · 7 or _

d

b 4

b

7

EXAMPLE

4

c

5

20

Divide by a Fraction or Whole Number

Find each quotient. Write in simplest form. Dividing By a Whole Number When dividing by a whole number, always rename it as an improper fraction first. Then multiply by its reciprocal.

5 1 _ a. _ ÷

5 b. _ ÷6

3 9 _1 ÷ _5 = _1 · _9 3 9 3 5 3 1 _ =_ ·9 3 5

8

_5 ÷ 6 = _5 ÷ _6

Multiply by the 5 9 reciprocal of _, _.

6 Write 6 as _. 1 8 1 5 _ 1 Multiply by the _ = · 6 1 8 6 reciprocal of _1 , _6 . 5 =_ Multiply. 48

8

9 5


1

3 =_

Simplify.

5

3

5 3 2B. _ ÷ -_

1 _ 7 2A. _ ÷

8

15

3 2C. _ ÷ 11

6 2D. -_ ÷ 12

4

4

7

To divide by a mixed number, you can rewrite the divisor as an improper fraction.

EXAMPLE

Divide by a Mixed Number

1 1 Find -7_ ÷ 2_. Write the quotient in simplest form. 2 10 1 1 15 _ 21 ÷ -7_ ÷ 2_ = -_ 2 2 10 10 15 10 = -_ · _ 2 21 5

5

2

21

Rename the mixed numbers as improper fractions. Multiply by the multiplicative inverse of _, _. 21 10 10 21

15 _ = -_ · 10 1

Divide out common factors.

7

25 4 = -_ or -3_ 7

7

Simplify.

Find each quotient. Write in simplest form.

3 1 3A. 6_ ÷ -4_ 8


4

4 2 3B. -6_ ÷ -2_ 5

5


You can divide algebraic fractions in the same way that you divide numerical fractions.

EXAMPLE

Divide by an Algebraic Fraction

3xy 2x Find _ ÷ _ . Write the quotient in simplest form. 8 4 3xy 3xy 8 2x 8 2x _ _ ÷ _ = _ · _ Multiply by the multiplicative inverse of _ , . 8 2x 4 8 2x 4 1

3xy 4

2

8 =_·_ 1

2x


1

6y = _ or 3y 2

Simplify.

Find each quotient. Write in simplest form. mn2 m2n 4B. _ ÷ _

5ab _ 10b 4A. _ ÷ 6

4

7

8

Dimensional Analysis Dimensional analysis is a useful way to examine the solution of division problems.

Real-World Link Organized cheerleading is over a hundred years old. In November of 1898, John Campbell led the crowd at the University of Minnesota football game in the first-ever organized cheer. Source: Official Cheerleader Handbook

CHEERLEADING How many cheerleading uniforms can be made with 3 7 74 _ yards of fabric if each uniform requires 2_ yards? 8 4 3 7 To find how many uniforms, divide 74_ by 2_ . Think: How many 2 _7 s are in 74_3 ? 3 299 23 7 74_ ÷ 2_ = _ ÷ _ 8

4

8 4 299 _ 8 _ = · 4 23 13

2

4

23

299 _ =_ · 8 1

= 26

8

8

4

4

Write 74_ and 2_ as improper fractions. 3 4

7 8

Multiply by the reciprocal of _, _. 23 8 8 23


1

Simplify.

So, 26 uniforms can be made.

CHECK

Use dimensional analysis to examine the units. yards uniforms

uniforms yards

yards ÷ _ = yards · _ Divide out the units. = uniforms

Simplify.

The result is expressed as uniforms.

3 2 5. BREAKFAST A box of cereal contains 15_ ounces. If a bowl holds 2_ ounces 5 5 of cereal, how many bowls of cereal are in one box? Personal Tutor at pre-alg.com Lesson 5-4 Dividing Rational Numbers Tony Anderson/Getty Images

247

Example 1 (p. 245)

Find the multiplicative inverse of each number. 4 1. _

Examples 2– 3 (p. 246)

(p. 247)

Example 5 (p. 247)

HOMEWORK

HELP

For See Exercises Examples 17–22 1 23–36 2 37–40 3 41–46 4 47, 48 5

8

Find each quotient. Use an area model if necessary.

5 2 5. -_ ÷ -_

6 1 _ 4. _ ÷

2 7 7 7. _ ÷ (-14) 9 8 1 ÷ 3_ 10. -_ 5 9

Example 4

1 3. 3 _

2. -16

5

3

6

4 8. _ ÷ (-2) 5 1 1 11. 2_ ÷ -1_ 6 5

8 4 _ 6. -_ ÷

9 5 1 9. 7_ ÷ 5 3 2 1 12. -5 _ ÷ 2_ 7 7

ALGEBRA Find each quotient. Write in simplest form. 14 _ 1 13. _ n ÷n

x2 _ ax 15. _ ÷

ab _ b 14. _ ÷ 6

4

5

2

16. CARPENTRY How many boards, each 2 feet 8 inches long, can be cut from a board 16 feet long if there is no waste?

Find the multiplicative inverse of each number. 6 17. _

1 18. -_

19. -7

20. 24

1 21. 5_

2 22. -3 _

11

5

9

4

Find each quotient. Use an area model if necessary. 3 1 _ 23. _ ÷

2 _ 1 24. _ ÷

5 1 _ 25. -_ ÷

6 4 ÷ -_ 26. _

8 _ 4 27. _ ÷

7 _ 14 28. _ ÷

3 _ 3 ÷ 29. _

2 2 30. _ ÷ -_

3 5 31. -_ ÷ -_

3 1 ÷ -_ 32. -_

4 33. 12 ÷ _

4 34. -8 ÷ _

5 ÷ (-4) 35. -_

2 36. 6_ ÷5

3 5 37. 3_ ÷ 1_

5

4

( 5)

11 4

9 9

1 1 ÷ -1_ 38. 7_

(

( 9)

5

)

9

5

6

10

1 2 39. -6_ ÷ 3_ 9

15

5

3

8

2

8

9

5

6

2

3

9

4

10

4

3

3 2 40. -10_ ÷ -2_

(

5

5

)

ALGEBRA Find each quotient. Write in simplest form. a a 41. _ ÷_ 7

EXTRA

PRACTIICE


42 5s _ 6rs 44. _ ÷ t t

10 _ 5 42. _ ÷ 3x

cd 43. _c ÷ _

2x

8

k3

2s _ st3 46. _ ÷ 2

k 45. _ ÷ _ 9

5

24

t

8

3 1 47. FOOD How many _ -pound hamburgers can be made from 2_ pounds of 4 4 ground beef? 1 cups of sugar. How many batches 48. COOKING A batch of cookies requires 1_ 2

1 cups of sugar? of cookies can be made from 7_ 2


ALGEBRA Evaluate each expression. 8 7 49. m ÷ n if m = -_ and n = _

3 1 50. r2 ÷ s2 if r = -_ and s = 1_

18

9

3

4

1 hours and earned $19.50. What was 51. BABY-SITTING Barbara baby-sat for 3_ 4 her hourly rate?

H.O.T. Problems

52. OPEN ENDED Write a division expression that can be simplified by using 7 . the multiplicative inverse of _ 5

3 1 _ 1 1 by _ , 1, _ , and _ . What happens to the quotient as 53. CHALLENGE Divide _ 4

2 4 8

12

the value of the divisor decreases? Make a conjecture about the quotient 3 by fractions that increase in value. Test your conjecture. when you divide _ 4

54.

Writing in Math

Explain how dividing by a fraction is related to multiplying. Illustrate your reasoning by including a model of a whole number divided by a fraction.

1 A promotional poster is printed on 16-inch by 24 _ -inch 2 posterboard and the space between the three sections and both top and bottom of the poster are shown.

55. If the total length of the three sections is 18 1 inches, 2 how long are each of the three equal sections? 1 in. A 2_ 3

5 C 4_ in. 8

4 B 3_ in.

1 D 6_ in.

15

IN

IN

IN

IN

6

1 inches, what is the 56. GRIDDABLE If the width of one section is 14 _ 4 area of one section? Round your answer to the nearest hundredth.

Find each product. Write in simplest form. (Lesson 5-3) 3 _ 57. _ ·1 5

3

15 2 58. -_ · -_ 9

16

4 _ 59. -2_ ·3 5

8

5 1 60. -_ · 1_ 12

7

Identify all sets to which each number belongs. (Lesson 5-2)

61. 16

62. -2.8888 …

63. 0. 8

64. 5.121221222 …

65. COMPUTERS In a survey, 17 students out of 20 said they use a computer as a reference source for school. Write 17 out of 20 as a decimal. (Lesson 5-1)

PREREQUISITE SKILL Write each improper fraction as a mixed number in simplest form. (Lesson 4-4) 9 66. _ 4

8 67. _ 7

17 68. _ 2

24 69. _ 5

Lesson 5-4 Dividing Rational Numbers

249

5-5

Adding and Subtracting Like Fractions

Main Ideas • Add like fractions. • Subtract like fractions.

Measures of different parts of an insect are shown. The sum of the parts is _6 inch. Use a ruler to find each 8 measure. 3 1 in. + _ in. a. _

5 in. 8

3 4 b. _ in. + _ in.

8 8 4 4 _ _ c. in. + in. 8 8

1 in. 8

6 in. 8

8 8 6 _ _ d. in. - 3 in. 8 8

Add Like Fractions Fractions with the same denominator are called like fractions.

Adding Like Fractions Words

To add fractions with like denominators, add the numerators and write the sum over the denominator.

a+b a b _ _ Symbols _ c + c = c , where c ≠ 0

EXAMPLE

1+2 3 1 2 Example _ + _ = _ or _ 5

5

5

5

Add Fractions

Find each sum. Write in simplest form. 3 5 +_ a. _

7 7 3+5 3 _ + _5 = _ 7 7 7 8 1 = _ or 1_ 7 7

Estimate 0 + 1 = 1 The denominators are the same. Add the numerators. Simplify and rename as a mixed number.

Compared to the estimate, the answer is reasonable. 5 -7 + _ b. _

( ) ( )

8 8 5 + (-7) 5 -7 _+ _ =_ 8 8 8 -2 1 = _ or -_ 8 4

1 1 Estimate _ + (-1) = -_ 2

2

The denominators are the same. Add the numerators. Simplify.

Compare your answer to the estimate. Is it reasonable?

5 4 1A. _ +_ 6

250 Chapter 5 Rational Numbers John Cancalosi/Stock Boston

6

6 4 1B. _ + -_ 7

( 7) Extra Examples at pre-alg.com

EXAMPLE

Add Mixed Numbers

5 1 Find 6 _ + 1_ . Write the sum in simplest form. Estimate 7 + 1 = 8 Alternative Method You can also stack the mixed numbers vertically to find the sum. 6_

8 8 5 5 1 1 6_ + 1_ = (6 + 1) + _ +_ 8 8 8 8 5+1 _ =7+ 8 6 3 _ = 7 or 7_ 8 4

(

)

Add the whole numbers and fractions separately. Add the numerators. Simplify. Compared to the estimate, the answer is reasonable.

5 8

_1

+1 8 _

3 6 7 _ or 7 _ 8 4

Find each sum. Write in simplest form. 3 1 4 1 2A. 3_ + 7_ 2B. 6 _ + 9_ 5

5

10

10

Subtract Like Fractions The rule for subtracting fractions with like denominators is similar to the rule for addition. Subtracting Like Fractions To subtract fractions with like denominators, subtract the numerators and write the difference over the denominator.

Words Symbols

a-b a b _ - _ = _, where c ≠ 0 c

c

EXAMPLE

5 5-1 1 4 Example _ - _ = _ or _

c

7

7

7

7

Subtract Fractions

9 13 Find _ -_ . Write the difference in simplest form. 20 20 9 - 13 9 13 _-_=_ 20 20 20 -4 1 = _ or -_ 5 20

1 1 Estimate _ - 1 = -_ 2

2

The denominators are the same. Subtract the numerators. Simplify.

Find each difference. Write in simplest form. 5 10 3 6 3A. _ -_ 3B. _ -_ 15

EXAMPLE Alternative Method You can check your answer by subtracting the whole numbers and fractions separately. 2 9_ - 5_ = (9 - 5) 6 6 2 1 + _ -_ 1

(6 6) 1 = 4 + (-_) 6

= 3_ 5 6

9

15

9

Subtract Mixed Numbers

1 2 Evaluate a - b if a = 9 _ and b = 5 _ . 6

1 2 a - b = 9_ - 5_

6 6 55 32 _ _ = 6 6 23 5 _ = or 3 _ 6 6

6

Estimate 9 - 5 = 4

2 Replace a with 9_ and b with 5_ . 1 6

6

Write the mixed numbers as improper fractions. Subtract the numerators. Simplify.

3 7 4. Evaluate x - y if x = 5 _ and y = 9 _ . 8

8

Personal Tutor at pre-alg.com Lesson 5-5 Adding and Subtracting Like Fractions

251

You can use the same rules for adding or subtracting like algebraic fractions as you did for adding or subtracting like numerical fractions.

EXAMPLE

Add Algebraic Fractions

n 5n Find _ +_ . Write the sum in simplest form. 8 8 n + 5n 5n n _+_=_ 8 8 8 6n 3n = _ or _ 8 4

The denominators are the same. Add the numerators. Add the numerators. Simplify.

Find each sum. Write in simplest form.

y 5y 5B. _ + _

4d 2d 5A. _ +_ 10

Examples 1–3 (pp. 250–251)

8

Find each sum or difference. Write in simplest form. 5 1 1. _ +_

9

10

10

2 4 3. _ + -_

3 3 2. _ +_

7

7

9

3 1 + 8_ 4. 2_

4 2 5. -2_ + -_

3 5 6. 3_ + 6_

3 11 -_ 7. _

5 1 8. -_ -_

4 10 9. _ - _

6

6

5

14

14

Example 4

8

10

8

5

12

8

8

8

12

1 inches tall at the end of school in June. He 10. MEASUREMENT Hoai was 62_ 8

7 inches tall in September. How much did he grow during the was 63_

(p. 251)

8

summer? 3 1 and v = 6_ . ALGEBRA Evaluate each expression if u = 7_ 7

11. u - v Example 5 (p. 252)

ALGEBRA Find each sum or difference. Write in simplest form. 6r 2r 13. _ + _ 11

HOMEWORK

HELP

For See Exercises Examples 15–18 1 19–22 2 23–26 3 27–36 4 37–40 5

7

12. v - u 19 12 _ 14. _ a - a ,a≠0

11

Find each sum or difference. Write in simplest form. 2 1 15. _ +_ 5

5

13 9 18. -_ + -_

21. 24. 27. 30.

16 16 5 7 5_ + 3_ 9 9 5 17 _ -_ 18 18 3 5 2_ - 1_ 8 8 6 5 -8_ - -2_ 11 11


3 3 17. -_ + -_

19.

20.

22. 25. 28.

3 7 16. _ +_

31.

10 10 2 2 7_ + 4_ 5 5 5 7 2_ + 2_ 12 12 1 7 _ - -_ 12 12 9 1 8_ - 6_ 10 10 5 3 -4_ -_ 8 8

23. 26. 29. 32.

4 4 9 17 5_ + 5_ 20 20 10 8 _ -_ 11 11 9 7 _ - -_ 20 20 5 4 7_ - 2_ 7 7 3 2 12_ - 13 _ 6 6

8 1 11 ALGEBRA Evaluate each expression if x = _ , y = 2_ , and z = _ . Write in 12 12 12 simplest form.

33. x + y

34. z + y

35. z - x

36. y - x

ALGEBRA Find each sum or difference. Write in simplest form. x 4x 37. _ +_ 8

3r 3r 38. _ +_

8

10

4 1 39. 5_ c - 3_ c 7

10

5 1 40. -2_ y + 8_ y

7

6

6

41. CARPENTRY A 3-foot long shelf is to be installed between two walls that 5 inches apart. How much of the shelf must be cut off so that it fits are 32_ 8

between the walls? Real-World Career Carpenter Carpenters must be able to make precise measurements and know how to add and subtract fractional measures.


PETS The table shows the weight of Leon’s dog during its first five years. 42. How much weight did Leon’s dog lose between

Age Weigh (years) (pounds)

ages 3 and 4? 43. How much weight did Leon’s dog gain between

1

2 17 _

2

5 18 _

8

8 4 8 3 _ 18 8 7 _ 20 8

19 _

3

years 1 and 5? 7 pounds between 44. Suppose Leon’s dog gained 2_

4

8

years 5 and 6. How much does it weigh now?

5

Find each sum or difference. Write in simplest form. 3 5 7 - 7_ + 2_ 45. 12_ 8

EXTRA

PRACTIICE


H.O.T. Problems

8

5 5 1 46. 5_ + 3_ - 2_

8

6

6

6

47. GARDENING Tate’s flower garden has a perimeter of 25 feet. He plans to add 2 feet 9 inches to the width and 3 feet 9 inches to the length. What is the new perimeter in feet? 48. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would add or subtract like fractions. 49. OPEN ENDED Write a subtraction expression in which the difference of two 18 . fractions is _ 25

3 1 and -4_ . Who is 50. FIND THE ERROR Kayla and Ethan are adding -2_ 8 8 correct? Explain your reasoning.

Kayla

-17 35 3 -2 _1 + -4 _ = _ + -_ 8 8 8 8

Ethan

3 35 17 1 -2_ + (-4_) = _ + (- _) 8

52 = -_ or -6 _1 8

2

CHALLENGE The 7-piece square puzzle at the right is called a tangram. 51. If the value of the entire puzzle is 1, what is the value of each piece? 52. How much is A + B? 53. How much is F + D? 54. How much is C + E? 55. Which pieces each equal the sum of E and G?

8

8 8 18 1 _ = - or -2_ 4 8

B C D

A E F

Lesson 5-5 Adding and Subtracting Like Fractions Tony Freeman/PhotoEdit

G

253

56.

Writing in Math Explain how fractions are important when taking measurements. Include in your answer some real-world examples in which fractional measures are used.

57. The average times it takes Miguel to cut his lawn and his neighbor’s lawn are given in the table. Last summer, he cut his lawn 10 times and his neighbor’s 5 times. About how many hours did he spend cutting both lawns? Lawn

_3

Neighbor’s

_2

2

16

15 layer of padding _ inch thick is 16

placed on top. What is the total thickness of the wood and the padding? 3 F 1_ in.

8 1 G 1_ in. 2 24 H 1_ in. 16 1 J 2_ in. 2

Time of Cut (hours)

Miguel’s

1 A 8_ h

9 58. A piece of wood is 1_ inches thick. A

4 4

B 9h

1 C 9_ h

D 10 h

2

Find each quotient. Write in simplest form. (Lessons 5-4) 3 1 59. _ ÷_ 6

4

5 1 60. -_ ÷_ 3

8

2 1 61. _ ÷ 1_ 5

2

4 62. 8_ ÷ -12 5 15

Find each product. Write in simplest form. (Lesson 5-3) 2 _ 63. _ ·3 5

4

8 1 64. _ · -_ 6

9

4 1 65. _ · 2_ 7

1 1 66. -1_ · 1_

3

7

3

67. ALGEBRA Find the product of 4y2 and 8y5. (Lesson 4-5) EXERCISE The table shows the amount of time Craig spends jogging every day. He increases the time he jogs every week. (Lesson 3-7) 68. Write an equation to show the number of minutes spent jogging m for each week w. 69. How many minutes will Craig jog during week 9?

7EEK

PREREQUISITE SKILL Use exponents to write the prime factorization of each number or monomial. (Lesson 4-2) 70. 60 73. 12n 254 Chapter 5 Rational Numbers

71. 175 74.

24s2

72. 112 75. 42a2b

4IME *OGGING MIN

Factors and Multiples Many words used in mathematics are also used in everyday language. You can use the everyday meaning of these words to better understand their mathematical meaning. The table shows both meanings of the words factor and multiple. Term

Everyday Meaning

Mathematical Meaning

factor

something that contributes to the production of result • The weather was not a factor in the decision. • The type of wood is one factor that contributes to the cost of the table.

one of two or more numbers that are multiplied together to form a product

multiple

involving more than one or shared by many • multiple births • multiple ownership

the product of a quantity and a whole number

Source: Merriam Webster’s Collegiate Dictionary

When you count by 2, you are listing the multiples of 2. When you count by 3, you are listing the multiples of 3, and so on, as shown in the table below. Number

Factors

Multiples

2

1, 2

2, 4, 6, 8, . . .

3

1, 3

3, 6, 9, 12, . . .

4

1, 2, 4

4, 8, 12, 16, . . .

Notice that the mathematical meaning of each word is related to the everyday meaning. The word multiple means many, and in mathematics, a number has infinitely many multiples.

Reading to Learn 1. Write your own rule for remembering the difference between factor and multiple. 2. RESEARCH Use the Internet or a dictionary to find the everyday meaning of each word listed below. Compare them to the mathematical meanings of factor and multiple. Note the similarities and differences. a. factotum

b. multicultural

c. multimedia

3. Make lists of other words that have the prefixes fact- or multi-. Determine what the words in each list have in common. Reading Math Factors and Multiples

255

CH

APTER

5


Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. (Lesson 5-1) 4 1. _ 25 1 3. 3_ 8

5 _ 13. _ · 4 18

15 1 _ 15. -1_ ·2 2 3

2 2. -_

9 5 4. 1_ 6

5. MULTIPLE CHOICE Which fraction is between _5 and _7 ? (Lesson 5-1) 7

Find each product or quotient. Write in simplest form. (Lessons 5-3 and 5-4)

8 C _

9 10 D _ 11

10 3 B _ 4

6. MANUFACTURING A garbage bag has a thickness of 0.8 mil, which is equal to 0.0008 inch. What fraction of an inch is this? (Lesson 5-2) Write each decimal as a fraction or mixed number in simplest form. (Lesson 5-2) 7. -6.75

8. 0.12

9. -0.5555 . . .

10. 3.08 3

1 of Earth’s 11. GEOGRAPHY Africa makes up _ 5 entire land surface. Use the table to find the fraction of Earth’s land surface that is made up by each of the other continents. Write each fraction in simplest form. (Lesson 5-2)

Antarctica

Decimal Portion of Earth’s Land 0.095

Asia

0.295

Europe

0.07

North America

0.16

Source: Incredible Comparisons

12. TRAVEL One of the fastest commuter trains is the Japanese Nozomi, which averages 162 miles per hour. About how many minutes would it take to travel 119 miles from Hiroshima to Kokura on the train? (Lesson 5-3) 256 Chapter 5 Rational Numbers

4

1 16. 3 _ ÷ (-4) 3

2

18. MULTIPLE CHOICE If the newsletter is 4 -inch by printed on 8_ 8 11-inch paper and the space between the columns and both ends of the page are shown, how wide are the three equal columns? (Lesson 5-4) 1 F 2_ in.

6 5 G 2_ in. 12

Î ° n

Ó ° n

Ó ° n

Î ° n

1 H 2_ in.

2 3 J 2_ in. 4

Find each sum or difference. (Lesson 5-5) 8 2 19. _ +_

6 11 20. _ -_

1 2 21. -2_ -_ 3 3

5 22. -3 + -4_

3 6 23. 5 _ + 2_

5 7 24. 2_ - 8_

15

7

Continent

8

17. SEWING How many 9-inch ribbons can be cut 1 yards of ribbon? (Lesson 5-4) from 1_

8

7 A _

7 1 14. _ ÷ -_

15

7

12

12

12

8

12

25. MULTIPLE CHOICE A pitcher of lemonade 9 full at the beginning of the party. It was _ 10 _ was only 1 full after the party ended. How 10 much lemonade was drunk during the party? (Lesson 5-5) 1 A _

10 3 B _ 10 3 C _ 5 _ D 4 5

5-6

Least Common Multiple

Main Ideas


• Find the least common multiple of two or more numbers.

A voter voted for both president and a senate seat in the year 2004. a. List the next three years in which the voter can vote for president.

• Find the least common denominator of two or more fractions.

b. List the next three years in which the voter can vote for the same senate seat.

New Vocabulary multiple common multiples least common multiple (LCM) least common denominator (LCD)

Candidate

Length of Term (years)

President

4

Senator

6

c. What will be the next year in which the voter has a chance to vote for both president and the same senate seat?

Least Common Multiple A multiple of a number is a product of that number and a whole number. Sometimes numbers have some of the same multiples. These are called common multiples. multiples of 4:

0, 4, 8, 12, 16, 20, 24, 28, …

multiples of 6:

0, 6, 12, 18, 24, 30, 36, 42, …

Some common multiples of 4 and 6 are 0, 12, and 24.

The least of the nonzero common multiples is called the least common multiple (LCM). So, the LCM of 4 and 6 is 12. When numbers are large, an easier way of finding the least common multiple is to use prime factorization. The LCM is the smallest product that contains the prime factors of each number.

EXAMPLE

Find the LCM

Find the LCM of 108 and 240. Number Prime Factorization

Prime Factors If a prime factor appears in both numbers, use the factor with the greatest exponent.

Exponential Form

108

2·2·3·3·3

22 · 33

240

2·2·2·2·3·5

24 · 3 · 5

The prime factors of both numbers are 2, 3, and 5. Multiply the greatest power of 2, 3, and 5 appearing in either factorization. LCM = 24 · 33 · 5 = 2160

1. Find the LCM of 120 and 180. Extra Examples at pre-alg.com

Lesson 5-6 Least Common Multiple

257

The LCM of two or more monomials is found in the same way as the LCM of two or more numbers.

The LCM of Monomials

EXAMPLE

Find the LCM of 18xy2 and 10y. 18xy2 = 2 · 32 · x · y2 10y = 2 · 5 · y

Find the prime factorization of each monomial. Highlight the greatest power of each prime factor.

LCM = 2 · 32 · 5 · x · y2 = 90xy2

Multiply the greatest power of each prime factor.

2. Find the LCM of 24 a3b and 30a.

Least Common Denominator The least common denominator (LCD) of two or more fractions is the LCM of the denominators.

EXAMPLE

Find the LCD

5 11 Find the LCD of _ and _ .

9 21 Write the prime factorization of 9 and 21.

9 = 32 21 = 3 · 7

Highlight the greatest power of each prime factor.

LCM = 32 · 7 = 63

Multiply.

5 11 The LCD of _ and _ is 63. 9

21

3 7 3. Find the LCD of _ and _ . 8

10

One way to compare fractions is to write them using the LCD. We can multiply the numerator and the denominator of a fraction by the same number, because it is the same as multiplying the fraction by 1.

EXAMPLE Replace

Compare Fractions

1 with , or = to make _

7 _ a true statement.

6

15

The LCD of the fractions is 2 · 3 · 5 or 30. Rewrite the fractions using the LCD and then compare the numerators. 5 1·5 5 _1 = _ =_ Multiply the fraction by _ to make the denominator 30. 5 6 2·3·5 30 7 · 2 14 7 2 _ = _ = _ Multiply the fraction by _ to make the denominator 30. 2 3·5·2 30 15 5 14 1 7 Since _ < _, then _ < _. 30 30 6 15

4. Replace


2 with , or = to make _ 3

_5 a true statement. 9

Order Rational Numbers TRAVEL The table shows the arrival times of four flights compared to their scheduled arrival times into Greensboro, North Carolina. Order the flights from least delayed to most delayed. (Hint: A negative fraction indicates a flight that arrived earlier than its scheduled arrival time.)

$EPARTURE #ITY

Step 1 Order the negative fractions first. The LCD of 6 and 8 is 24. 3 9 = -_ -_ 8

1 4 -_ = -_ 6

24

$IFFERENCE BETWEEN !RRIVAL 4IME AND 3CHEDULED 4IME H

!TLANTA

? n

$ALLAS

?

-IAMI

? n

3AN &RANCISCO

?

24

9 3 4 1 < -_, then -_ < -_. Compare the negative fractions. Since -_ Ordering Rational Numbers You can break ordering rational numbers into two steps since negative numbers are always less than positive numbers.

24

8

24

6

Step 2 Order the positive fractions. The LCD of 2 and 7 is 14. 5 10 1_ = 1_ 7

1 7 1_ = 1_ 2

14

14

10 5 7 1 < 1_, then 1_ < 1_. Compare the positive fractions. Since 1_ 14

2

14

7

5 3 1 Since -_ < -1 < 1_ < 1_, the order of the flights from least delayed to most 8

6

2

7

delayed are Atlanta, Miami, San Francisco, and Dallas. Order the fractions from least to greatest. 1 3 1 2 3 3 4 1 5A. -7_ , -6_ , -6_ , -7_ 5B. _, _, _, _ 5 5 4 4 6 20 10 7 Personal Tutor at pre-alg.com

Example 1 (p. 257)

Example 2 (p. 258)

Example 3 (p. 258)

Find the least common multiple (LCM) of each pair of numbers. 1. 6, 8

2. 7, 9

Find the least common multiple (LCM) of each pair of monomials. 4. 36ab, 4b

(p. 258)

Example 5 (p. 259)

5. 5x2, 12y2

6. 14e3, 8e2

Find the least common denominator (LCD) of each pair of fractions. 1 _ 7. _ ,3 2 8

Example 4

3. 10, 14

Replace each 1 10. _ 4

3 _ 16

3 _ 9. _ ,5

2 _ 8. _ , 7 3 10

5 8

with , or = to make a true statement. 10 11. _ 45

_2

5 12. _

9

7

13. WEATHER The table shows the amount of rain that fell during a rainstorm in four Kentucky cities. Order the cities from least to greatest amount of rainfall.

City Bowling Green Frankfort Lexington Louisville

_7 9

Rainfall (in.) 7 1

10 2 15 3 17 3 2 8


259

HOMEWORK

HELP

For See Exercises Examples 14–23, 29 1 24–28 2 30–37 3 38–43 4 44–47 5

Find the least common multiple (LCM) of each pair of numbers or monomials. 14. 4, 10

15. 20, 12

16. 2, 9

17. 16, 3

18. 15, 75

19. 21, 28

20. 14, 28

21. 20, 50

22. 18, 32

23. 24, 32

24. 20c, 12c

25. 16a2, 14ab

26. 7x, 12x

27. 75n2, 25n4

28. 20ef, 52f 3

29. AUTO RACING One driver can circle a one-mile track in 30 seconds. Another driver takes 20 seconds. If they both start at the same time, in how many seconds will they be together again at the starting line? Find the least common denominator (LCD) of each pair of fractions. 1 _ ,7 30. _

4 8 4 _ 34. _ , 5 9 12

Replace each

8 _ 31. _ ,1

4 _ 32. _ ,1

15 3 3 _ 35. _ ,5 8 6

2 _ 33. _ ,6

5 2 1 _ 36. _ ,4 3 7

5 7 5 _ 37. _ ,8 6 9


1 38. _

5 _ 2 12 21 _1 41. _ 100 5

7 39. _

_5 9 6 17 _ 1 42. _ 34 2

3 40. _

_4 5 7 12 _ 36 43. _ 17 51

Order the fractions from least to greatest. 5 _ 1 _ , 3, _ ,5 44. _

12 4 3 6 1 4 11 1 46. -2_ , -2_ , -2_ , -2_ 2 9 6 18

23 4 2 7 45. -_ , -_ , -_ , -_

30 5 3 10 5 _ 1 _ 47. 1_ , 1 3 , 1_ , 11 24 4 8 3

48. PETS In Brady’s math class, approximately 3 of the students have pets. 5 About 41 out of every 50 students in his school have pets. Do a greater fraction of students have pets in Brady’s math class or in his school? 49. ANALYZE TABLES The table shows the number of children who signed up to play soccer in the park district. Would you use the GCF or LCM to find the greatest number of teams that can be formed if each team must have the same number of 6-year-olds, 7-year-olds, and 8-year-olds? Explain your reasoning and then find the answer. How many 6-year-olds, 7-year-olds, and 8-year-olds are on each team?

Age

Number

6

60

7

96

8

24

Find the least common multiple (LCM) of each set of numbers. 50. 7, 21, 84 EXTRA

PRACTIICE


51. 9, 12, 15

52. 45, 30, 35

53. FITNESS Suppose you run on the treadmill every other day and lift weights every third day. After you add pushups to your routine, you do all three exercises every thirtieth day. How often do you do pushups? 54. Find two composite numbers between 10 and 20 whose least common multiple (LCM) is 36.


55. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would compare fractions. 56. OPEN ENDED Write two fractions whose least common denominator (LCD) is 35.

H.O.T. Problems

CHALLENGE Determine whether each statement is sometimes, always, or never true. Give an example or explanation to support your answer. 57. The LCM of three numbers is one of the numbers. 58. If two numbers do not contain any factors in common, then the LCM of the two numbers is 1. 59. The LCM of two numbers, except 1, is greater than the GCF of the numbers. 60. The LCM of two whole numbers is a multiple of the GCF of the same two numbers. 61.

Writing in Math Use the information about prime factors on page 257 to explain how to use them to find the LCM of two or more numbers.

62. A radio station is giving away two concert tickets to every sixteenth caller and a dinner for two to every twentieth caller. Which caller will receive both the concert tickets and the dinner?

63. A party goods store sells the party supplies in the table. In order to have the same number of cups, plates, and napkins, what is the least number of each that must be purchased? Quantity in One Package

A 32th

Supply

B 40th

cups

15

C 56th

plates

30

D 80th

napkins

20

F 40

G 48

H 60

J 64

Find each sum or difference. Write in simplest form. (Lesson 5-5) 3 7 64. _ -_ 8

8

9 5 65. 3_ -_ 11

11

13 3 66. _ +_ 14

14

5 1 67. 2_ + 4_ 6

6

ALGEBRA Find each quotient. Write in simplest form. (Lesson 5-4) 3 _1 68. _ n÷n

x x 69. _ ÷_ 8

6

ac 70. _ ÷ _c 5

d

6k 3 71. _ ÷_ 7m

14m

72. ALGEBRA Translate the sum of 7 and two times a number is 11 into an equation. Then find the number. (Lesson 3-6)

PREREQUISITE SKILL Estimate each sum. (page 751) 3 3 73. _ + _ 8

4

9 14 74. _ + _ 10

15

4 1 75. _ + 2_ 7

5

7 2 76. 5 _ +_ 8

3


261

EXTEND

5-6

Algebra Lab

Juniper Green

Juniper Green is a game that was invented by a teacher in England.

GETTING READY This game is for two people, so students should divide into pairs.

RULES OF THE GAME • The first player selects an even number from the hundreds chart and circles it with a colored marker. • The next player selects any remaining number that is a factor or multiple of this number and circles it. • Players continue taking turns circling numbers, as shown below. • When a player cannot select a number or circles a number incorrectly, then the game is over and the other player wins. 2nd move Player 2 circles 7 because it is a factor of 42.

1

1st move Player 1 circles 42.

2

3

4

5

6

7

8

9

10

11 12 13 14

15 16

17 18

19 20

21 22 23 24

25 26

27 28

29 30

31 32 33 34

35 36

37 38

39 40

41 42 43 44

45 46

47 48

49 50

51 52 53 54

55 56

57 58

59 60

61 62 63 64

65 66

67 68

69 70

71 72 73 74

75 76

77 78

79 80

81 82 83 84

85 86

87 88

89 90

91 92 93 94

95 96

97 98

99 100

3rd move Player 1 circles 70 because it is a multiple of 7.

ANALYZE THE RESULTS Play the game several times and then answer the following questions. 1. Why do you think the first player must select an even number? Explain. 2. Describe the kinds of moves that were made just before the game was over. Reprinted with permission from Mathematics Teaching in the Middle School, copyright (c) 1999, by the National Council of Teachers of Mathematics. All rights reserved.

262 Chapter 5 Rational Numbers Geoff Butler

5-7 Main Ideas • Add unlike fractions. • Subtract unlike fractions.

Adding and Subtracting Unlike Fractions 1 1 The sum _ +_ is modeled at the 2 3 right. We can use the LCM to find the sum.

£ Ó

a. What is the LCM of the denominators? b. If you divide the model into six parts, what fraction of the model is shaded?

£ Î

1 _ ? 1? c. How many parts are _ 2 3

d. Describe a model that you could 1 1 use to add _ and _ . Then use it to 3 4 find the sum.

£ Ó

£ Î

Add Unlike Fractions Fractions with different denominators are called unlike fractions. In the activity, you used the LCM of the denominators to rename the fractions. You can use any common denominator. Adding Unlike Fractions To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify as with like fractions.

Words

1 _ 2 1 5 2 3 _ + =_·_+_·_

Example

3

5

3

5

5

3

5 6 11 = _ + _ or _ 15

EXAMPLE LCD You can rename unlike fractions using any common denominator. However, it is usually simpler to use the least common denominator.

15

15

Add Unlike Fractions

Find 1 + 23. 4

_1 + _2 = _1 · _3 + _2 · _4 3

4

3 4 3 3 8 _ _ = + 12 12 11 _ = 12

Find each sum. 1 1 1A. _ +_ 2

5

4

Use 4 · 3 or 12 as the least common denominator. Rename each fraction with the common denominator. Add the numerators.

2 1 1B. _ +_ 3

8

Lesson 5-7 Adding and Subtracting Unlike Fractions

263

EXAMPLE

Add Fractions and Mixed Numbers

Find each sum. Write in simplest form. 3 -7 1 1 + _ Estimate _ - _ = 0 a. _

Negative Signs When adding or subtracting a negative fraction, place the negative sign in the numerator.

2 2 8 12 3 _ -7 -7 _ _3 + _ =_ ·3+ _ · 2 The LCD is 23 · 3 or 24. 2 8 3 12 12 8 9 -14 =_ + _ Rename each fraction with the LCD. 24 24 5 = -_ Add. Compare to the estimate. Is the answer reasonable? 24 1 2 b. 1_ + -2_ Estimate 1 + (-2) = -1 3 9 2 1 11 7 1_ + -2_ =_ + -_ Write the mixed numbers as improper fractions. 9 3 9 3 11 7 _ =_ + -_ · 3 Rename -_73 using the LCD, 9. 3 9 3 11 -21 =_+ _ Simplify. 9 9 -10 1 =_ or -1_ Add. Compared to the estimate, the answer is reasonable. 9 9

3 5 2A. _ +_ 4

9

3 5 2C. 3 _ + -4 _

5 8 2B. -_ +_

14

5

12

6

Subtract Unlike Fractions The rule for subtracting fractions with unlike denominators is similar to the rule for addition. Subtracting Unlike Fractions To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplify as with like fractions.

EXAMPLE Reasonableness Use estimation to check whether your answer is reasonable. 6

2 - ≈0-1 21 7

≈ -1 17 is close to -1. - 21

Subtract Fractions and Mixed Numbers

Find each difference. Write in simplest form. 6 1 -_ a. _

7 21 6 _ 1 1 _ _ - 6 =_ -_ ·3 7 7 3 21 21 18 1 =_ -_ 21 21 -17 17 _ = or -_ 21 21

3 8 3A. _ -_ 4


9

1 1 b. 6 _ - 4_ The LCD is 21. Rename using LCD.

5

2

5

Write as improper fractions.

2

5

13 _ 21 _ =_ · 5 -_ · 2 Rename using LCD.

2 5 5 65 42 =_-_ 10 10 23 3 = _ or 2 _ 10 10

Subtract.

5 1 3B. 7_ - 6_ 6

2

13 1 1 21 6_ - 4_ =_ -_

8

2

Simplify. Subtract.

5 1 3C. 5 _ - -4 _ 3

9

COMPUTERS To set up a computer network in an office, a 100-foot cable is cut and used to connect 3 computers to the server as shown. How much cable is left to connect the third computer?

You know that the 100-foot cable was used to connect two computers to the server.

Plan

Add the measures of the cables that were already used and subtract that sum from 100. Estimate your answer. 100 - (19 + 41) ≈ 100 - 60 or 40 feet

Solve

3 6 1 1 19_ + 40_ = 19_ + 40_ 4

8 7 = 59_ 8

8

FT 4FSWFSS

Explore

8

FT

3ERVER

Rename 40_ with the LCD, 8. 3 4

Simplify.

8 7 7 100 - 59_ = 99_ - 59_

8 Rename 100 with the LCD, 8, as 99_. 8 8 8 1 = 40_ Simplify. 8 1 There is 40_ feet of cable left to connect the third computer. 8 1 Since 40_ is close to 40, the answer is reasonable. 8

8

Check

3 4. INSECTS The speed of a hornet is 13_ miles per hour. The speed of a 10

4 miles per hour. How much faster is the dragonfly than dragonfly is 17 _

the hornet?

5


Examples 1, 2 (pp. 263–264)

Find each sum. Write in simplest form. 3 1 1. _ +_

1 1 2. _ +_

5 15 5 1 4. 8_ + 11_ 12 4

Example 3 (p. 264)

3 10 3 5 5. 4_ + 10_ 8 12

1 2 7. _ -_ 3

7 - 2 8. -

10 15 5 1 11. 6_ - 2_ 6 3

3 1 10. -9_ - -5_ 2 4

(p. 265)

6

18

3 4 6. 6_ + -1_ 5

4

Find each difference. Write in simplest form. 4

Example 4

7 1 3. -_ +_

5 7 9. _ -_ 8

12

3 1 12. 12_ - 6_ 2 8

5 1 yards of fabric to make a skirt and 14_ yards to 13. SEWING Jessica needs 5_ 8

make a coat. How much fabric does she need in all?


2


265

HOMEWORK

HELP

For See Exercises Examples 14–19 1, 2 20–23 2 24–31 3 32–33 4

Find each sum or difference. Write in simplest form. 3 3 14. _ +_ 5 4

8

1 4 20. 8_ + 3_ 2 5 3

7 1 21. _ + 4_

1 11 22. -4_ + -7_

5 1 24. _ -_

7 2 25. _ -_

1 2 27. -_ -_

8 2 28. -6_ -_

8

2 7 23. -10_ + 9_

7

21 3 1 19. -_ +_ 7 4

3

8

3 5 5 1 30. 16_ - 12_ 6 3

16

7 29. 216 -_ 30 15

6

24

8

12

3 7 26. _ - -_ 8

5 10 16. _ + -_

13 3 1 18. -_ +_ 8 2

3 5 17. _ + -_

9 3 15. _ +_ 26

10

18

5

3

9

1 1 31. 3_ - -7_ 2 3

For Exercises 32–35, select the appropriate operation. Justify your selection. Then solve. 32. EARTH SCIENCE Did you know that water has a greater density than ice? Use the information in the table to find how much more water weighs per cubic foot.

1 Cubic Foot

Weight (lb)

water

1 2 9 56 10

33. PUBLISHING The length of a page in a yearbook is

ice

1 inch, and the bottom 10 inches. The top margin is _ 2 3 _ margin is inch. What is the length of the page inside

62

4

the margins? 1 of the votes and Sara 34. VOTING In the class election, Murray received _ 3

2 of the votes. Makayla received the rest. What fraction of the received _ 5

votes did Makayla receive? 35. ANALYZE TABLES Use the table to find the sum of precipitation that fell in Columbia, South Carolina, in August, September, and October.

EXTRA

PRACTICE


36. RESEARCH Use the Internet or another source to find out the monthly rainfall totals in your community during the past year. How much rain fell in August, September, and October during the past year?

!MOUNT OF 0RECIPITATION IN

!UG

??

3EPT

??

/CT

?

Find each difference. Write in simplest form.

3 3 - -4_ 37. -19_ 8

H.O.T. Problems

-ONTH

5

5 13 39. 8_ - -12_

2 4 38. -3_ - -2_

4

7

12

40. OPEN ENDED Write a real-world problem that you could solve by 3 1 from 15_ . subtracting 2_ 8

4

3 cup, 41. CHALLENGE A set of measuring cups has measures of 1 cup, _ 4

1 1 1 1 _ cup, _ cup, and _ cup. How could you get _ cup of milk by using 2

3

these measures? 266 Chapter 5 Rational Numbers

4

6

18

9 7 42. FIND THE ERROR Roberto and Daniel are finding _ +_ . Who is correct so 10 12 far? Explain your reasoning.

Daniel 9 + 7 = 9+7 10 12 10 + 12

Roberto

7 7 9 9 12 _ +_=_·_ + _ · 10 12

10

43.

10

12

12

10

Writing in Math Explain how to add and subtract fractions with different denominators. Illustrate your answer with an example using the LCM and an explanation of how prime factorization can be used to add and subtract unlike fractions.

45. The results of a grocery store survey are listed in the table. Find the fraction of families who grill out more than one time per month.

44. For an art project, Halle needs 113 inches of red ribbon and 67 inches 8 9 of white ribbon. Which is the best estimate for the total amount of ribbon that she needs? A 8 in. B 10 in. C 18 in.

2 F _

D 26 in.

25

How Often Do You Grill Out? Times per Fraction of Month People Less than 1

11 50

1

2 25

2–3

4 25

4 or more

27 50

23 G _

7 H _

100

39 J _ 50

10

Find the LCD of each pair of fractions. (Lesson 5-6) 4 _ 46. _ , 7

5 _ 47. _ , 3

9 12

3 _ 48. _ , 2

8 14

1 _ 49. _ , 73

15t 5t

3n 6n

Find each sum or difference. Write in simplest form. (Lesson 5-5) 3 3 50. 2_ + 6_ 4

4

3 2 51. 3_ -_ 5

5 1 52. 4_ + 5_

5

6

6

5 1 53. 6_ - -8_ 4

15

54. MOVIES A movie is made up of hundreds of thousands of individual pictures called frames. The frames are shown through a projector at a rate of 24 frames per second. How many frames would be needed for a 30-minute scene? (Lesson 1-1)

PREREQUISITE SKILL Find each quotient. Round to the nearest tenth, if necessary. (Page 749) 55. 25.6 ÷ 3

56. 37 ÷ 4.7

57. 30.5 ÷ 11.2

58. 46.8 ÷ 15.6

59. 34.8 ÷ 5.8

60. 63 ÷ 7.5


267

5-8

Solving Equations with Rational Numbers

Main Idea • Solve equations containing rational numbers.

Musical sounds are made by vibrations. If n represents the number of vibrations for middle C, then the approximate vibrations for the other notes going up the scale are given below. Notes

Middle C

D

E

F

G

A

B

C

n

9 n 8

5 n 4

4 n 3

3 n 2

5 n 3

15 n 8

2 n 1

Number of Vibrations

a. A guitar string vibrates 440 times per second to produce the A above middle C. Write an equation to find the number of vibrations per second to produce middle C. If you multiply each side by 3, what is the result? b. How would you solve the second equation you wrote in part a? c. How can you combine the steps in parts a and b into one step? d. How many vibrations per second are needed to produce middle C?

Solve Addition and Subtraction Equations You can solve equations with rational numbers using the same properties you used to solve equations with integers.

EXAMPLE Look Back

Solve by Using Addition and Subtraction

a. Solve 2.1 = t - 8.5.

To review solving equations, see Lessons 3-3 and 3-4.

2.1 = t - 8.5

Write the equation.

2.1 + 8.5 = t - 8.5 + 8.5 Add 8.5 to each side. 10.6 = t 3 2 b. Solve x + _ =_ . 5 3 3 2 x+_ =_ 3 5 3 3 3 2 x+_ -_ =_ -_ 5 5 3 5 9 10 1 _ x= -_ or _ 15 15 15

Solve each equation. 1A. n - 9.7 = -13.9 268 Chapter 5 Rational Numbers

Simplify.

Write the equation. Subtract _ from each side. 3 5

Rename the fractions using the LCD and subtract.

7 2 1B. _ +p=_ 10

5

Solve Multiplication and Division Equations Use the same process to solve these equations as those involving integers.

EXAMPLE

Solve by Using Division

Solve -3y = 1.5. Check your solution. -3y = 1.5

Write the equation.

-3y 1.5 _ =_ -3 -3


y = -0.5

Simplify. Check the solution.

Solve each equation. Check your solution. 2A. 6a = -8.4 2B. -36 = -5z 1 1 To solve _ x = 3, you can divide each side by _ or multiply each side 2

1 by the multiplicative inverse of _ , which is 2.

2

2

Reciprocals

_1 x = 3 2

Recall that dividing by a fraction is the same as multiplying by its multiplicative inverse.

1 2·_ x =2·3

The product of any number and its multiplicative inverse is 1.

EXAMPLE

2

x =6

Write the equation. Multiply each side by 2. Simplify.

Solve by Using Multiplication

-2 Solve _ x = -7. Check your solution. 3

-2 _ x = -7

3 -3 _ -3 -2 _ x =_ (-7) 2 2 3 1 21 x=_ or 10_ 2 2

Write the equation. Multiply each side by -3 . 2 Simplify. Check the solution.

Solve each equation. Check your solution. 5 1 3A. 10 = _ h 3B. _ m = -3 6

8

3 7 PETS Oscar feeds his dog _ cup of dog food in the morning and _ cup 8 4 of dog food in the evening. If a bag of dog food contains 50 cups, how many days will the bag last?

The amount of dog food that Oscar feeds his dog each day is

_7 + _3 = _7 + _6 or 1_5 cups. 8

4

8

8

8

(continued on the next page) Extra Examples at pre-alg.com

Lesson 5-8 Solving Equations with Rational Numbers

269

1_ cups per day 5 8

Words

times

d days

equals

50 cups of dog food

d

=

50

Let d the number of days.

Variable

1_ 5 8

Equation

·

5 1_ d = 50

8 13 _ d = 50 8 8 _ 8 _ · 13 d = 50 · _ 13 8 13 400 d=_ ≈ 30.8 13

Write the equation. Rename 1_ as an improper fraction. 5 8

Multiply each side by _. 8 13

Simplify.

The bag of dog food will last approximately 31 days.

4. RETAIL A pair of shoes that normally costs $45 now costs $30. What fraction of the regular price is the reduced price? Personal Tutor at pre-alg.com

Solve each equation. Check your solution. Example 1 (p. 268)


1. y + 3.5 = 14.9 4. b - 5 = 13.7

5 5 3 _ _ 5. c - = 6 5

7. -8.4 = -6f

8. 3.5a = 7

Example 4

HOMEWORK

HELP

For See Exercises Examples 14–19 1 20–25 2 26, 27 3 28–31 4 32, 33 5

1 1 3. 4 _ = r + 6_

2

6

1 s = 15 10. -_

4

6. a - 2.7 = 3.2 9. -3.4 = 0.4x

3 11. 9 = _ g 4

6

(pp. 269–270)

3 3 2. _ =w+_

2 12. _ p = -22 3

13. SPACE The weight of an object on the Moon is one-sixth its weight on Earth. If an object weighs 54 pounds on the Moon, how much does it weigh on Earth? Write and solve a multiplication equation to determine the weight of an object on Earth.

Solve each equation. Check your solution. 14. y + 7.2 = 21.9

15. 4.7 = a + 7.1

2 1 16. _ =_ +b

5 7 17. m + _ = -_

3 5 18. 3_ + n = 6_

1 1 19. y + 1_ = 3_

12

18

8

4

3

8

3

18

20. x - 5.3 = 8.1

21. n - 4.72 = 7.52

8 2 23. x - _ = -_

1 1 24. b - 1_ = 4_

3 1 22. n - _ =_ 8 6 1 2 _ 25. 7 = r - 5_

26. 4.1p = 16.4

27. -0.4y = 2

2 28. 8 = _ d

1 29. _ t=9

1 30. 4 = -_ q

3 31. -6 = _ a

5

5


15

2

8

4

2

3

5

3

-

Þ

33. BUSINESS A store is going out of business. Items that normally cost $24.99 now cost $16.66. What type of discount is the store offering on those items?

Î Ó £ ä

Õ Ì À i > > Ì Ã À > `

> Þ " Þ « V 9 Ãi , Ìi V Þ Õ Ì > Ã 9i ÜÃ Ì i

.UMBER OF 6ISITORS MILLIONS

For Exercises 32 and 33, write an equation and solve the problem. 32. ANALYZE GRAPHS The graph )''* Dfjk M`j`k\[ shows the most visited national L%J% EXk`feXc GXibj £ä parks in the United States in 2003. The Great Smoky n Mountains had 5.25 million Ç more visitors than the Grand È Canyon. How many people x { visited the Grand Canyon?

53 .ATIONAL 0ARKS

-ÕÀVi\ 4OP 4EN OF %VERYTHING

Solve each equation. Check your solution. 1 2 n=_ 34. _ 3

5 1 35. _ = -_ r

9

8

28 7 36. -_ t = -_

2

9

36

For Exercises 37–40, write an equation and solve the problem. 37. METEOROLOGY When a storm struck, the barometric pressure was 28.79 inches. Meteorologists said that the storm caused a 0.36-inch drop in pressure. What was the pressure before the storm? 38. MUSIC Carla downloaded some songs onto her digital music player and 5 1 full. If the player was _ full before the download, what now the player is _ 6

5

fraction of the space on the player do the new songs occupy? 1 batches of cookies for a bake sale and used 39. COOKING Gabriel made 2_ 2

3 cups of sugar. How much sugar is needed for one batch of cookies? 3_ 4

1 40. PUBLISHING A newspaper is 12_ inches wide and 22 inches long. This is 4

1 inches narrower and one-half inch longer than the old edition. What 1_ 4

were the previous dimensions of the newspaper? EXTRA

PRACTICE


H.O.T. Problems

3 square inches. 41. GEOMETRY The area A of the triangle is 33_ 4

1 bh to find the height h of the Use the formula A = _ 2

triangle with the given base b.

h

42. FIND THE DATA Refer to the United States Data File on 9 in. pages 18–21. Choose some data and write a real-life problem in which you would solve equations with rational numbers. 43. FIND THE ERROR Grace and Ling are solving 0.3x = 4.5. Who is correct? Explain your reasoning. Grace 0.3x = 4.5 0.3x _ _ = 4.5 3 3 x = 1.5

Ling 0.3x = 4.5 0.3x _ _ = 4.5 0.3 0.3 x = 15

Lesson 5-8 Solving Equations with Rational Numbers

271

44. Which One Doesn’t Belong? Identify the equation that does not belong with the other three. Explain your reasoning.

_5 j = 20

c + 3.17 = -3.17

3 2 4_ = w - 2_ 9 4

-0.8k = 10

8

45. CHALLENGE The denominator of a fraction is 4 more than the numerator. If both the numerator and denominator are increased by 1, the resulting 1 . Find the original fraction. fraction equals _ 2

46.

Writing in Math Explain how fractions are used to compare musical notes. How are reciprocals useful in finding the number of vibrations per second needed to produce certain notes? Give an example.

47. GRIDDABLE A hamburger is formed into the shape of a circle with a radius of 3 inches. If a grill is 1_ 4 28 inches wide, how many hamburgers can fit across the grill?

48. Mary and Tabitha ran in a race. Mary’s time was 12 minutes, which 3 of Tabitha’s time. Using t for was _ 4 Tabitha’s time, which equation represents the situation?

Î

£ °

3 A _ t = 12

3 C t-_ = 12

3 B t+_ = 12

3 D 12t = _

4

4

4

4

Find each sum or difference. Write in simplest form. (Lesson 5-7) 3 1 49. _ +_ 5

5 1 50. _ -_

3

6

5 1 + -_ 52. _ 9

1 55. _ 2

1 1 51. -4_ -_

4

3 1 53. -3_ + -2_ 4

12

Replace each

8

6

9 1 54. 8_ - 1_

8

10

6

with , or = to make a true statement. (Lesson 5-6)

5 _ 12

56.

16 _ 9 _ 50 30

4 57. _ 5

48 _ 60

58.

7 _3 _ 8

12

1 59. ALGEBRA Evaluate a - b if a = 9_ and b = 1_ . (Lesson 5-5) 5 6

6

60. GEOMETRY Express the area of the rectangle as a monomial. (Lesson 4-5) 61. HEALTH According to the National Sleep Foundation, teens should get approximately 9 hours of sleep each day. What fraction of the day is this? Write in simplest form. (Lesson 4-4)

PREREQUISITE SKILL Find each sum. (Lesson 2-2) 62. 24 + (-12) + 15

63. (-2) + 5 + (-3)

64. 4 + (-9) + (-9) + 5

65. -10 + (-9) + (-11) + (-8)


{X ÓY

XY Ó

EXPLORE

5-9

Algebra Lab

Analyzing Data Often, it is useful to describe or represent a set of data by using a single number. The table shows the daily maximum temperatures for twenty days during a recent September in Phoenix, Arizona. One number to describe this data set might be 96. Some reasons for choosing this number are listed below. • It occurs four times, more often than any other number. • If the numbers are arranged in order from least to greatest, 96 falls in the center of the data set.

There is an equal number of data above and below 96.

80 89 90 94 94 94 95 96 96 96 96 97 97 98 98 98 98 98 99 99 So, if you wanted to describe a typical high temperature for Austin during August, you could say 96°F.

COLLECT THE DATA Collect a group of data. Use one of the suggestions below, or use your own method. • Research data about the weather in your city or in another city, such as temperatures, precipitation, or wind speeds. • Find a graph or table of data in the newspaper or a magazine. Some examples include financial data, population data, and so on. • Conduct a survey to gather some data about your classmates. • Count the number of raisins in a number of small boxes.

ANALYZE THE RESULTS 1. Choose a number that best describes all of the data in the set. 2. Explain what your number means, and explain which method you used to choose your number. 3. Describe how your number might be useful in real life. Explore 5-9 Algebra Lab: Analyzing Data Geoff Butler

273

5-9

Measures of Central Tendency

Main Ideas • Use the mean, median, and mode as measures of central tendency. • Choose an appropriate measure of central tendency and recognize measures of statistics.

New Vocabulary measures of central tendency mean median mode

The Iditarod is a 1150-mile dogsled race across Alaska. The winning times for 1977–2004 are shown.

Winning Times (days)

a. Which number appears most often? b. If you list the data in order from least to greatest, which number is in the middle?

17

15

15

14

12

16

13

13

18

12

11

11

11

11

13

11

11

11

9

9

9

9

10

9

9

8

9

9

Source: Anchorage Daily News

c. What is the sum of all the numbers divided by 28? d. If you had to give one number that best represents the winning times, which would you choose? Explain.

Mean, Median, and Mode When you have a list of numerical data, it is often helpful to use one or more numbers to represent the whole set. These numbers are called measures of central tendency. Measures of Central Tendency mean Mean, Median, Mode • The mean and median do not have to be part of the data set. • If there is a mode, it is always a member of the data set.

the sum of the data divided by the number of items in the data set

median the middle number of the ordered data, or the mean of the middle two numbers mode

the number or numbers that occur most often

a. SPORTS The heights of the players on the girls’ basketball team are shown. Find the mean, median, and mode. sum of heights number of players 63 + 61 + . . . + 59 = __ 12 732 _ or 61 The mean height is 61 inches. = 12

mean = __

Height of Players (in.) 63 58 61 60 61 59 68 55 63 59 66 59

To find the median, order the numbers from least to greatest. 55, 58, 59, 59, 59, 60, 61, 61, 63, 63, 66, 68 60 + 61 _ = 60.5 2

There is an even number of items. Find the mean of the two middle numbers.

The median height is 60.5 inches. The height 59 inches appears three times so 59 is the mode. 274 Chapter 5 Rational Numbers

b. HURRICANES The line plot shows the number of Atlantic hurricanes that occurred each year from 1974 to 2004. Find the mean, median, and mode.

1

⫻

⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻ ⫻

2

3

4

5

⫻ ⫻

⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻

⫻

⫻

6

7

8

9

10

11

12

Source: National Weather Service

2 + 3(4) + 4(6) + 5(5) + 6(2) + 7(4) + 8(3) + 9(4) + 10 + 11 31

mean = _____ ≈ 5.9 Real-World Link A hurricane can be up to 600 miles in diameter and can reach 8 miles in the air. Source: sptimes.com

There are 31 numbers. So the median is the 16th number, or 5. You can see from the graph that 4 occurs most often. So 4 is the mode.

1 , 11, 5, 1. SHOES The shoe sizes of students in Ms. Alberti’s classroom are 10 _ 1 , 6, 6 _ 1 , 11, 7, 7 _ 1 , 8, 9, 5 _ 1 , 10 _ 1 , 4, 10 _ 1 , 10, 5, 14, and 12 _ 1 . Find2the 6, 10 _ 2 2 2 2 2 2 2 mean, median, and mode.

Choose Appropriate Measures Different circumstances determine which of the measures of central tendency are most useful. Using Mean, Median, and Mode mean

• the data set has no extreme values (values that are much greater or much less than the rest of the data)

median • the data set has extreme values • there are no big gaps in the middle of the data mode

• the data set has many repeated numbers

Choose an Appropriate Measure WEATHER The table shows daytime high temperatures for a week. Which measure of central tendency best represents the data? Then find the measure of central tendency. Since the set of data has no extreme values or numbers that are identical, the mean would best represent the data.

Day

Temperature

Sun.

84°F

Mon.

83°F

Tues.

89°F

Wed.

90°F

Thurs.

91°F

84 + 83 + . . . + 80 602 or 86 mean: __ = _

Fri.

85°F

Sat.

80°F

7

7

The temperature 86°F best represents the data.

2. EXERCISE The following set of data shows the number of sit-ups Pablo had done in one minute for the past 6 days: 40, 37, 45, 49, 50, 56. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. Extra Examples at pre-alg.com NASA

Lesson 5-9 Measures of Central Tendency

275

Using measures of central tendency can help you analyze the data from fast-food restaurants. Visit pre-alg.com to continue work on your project.

NUTRITION The table shows the number of Calories per serving of each vegetable. Tell which measure of central tendency best represents the data. Then find the measure of central tendency.

Vegetable

There is one value that is much greater than the rest of the data, 66. Also, there does not appear to be a big gap in the middle of the data. There is only one set of identical numbers. So, the median would best represent the data.

Calories

Vegetable

Calories

asparagus

14

cauliflower

10

beans

30

celery

17

bell pepper

20

corn

66

broccoli

25

lettuce

9

cabbage

17

spinach

9

carrots

28

zucchini

17

9, 9, 10, 14, 17, 17, 17, 20, 25, 28, 30, 66 The median is 17 Calories.

CHECK You can check whether the median best represents the data by finding mean with and without the extreme value. mean with extreme value 262 sum of values __ =_ 12 number of values

Interpreting Data You need to interpret information carefully so that you do not give a false impression for a set of data. As you have seen, extreme values affect how a set of data is perceived.

≈ 21.8

mean without extreme value sum of values 196 __ =_ 11 number of values

≈ 17.8

The mean without the extreme value is closer to the median. The extreme value increases the mean by about 4. Therefore, the median best represents the data.

3. RETAIL An electronics store recorded the number of customers it had each hour during the day. 86, 71, 79, 86, 79, 32, 88, 86, 82, 69, 71, 70 Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency.

Measures of central tendency can be used to show different points of view.

The average wait times for 10 different rides at an amusement park are 65, 21, 17, 52, 25, 17, 11, 22, 60, and 44 minutes. Which measure of data would the park advertise to show the wait times for its rides are short? A Mode B Median

C Mean D Cannot be determined

Read the Test Item To find which measure of central tendency to use, find the mean, median, and mode of the data and select the least measure. 276 Chapter 5 Rational Numbers

Solve the Test Item Analyzing Data Use these clues to help you analyze data. • Extremely high or low values affect the mean. • A value with a high frequency affects the mode. • Data that is clustered affect the median.

65 + 21 + … + 44 10

334 or 33.4 Mean: __ = _

Mode: 17

10

Median: 11, 17, 17, 21, 22, 25, 44, 52, 60, 65 22 + 25 _ or 23.5 2

The mode is the least measure. So the answer is A.

4. Serena received the following scores on her first six math tests: 90, 68, 89, 94, 60, and 93. Which measure of data might she want to use when describing how she is doing in math class? F Mode

G Median

H Mean

J Cannot be determined


Example 1 (pp. 274–275)

Find the mean, median, and mode for each set of data. If necessary, round to the nearest tenth. 1. 4, 5, 7, 3, 9, 11, 23, 37 ⫻ ⫻

⫻ ⫻ ⫻

⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻

1

2

3

4

3.

Example 2 (p. 275)

Example 3 (p. 276)

2. 7.2, 3.6, 9.0, 5.2, 7.2, 6.5, 3.6

5

⫻ ⫻ ⫻

⫻

6

7

8

4. VACATIONS The table shows the number of annual vacation days for nine countries. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 5. BOOKS The number of books sold during the past week is shown below. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 53, 61, 46, 59, 61, 55, 49

Example 4 (pp. 276–277)

Annual Vacation Days Country

Number of Days

Brazil

34

Canada

26

France

37

Germany

35

Italy

42

Japan

25

Korea

25

United Kingdom

28

United States

13

Source: World Tourism Organization

6. MULTIPLE CHOICE Suppose 83 books sold on the eighth day in Exercise 5. Which measure of central tendency would change the most? A The mean B The median C The mode D All measures were affected equally. Lesson 5-9 Measures of Central Tendency

277

HOMEWORK

HELP

For See Exercises Examples 7–12 1 13–14 2, 3 15–17 4

Find the mean, median, and mode for each set of data. Round to the nearest tenth, if necessary. 7. 41, 37, 43, 43, 36

8. 2, 8, 16, 21, 3, 8, 9, 7, 6

9. 14, 6, 8, 10, 9, 5, 7, 13 11. 15

⫻ ⫻ ⫻ 16

10. 7.5, 7.1, 7.4, 7.6, 7.4, 9.0, 7.9, 7.1

⫻

⫻ ⫻ ⫻

⫻ ⫻

⫻ ⫻ ⫻

⫻

17

18

19

20

21

12.

⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻ ⫻

⫻ ⫻

⫻

4.1

4.2

4.3

4.4

⫻

22 4.5

4.6

4.7

4.8

For Exercises 13–14, which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 13.

University Michigan

All-Time Football Wins 833

Notre Dame

796

Nebraska

781

Texas

776

Alabama

758


14.

2003 Corn Production State

Bushels (millions)

CA

27.2

GA

36.8

MD

50.4

MS

71.6

TX

194.7

Source: U.S. Dept. of Agriculture

15. BASKETBALL Refer to the cartoon at the right. Which measure of central tendency would make opponents believe that the height of the team is much taller than it really is? Explain.

EXTRA

PRACTICE

16. TESTS Which measure of central tendency best summarizes the test scores shown below? Explain. 97, 99, 95, 89, 99, 100, 87, 85, 89, 92, 96, 95, 60, 97, 85


H.O.T. Problems

17. ICE SKATING Sasha needs to average 5.8 points from 14 judges to win the competition. The mean score of 13 judges was 5.9. What is the lowest score Sasha can have from the 14th judge and still win? 18. OPEN ENDED Write a set of data with at least four numbers that has a mean of 8 and a median that is not 8. 19. CHALLENGE A real estate guide lists the “average” home prices for counties in your state. Do you think the mean, median, or mode would be the most useful average for homebuyers? Explain. 20.

Writing in Math Explain how measures of central tendency are used in the real world. Include in your answer examples of real-world data from home or school that can be described using the mean, median, or mode.


21. The graph shows the number of siblings that Ms. Cantor’s students have. Which measure of data best represents the data?

22. If 18 were added to the data set below, which statement is true? 16, 14, 22, 16, 16, 18, 15, 25 F The mode increases.

ÀÌiÀÃ >` -ÃÌiÀÃ

G The mean increases.

ÕLiÀ v -ÌÕìÌÃ

£ä

H The mode decreases.

n

J The median increases.

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23. The hourly salaries of employees in a small store are shown. Which measure of data would the store are shown use to attract people to work there?

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HOMEWORK

xÈ

xä {ä Îä £x

Óä £ä

12. HEALTH Seven of the 28 students in math ä class have the flu. Is this sampling of the students who have the flu representative of the entire school? If so, how many of the 464 students who attend the school have the flu?

Real-World Link Approximately 170 million pounds of milk are produced annually in the United States. Source: National Agriculture Statistics Service

EXTRA

PRACTICE


,iÃ«Ãi

13. CONCERT As teenagers leave a concert, every 10th person is surveyed. They are asked if they would buy a T-shirt. One hundred forty out of a total of 800 people surveyed said yes. Is this sampling method valid? If so, how many people would you expect to buy T-shirts at the next concert if 7000 attend? Explain your reasoning. 14. ANALYZE TABLES Every hour, twenty customers in a Milk grocery store are randomly selected and surveyed on skim their milk preference. The results are shown in the table. low-fat After reviewing the data, the store manager decided whole that 40% of his total milk stock should be low-fat milk. Is this a valid conclusion? If it is not, what information should the store manager review to make a better conclusion?

Number 88 92 60

15. VIDEOS A video store is considering adding an international movie section. They surveyed 300 random customers, and 80 customers agree the international movie section is a good idea. Should the store add this section? Explain.

346 Chapter 6 Ratio, Proportion, and Percent Lester Lefkowitz/Getty Images

9iÃ

H.O.T. Problems

16. OPEN ENDED Give an example of a biased survey. 17. CHALLENGE Suppose you are a farmer and want to know if your corn crop is ready to be harvested. Describe an unbiased way to determine whether the crop is ready to harvest. 18.

Writing in Math

Why is sampling an important part of the manufacturing process? Illustrate your answer with an unbiased and biased sampling method you can use to check the quality of DVDs.

19. A real estate agent surveys people about their housing preferences at an open house for a luxury townhouse. Which is the best explanation for why the results of this survey might NOT be valid? A The survey is biased because the agent should have conducted the survey by telephone. B The survey is biased because the sample consisted of only people who already are interested in townhouses. C The survey is biased because the sample was a voluntary response sample. D The survey is biased because the agent should have conducted the survey at a single-family home.

20. An online survey produced the following results. If about 38,000 children participated in the survey, about how many drink two cans of soda or less per day? Ü >Þ >Ã v -`> 9Õ À > >Þ¶ vÛi À Ài £¯ vÕÀ Î¯ ÌÀii Ç¯

âiÀ £¯

ÌÜ £x¯

i À iÃÃ ÎÇ¯

Source: pbskids.org

F 5700

H 14,060

G 8993

J 26,980

21. Find the percent of change from 32 feet to 79 feet. Round to the nearest tenth, if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. (Lesson 6-9) Solve each problem using the percent equation. Round to the nearest tenth. (Lesson 6-8) 22. 7 is what percent of 32? 23. What is 28.5% of 84? ALGEBRA Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 4-4) 17g2h 6r 30x2 12cd 24. _ 25. _ 26. _ 27. _ 15rs 19e 51xy 51g 28. WEATHER During a 10-hour period, the temperature in Browning, Montana, changed at a rate of -10°F per hour, starting at 44°F. What was the ending temperature? (Lesson 2-4) Lesson 6-10 Using Sampling to Predict

347

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APTER

6

ownload Vocabulary view from pre-alg.com


&RACTION $ECIMAL 0ERCENT

Key Concepts Ratios and Rates

(Lesson 6-1)

• A ratio is a comparison of numbers by division. • A unit rate is a simplified rate whose denominator is 1.

Proportions


constant of proportionality (p. 298) convenience sample (p. 343) cross product (p. 302) discount (p. 333) interest (p. 334) nonproportional (p. 297) percent (p. 313) percent equation (p. 332) percent of change (p. 338) percent proportion (p. 322)

population (p. 343) proportion (p. 302) proportional (p. 297) ratio (p. 292) rate (p. 293) sample (p. 343) scale (p. 308) scale drawing (p. 308) scale factor (p. 308) scale model (p. 308) unit rate (p. 293)

• A proportion is an equation stating two ratios or c a rates are equal. So, if _ = _ , then ad = bc. b

d

• A proportional relationship exists when the ratios of related terms are equal.

Scale Drawings and Models

(Lesson 6-4)

• A scale drawing or model represents an object that is too large or too small to be drawn or built at actual size. • The ratio of a length on a scale drawing or model to the corresponding length on the real object is the scale factor.

Fractions, Decimals, and Percents

(Lesson 6-5)

• Percent is a ratio of a number to 100. • Fractions, decimals, and percents are all different ways to represent the same number.

Percents

(Lessons 6-6 through 6-9)

part • A percent proportion is _ = percent, where whole

the percent is written as a fraction. • A percent equation is equivalent to a percent proportion except the percent is written as a decimal. • A percent of increase, or decrease, tells how much an amount has increased, or decreased, in relation to the original amount.

348 Chapter 6 Ratio, Proportion, and Percent

Vocabulary Check Complete each sentence with the correct term. Choose from the list above. 1. A statement of equality of two ratios or rates is called a _________. 2. A ________ is a subgroup or subset of the population. 3. A ________ is a ratio of two measurements having different units. 4. The ________ is the amount by which the regular price of an item is reduced. 5. ________ is the amount of money paid or earned for the use of money. 6. Proportional relationships can be described by using the equation y = kx, where k is the ________. 7. A ________ is a type of biased sample. Vocabulary Review at pre-alg.com


Ratios and Rates

(pp. 292–296)

8. 30 hours to 18 hours

Example 1 Express the ratio 2 meters to 35 centimeters as a fraction in simplest form.

9. 10 inches to 4 feet

First, convert 2 meters to centimeters.

Express each ratio or rate as a fraction in simplest form.

200 cm 2m _ =_

10. 5 quarts to 5 gallons

35 cm

11. 2 tons to 1800 pounds

200 cm ÷ 5 40 cm 40 _ =_ or _

12. BASEBALL Jean got 12 hits out of 16 times at bat. Express this rate as a fraction in simplest form.

6–2

35 cm ÷ 5

Proportional and Nonproportional Relationships Determine whether the set of numbers in each table forms a proportion. 13. Boxes 1 2 3 4 Pens

8

14. Number of People Brownies Eaten

16

24

32

2

4

6

8

2

5

7

10

15. FESTIVALS A customer at the ring toss booth gets 8 rings for $2. Write an equation relating the cost to the number of rings. At this same rate, how much would a customer pay for 11 rings? for 20 rings?

6–3

Using Proportions

3 12 9 22.5 18. _ =_ 7 y

7 cm

7

(pp. 297–300)

Example 2 Determine whether the set of numbers in the table forms a proportion. Distance (meters)

30

56

69

80

Time (minutes)

1

2

3

4

Write the rate of distance to time for each minute in simplest form. 30 _

56 _ _ = 28

1

2

1

69 _ _ = 23 3

1

80 _ _ = 20 4

1

Since the rates are not equal, the set of numbers do not form a proportion.

(pp. 302–306)

3 15 =_ . Example 3 Solve _

Solve each proportion. n 4 16. _ =_

35 cm

Next, divide the numerator and denominator by the GCF, 5.

84 21 17. _ =_ x

120 5 0.6 19. _ =_ 7.5 k

20. REAL ESTATE A homeowner whose house is assessed for $120,000 pays $1800 in taxes. At the same rate, what is the tax on a house assessed at $135,500?

15 _3 = _ 7

x

x 7 Write the proportion.

3 · x = 7 · 15 Cross products 3x = 105

Multiply.

3x 105 _ =_


3

3

x = 35

The solution is 35.


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Scale Drawings and Models

(pp. 308–312)

On the model of a ship, the scale is 1 inch = 12 feet. Find the actual length of each room. 21. 22. 23.

Room Stateroom Galley Gym

Model Length 0.9 in. 3.8 in. 6.0 in.

0.25 in. 1.75 in. drawing length drawing length _ =_ actual length 1 ft actual length x ft

0.25 · x = 1 · 1.75

24. MAPS The length of the expressway is 900 miles. If 0.5 inch on a map represents 50 miles, what is the length of the expressway on the map?

6–5

Fractions, Decimals, and Percents

26. 8.8%

27. 120% 28. 87.5%

Express each decimal or fraction as a percent. Round to the nearest tenth percent if necessary. 29. 0.24

30. 1.9

2 31. _ 5

Cross multiply.

0.25x = 1.75

Simplify.

x=7

Divide.

The actual length of the pond is 7 feet.

(pp. 313–318)

Express each percent as a fraction or mixed number in simplest form and as a decimal. 25. 35%

Example 4 A scale drawing shows a pond that is 1.75 inches long. The scale on the drawing is 0.25 inch = 1 foot. What is the length of the actual pond?

6 32. _ 80

Example 5 Express 60% as a fraction in simplest form and as a decimal. 60 3 or _ 60% = _

060% = 0.60 or 0.6

5

100

Example 6 Express 0.38 as a percent. 0.38 = 0.38 or 38% 5 Example 7 Express _ as a percent.

_5 = 0.625 or 62.5%

8

8

33. PETS In a survey, 0.2 of American households own a dog, one-fourth own cats, and 7% own a bird. Which group is largest? Explain.

6–6

Using the Percent Proportion

(pp. 322–326)

Use the percent proportion to solve each problem. 34. 18 is what percent of 45? 35. What is 74% of 110? 36. 23 is 92% of what number? 37. MUSIC Thirty percent of the music that Meghan owns is classical. If Meghan owns 120 albums, how many are classical? 350 Chapter 6 Ratio, Proportion, and Percent

Example 8 Forty-eight is 32% of what number? 48 32 _ =_ b

100

Write the percent proportion.

48 · 100 = b · 32 Find the cross products. 4800 = 32b 150 = b


So, 48 is 32% of 150.



6–7

Finding Percents Mentally

(pp. 327–331)

Find the percent of each number mentally. 38. 50% of 86

39. 20% of 55

1 40. 33_ % of 24

41. 90% of 60

3

Example 9 Find 20% of $45 mentally. 1 of 45 20% of 45 = _ 5

Think: 20% = _. 1 5

Think: 15 of 45 is 9.

=9 So, 20% of $45 is $9.

Estimate. Explain which method you used to estimate.

Example 10 Estimate 32% of 150.

42. 48% of 32

43. 67% of 30

1 1 32% is about 33_ % or _ .

1 44. _ % of 304

45. 147% of 200

_1 of 150 is 50.

3

3

3

3

46. BASKETBALL Tito has 244 free throw attempts in his high school career. If he was successful 77% of the time, about how many free throws did he make?

So, 32% of 150 is about 50.

47. GEOGRAPHY The United States has 88,633 miles of shoreline. Of the total amount, 35% is located in Alaska. About how many miles of shoreline are located in Alaska?

6–8

Using Percent Equations

(pp. 332–336)

Solve each problem using the percent equation.

Example 11 119 is 85% of what number?

48. 24 is what percent 50?

The part is 119, and the percent is 85%. Let n represent the whole.

49. What is 12.5% of 68?

119 = 0.85n

Write 85% as the decimal 0.85.

50. 56 is 28% of what number?

119 0.85n _ =_ 0.85 0.85

Divide each side by 0.85.

51. 35.7 is what percent of 17?

140 = n

So, 119 is 85% of 140.

52. SHOPPING A jersey is on sale for 50% off the original price. A week later, the manager takes another 50% off the sale price. Is the jersey now free? Explain. 53. INVESTMENTS What is the interest on 1 years? $10,000 invested at 9% for 1_ 2 Round to the nearest cent.


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Percent of Change

(pp. 338–342)

Find the percent of change. Round to the nearest tenth, if necessary. Then state whether each percent of change is a percent of increase or a percent of decrease. 54. from 40 ft to 12 ft

net weight - amount of change = ___ original weight

56. from 80 lb to 77 lb 57. from 29 min to 54 min

≈ -0.611 or -61.1%

58. CLUBS The number of members in the recycling club increased by 15 people. If the club had 12 members previously, what was the percent of increase of the members in the club?

Using Sampling to Predict

percent of change

14 - 36 =_ 36 -22 _ = 36

55. from 80 cm to 96 cm

6–10

Example 12 Find the percent of change from 36 pounds to 14 pounds.

The percent of decrease is about 61.1%.

(pp. 343–347)

Identify each sample as biased or unbiased and describe its type. Explain your reasoning. 59. To determine the weekly top ten songs, the local radio station asks people to log onto their Web site and vote for their favorite song. 60. To determine what type of dessert people in a community like, Sara surveys 20% of the people who enter three different chocolate shops. 61. MUSIC Sixty-three out of the 105 students in the band said that their favorite class was music. Is this sampling representative of the entire school? If so, how many of the 848 students who attend the school would say music is their favorite class?

Example 13 AUTOMOBILES From a batch of 100,000 cars, the manufacturer tests the exhaust on every 500th car. The manufacturer found that 1 car was below standards. Is this sampling method valid? If so, find how many of the 100,000 cars you can expect to be below standards. Explain your reasoning. This is a systematic random sample because the samples are selected according to a specific interval. So, this sampling method is reasonable and will produce a valid prediction. Since every 500 cars were sampled, there were a total of 100,000 ÷ 500 or 200 cars sampled and 1 was substandard. The number of cars that were below standards 1 or 0.5%. were _ 200

Find 0.5% of 100,000. n = 0.005 × 100,000 or 500 Multiply. So, there are approximately 500 substandard cars in the batch. 352 Chapter 6 Ratio, Proportion, and Percent

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6

Practice Test

Express each rate as a unit rate. Round to the nearest tenth. 1. 145 miles in 3 hours 2. 245 miles every 6 hours 8.4 1.2 =_ a 3. What value of y makes _ y 1.1 proportion?

20. GIFTS Gifts, Inc. is selling their bobbleheads for 25% off their regular price. If a bobblehead costs $49.95, for how much is it on sale? Find the percent of change. Round to the nearest tenth, if necessary.

4. WATER Which bottle of water costs more per ounce: $1.25 for 12 ounces or $1.50 for 16 ounces?

21. 175 pounds to 140 pounds 22. 1 hour to 1 hour 10 minutes

5. TAXI The taxi cab company charges a $2 fee plus $1.10 for each mile driven. Complete the table and determine whether the pattern forms a proportion.

23. MULTIPLE CHOICE A builder is designing a swimming pool that is 8.5 inches in length on the scale drawing. The scale of the drawing is 1 inch = 6 feet. What is the length of the actual swimming pool?

Miles Driven Cab Fare

1


7. 225%

8. 0.6%

Express each decimal or fraction as a percent. Round to the nearest tenth, if necessary. 9. 0.47

30 11. _

10. 0.025

A 44 ft

C 49 ft

B 47 ft

D 51 ft

24. TOASTERS To determine the quality of toasters coming off an assembly line, the manager pulls every fiftieth toaster off the line and toasts a piece of bread. Identify this sample as biased or unbiased and describe its type. Explain your reasoning.

22

Use the percent proportion to solve each problem.

25. MULTIPLE CHOICE The table lists the reasons shoppers use online customer service.

12. 36 is what percent of 80? 13. 35.28 is 63% of what number? 14. Find 35% of 200. Estimate. Explain which method you used to estimate. 15. 25% of 82

16. 62% of 77

1 % of 2453 17. _

18. 439% of 61

12

19. INVESTMENTS Find the interest on $2700 that 1 years. is invested at 4% for 2_ 2


Reasons

Percent

Track Delivery Product Information Verify Shipping Charges Transaction Help

54 24 17 5

Out of 350 shoppers who own a computer, how many would you expect to say they use online customer service to track packages? F 189

H 84

G 154

J 19


353

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6


Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. Of the coins in a piggy bank, 20% are quarters, 35% are dimes, 15% are nickels, and 30% are pennies. There are 200 coins in the bank altogether. Which proportion can be used to find q, the total number of quarters in the piggy bank? 20 200 A_ =_ q 100 q 20 =_ B _ 200 100 20 _ = 200 C _ q

3. GRIDDABLE Ana earns $6.80 per hour when she works on weekdays. She earns twice that amount per hour when she works on weekends. If Ana worked 4 hours on Tuesday, 4 hours on Thursday, and 5 hours on Saturday, then how much did she earn in dollars? 4. The table shows the number of new car sales at a car dealership over the past several months. Month

4

5

6

7

8

9

10

11

Number of Sales

28

35

40

37

33

31

29

41

Suppose the dealership has a special promotion and sells 84 cars during December. Which measure of data will change the most?

100 q 20 _ =_ D 300 100

2. David surveyed 50 golfers on a Saturday morning about their favorite outdoor activity. The results are shown in the table below.

A The mean B The median C The mode D All measures will be affected equally.

Favorite Outdoor Activity Activity Hiking Golf Swimming Other

Number of Votes 8 20 9

Question 4 If you are unsure of the correct answer, eliminate the choices you know are incorrect. Then consider the remaining choices.

13

Based on these results, David concluded that playing golf is the favorite outdoor activity among people in his city. Which is the best explanation for why his conclusion might not be valid? F The survey should have been done on different days of the week. G The survey should have been done with men and women golfers. H The sample was not representative of all the people in the city. J There are more golfers on the weekend than during the week. 354 Chapter 6 Ratio, Proportion, and Percent

5. A statistician is organizing the winning percentages of the top hockey teams in the league. Choose the group of percentages that is listed in order from greatest to least. F 0.518, 0.517, 0.524, 0.508 G 0.524, 0.518, 0.517, 0.508 H 0.508, 0.524, 0.518, 0.517 J 0.508, 0.517, 0.518, 0.524 6. GRIDDABLE Nina wants to buy a new pair of inline skates. The regular price of the skates is $90, but they are on sale this week for 15% off. What is the sale price of the skates in dollars? Standardized Test Practice at pre-alg.com


7. A commemorative coin was worth $4.50 when it was issued in 2000. The table shows the value of the coin several years after it was issued.

10. Four-fifteenths of a flower bouquet had 6 of the bouquet had yellow flowers and _ 15 red flowers. What part of the bouquet was NOT yellow or red? 4 F _

Value of Commemorative Coin Year

Value of Coin

2000

$4.50

2001

$5.00

2002

$5.75

2003

$6.75

2004

$8.00

2005

$9.50

3 H_

5 _ G 11 15

5

1 J _ 3

11. In 2003, a new planet was discovered beyond Pluto. This new planet is 1010 miles from the Sun. Which of the following represents this number in standard notation? A 100 mi B 10,000 mi

Based on the information in the table, what is a reasonable prediction for the value of the coin in 2009? A $15.50

C $18.75

B $18.00

D $19.25

C 10,000,000 mi D 10,000,000,000 mi

Pre-AP Record your answers on a sheet of paper. Show your work.

8. The Milky Way galaxy is made up of about 200 billion stars, including the Sun. Write this number in scientific notation. F 2.0 × 108

H 2.0 × 1010

G 2.0 × 109

J 2.0 × 1011

12. An electronics store is having a sale on certain models of televisions. Mr. Castillo would like to buy a television that is on sale. This television normally costs $679.

9. The total number of points scored by 12 players on a basketball team is 870 points for the season. Craig scored 184 points. Which equation can be used to find p, the average number of points scored by Craig’s teammates?

a. What price, not including tax, will Mr. Castillo pay if he buys the television on Saturday?

870 + 184 A p=_ 11 870 - 184 B p=_

b. What price, not including tax, will Mr. Castillo pay if he buys the television on Wednesday?

11 870 - 184 C p=_ 11 184 D p = 870 + _ 11

c. How much money will Mr. Castillo save if he buys the television on Saturday?


1

2

3

4

5

6

7

8

9

10

11

12

Go to Lesson...

6-2

6-10

1-1

5-9

5-2

6-8

1-7

4-8

5-9

5-5

4-8

6-8


355

Functions and Graphing

7 •

Identify proportional or nonproportional linear relationships in problem situations and solve problems.

•

Make connections among various representations of a numerical relationship.

•


Key Vocabulary direct variation (p. 378) function (p. 359) rate of change (p. 371) slope (p. 384)

Real-World Link INSECTS If x is the number of chirps a cricket makes every 15 seconds, the equation y = x + 40 can help you estimate y, the outside temperature in degrees Fahrenheit.

Functions and Graphing Make this Foldable to collect examples of functions and graphs. Begin with an 11’’ × 17’’ sheet of paper.

1 Fold the short sides so they meet in the middle.

3 Open and Cut along the second fold to make four tabs. Staple a sheet of grid paper inside.

356 Chapter 7 Functions and Graphing Cisca Casteljins/Foto Natura/Minden Pictures


4 Add axes as shown. Label the quadrants on the tabs.

X

/

s

W



Option 1


Express each relation as a table. Then determine the domain and range. (Used in Lesson 7-1)

1. {(0, 4), (-3, 3)}

2. {(-5, 11), (2, 1)}

3. {(6, 8), (7, 10), (8, 12)}

Example 1

Express the relation {(1, 2), (2, 1), (3, 12)} as a table. Then determine the domain and range. The domain is {(1, 2, 3)}. x y

4. MARATHON Kelly ran at a rate of 8 feet per second. What is the distance Kelly ran in 24 seconds? in 1 minute 15 seconds?

The range is {(2, 1, 12)}.

1

2

2

1

3

12

Use the coordinate plane to name the point for each ordered pair. (Used in Lesson 7-5) y 5. (-3, 0) B 6. (3, -2) A 7. (-4, -2) 8. (0, 4) C F 9. (4, 6) x O 10. (4, 0) D E

Example 2

11. DIRECTIONS On a coordinate plane, the movie theater is located at (5, -6) and the grocery store is located at (-2, 6). Write directions on how to walk from the movie theater to the grocery store.

Point N is at (-3, 2).

Write an equation that describes each sequence. (Used in Lesson 7-8) 12. 4, 5, 6, 7, . . . 13. 13, 14, 15, 16, . . .

Example 3

Use the coordinate plane to name the point for (-3, 2). y Step 1 Start at (0, 0). K

Step 2 Step 3

Move 3 units to the left.

M

N

x

O

Move 2 units up.

L J

14. FITNESS Becky started an exercise program that calls for 12 minutes of jogging each day during the first week. Each week thereafter, Becky increases the time she jogs by 5 minutes. In which week will she first jog more than 30 minutes?

Write an equation that describes the sequence 2, 5, 8, 11, 14, . . . Term Number (n)

1

2

3

4

Term (t)

2

5

8

11

The difference of the term number is 1. The difference in the terms is 3. So, common difference is 3 times the difference in the term numbers minus 1. So, t = 3n - 1. Chapter 7 Get Ready for Chapter 7

357

EXPLORE

7-1

Algebra Lab

Input and Output In a function, there is a relationship between two quantities or sets of numbers. You start with an input value, apply a function rule of one or more operations, and get an output value. In this way, each input is assigned exactly one output.

ACTIVITY Step 1 To make a function machine, draw three squares in the middle ⫺5 of a 3-by-5-inch index ⫺4 Input Rule Output card, shown here in ⫺3 blue. ⫺5 ⫻2⫹3 ⫺7 ⫺2 Step 2 Cut out the square on ⫺1 the left and the square 0 on the right. Label the 1 left “window” INPUT 2 and the right 0 3 “window” OUTPUT. 1 4 Step 3 Write a rule such as 2 “× 2 + 3” in the center 3 square. 4 Step 4 On another index card, list the integers from -5 to 4 in a column close to the left edge. Step 5 Place the function machine over the number column so that -5 is in the left window. Step 6 Apply the rule to the input number. The output is -5 × 2 + 3, or -7. Write -7 in the right window.

ANALYZE THE RESULTS 1. Slide the function machine down so that the input is -4. Find the output and write the number in the right window. Continue this process for the remaining inputs. 2. Suppose x represents the input and y represents the output. Write an algebraic equation that represents what the function machine does. 3. Explain how you could find the input if you are given a rule and the corresponding output. 4. Determine whether the following statement is true or false. Explain. The input values depend on the output values. 5. Write an equation that describes the relationship between the input value x and output value y in each table.

Input -1 0 1 3

Output ⫺2 0 2 6

Input ⫺2 ⫺1 0 1

6. Write your own rule and use it to make a table of inputs and outputs. Exchange your table of values with another student. Use the table to determine each other’s rule. 358 Chapter 7 Functions and Graphing

Output 2 3 4 5

7-1

Functions

Main Ideas • Determine whether relations are functions. • Use functions to describe relationships between two quantities.

New Vocabulary function vertical line test

The table shows the time it should take a scuba diver to ascend to the surface from several depths to prevent decompression sickness. a. On grid paper, graph the depths and times as ordered pairs (depth, time). b. Describe the relationship between the two sets of numbers.

Depth (ft)

Time (s)

7.5 15 22.5 30

15 30 45 60

Source: diverssupport.com

c. If a scuba diver is at 45 feet, what is the best estimate for the amount of time she should take to ascend? Explain.

Relations and Functions Recall that a relation is a set of ordered pairs. A function is a special relation in which each member of the domain is paired with exactly one member in the range. Not a Function {(-2, 1), (-2, 3), (-5, 4), (-9, 7)}

Function {(-2, 1), (-4, 3), (-5, 4), (-9, 7)} Ó { x

£ Î { Ç

Ó

£ Î { Ç

x

Review Vocabulary

This is a function because each domain value is paired with exactly one range value

domain the set of x-coordinates in a relation; Example: The domain of {(1, 4), (-3, -7)} is {1, -3}.

Since functions are relations, they can be represented using ordered pairs, tables, or graphs.

range the set of y-coordinates in a relation; Example: The range of {(1, 4), (-3, -7)} is {4, -7}. (Lesson 1-6)

EXAMPLE

This is not a function because -2 in the domain is paired with two range values, 1 and 3.

Ordered Pairs and Tables as Functions

Determine whether each relation is a function. Explain. a. {(-3, 1), (-2, 4), (-1, 7), (0, 10), (1, 13)} This relation is a function because each element of the domain is paired with exactly one element of the range. b.

x y

5 1

3 3

2 1

0 3

-4 -2

1A. {(5, 1), (6, 3), (7, 5), (8, 0)}

-6 2

This is a function because for each element of the domain, there is only one corresponding element in the range. 1B.

x y

-1

-6

-3

-1

-5

-2

7

6

2

8

-2

1

Personal Tutor at pre-alg.com Lesson 7-1 Functions

359

Vocabulary Link Function Everyday Use a relationship in which one quality or trait depends on another. Height is a function of age. Math Use a relationship in which a range value depends on a domain value, y is a function of x.

Another way to determine whether a relation is a function is to apply the vertical line test to the graph of the relation. Use a pencil or straightedge to represent a vertical line.

y

Place the pencil at the left of the graph. Move it to the right across the graph. If, for each value of x in the domain, it passes through no more than one point on the graph, then the graph represents a function.

EXAMPLE

x

O

Use a Graph to Identify Functions

Determine whether the graph at the right is a function. Explain your answer.

y

The graph represents a relation that is not a function because it does not pass the vertical line test. At least one input value has more than one output value. By examining the graph, you can see that when x = 2, there are three different y values.

O

x

2. Determine whether the graph of times and distances below is a function. Explain your answer.

Time (min) 4 12 16 20 28

Distance (m) 1 3 4 5 7

Distance (mi)

Describe Relationships A function describes the relationship between two quantities such as time and distance. For example, the distance you travel on a bike depends on how long you ride the bike. In other words, distance is a function of time. 9 8 7 6 5 4 3 2 1 0

y

x 4 8 12 16 20 2428 32 36 Time (min)

SCUBA DIVING The table shows the water pressure as a scuba diver descends.

Depth (ft)

a. Do these data represent a function? Explain. Real-World Link Most of the ocean’s marine life and coral live and grow within 30 feet of the surface. Source: scuba.about.com

This relation is a function because at each depth, there is only one measure of pressure. b. Describe how water pressure is related to depth. Water pressure depends on the depth. As the depth increases, the pressure increases.

360 Chapter 7 Functions and Graphing Peter/Stef Lamberti/Getty Images

0 1 2 3 4 5

Water Pressure (lb/ft2) 0 62.4 124.8 187.2 249.6 312.0



3. SALES Do these data in the table represent a function? Describe how price is related to the number of balloons purchased. Number of Balloons Price per Balloon

100 $0.99

200 $0.90

300 $0.79

400 $0.60

500 $0.50

Determine whether each relation is a function. Explain. Example 1 (p. 359)

1. {(13, 5), (-4, 12), (6, 0), (13, 10)}

2. {(9.2, 7), (9.4, 11), (9.5, 9.5), (9.8, 8)}

3. Domain Range

4.

3 -2 5 -4 3

-3 -1 0 1 2

Example 2

5.

x 5 2 -7 2 5

y 4 8 9 12 14

6.

y

y

(p. 360)

O

Example 3 (pp. 360–361)

HOMEWORK

HELP


x

x

O

MEASUREMENTS For Exercises 7 and 8, use the data in the table. 7. Do the data represent a function? Explain. 8. Is there any relation between foot length and height?

Name Remana

Foot Length (cm) 24

Height (cm) 163

Enrico

25

163

Jahad

24

168

Cory

26

172

Determine whether each relation is a function. Explain. 9. {(-1, 6), (4, 2), (2, 36), (1, 6)}

10. {(-2, 3), (4, 7), (24, -6), (5, 4)}

11. {(9, 18), (0, 36), (6, 21), (6, 22)}

12. {(5, -4), (-2, 3), (5, -1), (2, 3)}

13. Domain

14.

-4 -2 0 3

15.

x -7 0 11 11 0

y 2 4 6 8 10

Range -2 1 2 1

16.

Domain -1 -2 -2 -6 x 14 15 16 17 18

Range 5 5 1 1

y 5 10 15 20 25 Lesson 7-1 Functions

361

Determine whether each relation is a function. Explain. 17.

x

O

19.

The lowest wind chill temperature ever recorded at an NFL game was -59°F in Cincinnati, Ohio, on January 10, 1982. Source: Southern AER

EXTRA

PRACTICE


H.O.T. Problems

y

x

FARMING For Exercises 21–24, use the table that shows the number and size of farms in the United States every decade from 1950 to 2000. 21. Is the relation (year, number of farms) a function? Explain. 22. Describe how the number of farms is related to the year. 23. Is the relation (number of farms, average size of farms) a function? Explain. 24. Describe how the average size of farms is related to the year.

x

O

20.

y

O

Real-World Link

18.

y

y

x

O

Farms in the United States Number

Average Size

(millions)

(acres)

1950

5.6

213

1960

4.0

297

1970

2.9

374

1980

2.4

426

1990

2.1

460

2000

2.2

434

Year

Source: U.S. Dept. of Agriculture

Tell whether each statement is always, sometimes, or never true. Explain. 25. A function is a relation. 26. A relation is a function. ANALYZE TABLES For Exercises 27 and 28, use the table that shows how various wind speeds affect the actual temperature of 15°F. 27. Do the data represent a function? Explain. 28. Describe how wind chill temperatures are related to wind speed.

Wind Speed Wind Chill (mph) Temperature (°F) 0 15 10 3 20 -2 30 -5 40 -8

29. RESEARCH Use the Internet or another source to Source: National Weather Service find the complete wind chill table. Does the data for actual temperature and wind chill temperature for a specific wind speed represent a function? Explain.

30. OPEN ENDED Draw the graph of a relation that is not a function. Explain why it is not a function. 31. REASONING Describe three ways to represent a function. Show an example of each. Then describe three ways to represent a relation that is not a function and show an example of each.

362 Chapter 7 Functions and Graphing Craig Tuttle/CORBIS

CHALLENGE The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. 32. Determine whether the inverse of the relation {(4, 0), (5, 1), (6, 2), (6, 3)} is a function. 33. Is the inverse of a function always, sometimes, or never a function? Give an example to explain your reasoning. 34.

Writing in Math

How can the relationship between water depth and time to ascend to the water’s surface be a function? Include a discussion about whether water depth can ever have two corresponding times to ascend to the water’s surface.

35. Which statement is true about the data in the table? A The data represent a function. B The data do not represent a function. C As the value of x increases, the value of y increases.

x -4 2 5 10 12

y -4 16 8 -4 15

D A graph of the data would not pass the vertical line test.

36. The table shows the water temperatures at various depths in a lake. Describe how temperature is related to the depth. Depth (ft) Temperature (°F )

0 74

10 72

20 71

30 61

40 55

50 53

F The water temperature stays the same as the depth increases. G The water temperature decreases as the depth increases. H The water temperature increases as the depth increases. J The water temperature decreases as the depth decreases.

37. TECHNOLOGY The table shows the results of a survey in which middle school students were asked whether they ever used the Internet for the activities listed. Use the data to predict how many students in a middle school of 650 have used the Internet to do research for school. (Lesson 6-10)

Activity E-mail research for school instant message games

Percent 71% 70% 61% 71%

Source: Atlantic Research and Consulting

Find each percent of change. Round to the nearest tenth, if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. (Lesson 6-9) 38. from $56 to $49

39. from 110 mg to 165 mg

40. SCIENCE The length of a DNA strand is 0.0000007 meter. Write the length of a DNA strand using scientific notation. (Lesson 4-7)

Evaluate each expression if x = 4 and y = -1. (Lesson 1-3) 41. 3x + 1 42. 2y 43. y + 6

44. 20 - 4x Lesson 7-1 Functions

363


EXTEND

7-1

Function Tables

You can use a TI-83/84 Plus graphing calculator to create function tables. By entering a function and the domain values, you can find the corresponding range values.

ACTIVITY Use a function table to find the range of y = 3n + 1 if the domain is {-5, -2, 0, 0.5, 4}. Step 1 Enter the function.

Step 2

• The graphing calculator uses X for the domain values and Y for the range values. So, Y = 3X + 1 represents y = 3n + 1.

• Use TBLSET to select Ask for the independent variable and Auto for the dependent variable. Then you can enter any value for the domain.

• Enter Y = 3X + 1 in the Y= list. KEYSTROKES:

3 X,T,␪,n

Format the table.

KEYSTROKES:

1

2nd [TBLSET]

ENTER

ENTER

Step 3 Find the range by entering the domain values. • Access the table. KEYSTROKES:

2nd [TABLE]

y 3(5) 1 14

• Enter the domain values. KEYSTROKES: -5

ENTER -2 ENTER . . . 4 ENTER

The range is {-14, -5, 1, 2.5, 13}.

EXERCISES Use the [TABLE] option on a graphing calculator to complete each exercise. 1. Suppose you are using the formula d = rt to find the distance d a car travels for the times t in hours given by {0, 1, 3.5, 10}. a. If the rate is 60 miles per hour, what function should be entered in the Y= list? b. Make a function table for the given domain. c. Between which two times in the domain does the car travel 150 miles? d. Describe how a function table can be used to estimate the time it takes to drive 150 miles. 2. Serena is buying one packet of pencils for $1.50 and a number of fancy folders x for $0.40 each. The total cost y is given by y = 1.50 + 0.40x. a. Use a function table to find the total cost if Serena buys 1, 2, 3, 4, and 12 folders. b. Suppose plain folders cost $0.25 each. Enter y = 1.50 + 0.25x in the Y = list as Y2. How much does Serena save if she buys pencils and 12 plain folders rather than pencils and 12 fancy folders? 364 Chapter 7 Functions and Graphing

7-2

Representing Linear Functions

Main Ideas


• Solve linear equations with two variables.

Peaches cost $1.50 per can.

• Graph linear equations using ordered pairs.

a. Complete the table to find the cost of 2, 3, and 4 cans of peaches.

New Vocabulary linear equation

Number of Cans (x )

1.50x

Cost (y )

1

1.50(1)

1.50

2

3 b. On grid paper, graph the ordered pairs 4 (number, cost). Then draw a line through the points.

c. Write an equation representing the relationship between number of cans x and cost y.

Solutions of Equations An equation such as y = 1.50x is called a linear equation. A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. Reading Math Input and Output The variable for the input is called the independent variable because the values are chosen and do not depend upon the other variable. The variable for the output is called the dependent variable because it depends on the input value.

Solutions of a linear equation are ordered pairs that make the equation true. One way to find solutions is to make a table. Consider y = -x + 8. y = -x + 8 x Step 1 Choose any convenient values for x.

⎧ -1 0 ⎨ 1 2 ⎩

y = -x + 8

y

(x, y)

y = -(-1) + 8

9

(-1, 9)

y = -(0) + 8

8

(0, 8)

y = -(1) + 8

7

(1, 7)

y = -(2) + 8

6

(2, 6)

Step 2 Substitute the values for x.

⎫ ⎬ ⎭

Step 4 Write the solutions as ordered pairs.

Step 3 Simplify to find the y-values.

So, four solutions of y = -x + 8 are (-1, 9), (0, 8), (1, 7), and (2, 6).

EXAMPLE

Use a Table of Ordered Pairs

Find four solutions of y = 2x - 1. Choose four values for x. Then substitute each value into the equation and solve for y. Four solutions are (0, -1), (1, 1), (2, 3), and (3, 5).

x

y = 2x - 1

y

(x, y)

0

y = 2(0) - 1

1

(0, -1)

1

y = 2(1) - 1

1

(1, 1)

2

y = 2(2) - 1

3

(2, 3)

3

y = 2(3) - 1

5

(3, 5)

1. Find four solutions of y = x + 5. Lesson 7-2 Representing Linear Functions

365

Solve an Equation for y CELL PHONES Games cost $8 to download onto a cell phone. Ring tones cost $1. Find four solutions of 8x + y = 20 in terms of the numbers of games x and ring tones y Darcy can buy with $20. Explain each solution. First, rewrite the equation by solving for y. 8x + y = 20 Write the equation. 8x + y - 8x = 20 - 8x Subtract 8x from each side. y = 20 - 8x Simplify. Choose four x values and substitute them into y = 20 - 8x. Choosing x-Values

(1, 12)

It is often convenient to choose 0 as an x value to find a value for y.

(2, 4)

→ She can buy 1 game and

12 ring tones. → She can buy 2 games and

4 ring tones.

(_14 , 18)

x

y = 20 - 8x

y

(x, y)

1

y = 20 - 8(1)

12

(1, 12)

2

y = 20 - 8(2)

4

(2, 4)

_1 4

1 y = 20 - 8 _

18

(_14 , 18)

5

y = 20 - 8(5)

-20

(5, -20)

(4)

→ This solution does not make sense in the situation because there

cannot be a fractional number of games.

(5, -20) → This solution does not make sense in the situation because there cannot be a negative number of ring tones.

2. SHOPPING Fancy goldfish x cost $3, and regular goldfish y cost $1. Find three solutions of 3x + y = 8 in terms of the number of each type of fish Tyler can buy for $8. Describe what each solution means. Personal Tutor at pre-alg.com

Graph Linear Equations A linear equation can also be represented by a graph. Linear Equations

Reading Math Linear Equations Graphs of all “linear” equations are straight lines. The coordinates of all points on a line are solutions of the equation.

y

y

y y 1x 3

x

O yx1

x

O

x

O

y 2x

Nonlinear Equations y

y

y

y 2x 3

y x2 1 O

x

O

x

x

O y 3 x

366 Chapter 7 Functions and Graphing

EXAMPLE

Graph a Linear Equation

Graph y = x + 1 by plotting ordered pairs. First, find ordered pair solutions. Four solutions are (-1, 0), (0, 1), (1, 2), and (2, 3).

Plotting Points It is best to find at least three points. You can also graph just two points to draw the line and then graph one point to check.

x

y=x+1

y

(x, y)

-1

y = -1 + 1

0

(-1, 0)

0

y=0+1

1

(0, 1)

1

y=1+1

2

(1, 2)

2

y=2+1

3

(2, 3)

Plot these ordered pairs and draw a line through them. Note that the ordered pair for any point on this line is a solution of y = x + 1. The line is a complete graph of the function.

y

(1, 0)

x

O

CHECK It appears from the graph that (-2, -1) is also a solution. Check this by substitution. y=x+1 -1 -2 + 1 -1 = -1

(2, 3) (1, 2) (0, 1)

yx1

Write the equation. Replace x with -2 and y with -1. Simplify.

3. Graph y = 2x - 1 by plotting ordered pairs. A linear equation is one of many ways to represent a function. Representing Functions Words Table of Ordered Pairs

Equation

Example 1 (p. 365)

(p. 366)

Example 3 (p. 367)

x

y

0

-3

1

-2

2

-1

3

0

Graph

y

x

O

y x3

y=x-3

Find four solutions of each equation. Show each solution in a table of ordered pairs. 1. y = x + 8

Example 2

The value of y is 3 less than the corresponding value of x.

2. y = 4x

3. y = 2x - 7

4. -5x + y = 6

5. SCIENCE The distance in miles d that light travels in t seconds is given by the linear function d = 186,000t. Find two solutions of this equation and describe what they mean. Graph each equation by plotting ordered pairs. 6. y = x + 3


7. y = 2x - 1

8. x + y = 5

Lesson 7-2 Representing Linear Functions

367

HOMEWORK

HELP


Copy and complete each table. Use the results to write four solutions of the given equation. Show each solution in a table of ordered pairs. 9. y = x - 9 x

x-9

-1

-1 - 9

10. y = 2x + 6 y

x

2x + 6

-4

2(-4) + 6

0

0

4

2

7

4

y

Find four solutions of each equation. Show each solution in a table of ordered pairs. 11. y = x + 4

12. y = x - 7

13. y = 3x

14. y = -5x

15. y = 2x - 3

16. y = 3x + 1

17. x + y = 9

18. x + y = -6

19. 4x + y = 2

20. 3x - y = 10

21. 2x - y = -4

22. -5x + y = 12

MEASUREMENT The equation y = 0.62x describes the approximate number of miles y in x kilometers. 23. Describe what the solution (8, 4.96) means. 24. About how many miles is a 10-kilometer race? FITNESS During a workout, a target heart rate y in beats per minute is represented by y = 0.7(220 - x), where x is a person’s age. 25. Compare target heart rates of people 20 years old and 50 years old. 26. In which quadrant(s) would the graph of y = 0.7(220 - x) make sense? Explain your reasoning. Graph each equation by plotting ordered pairs. 27. y = x + 2

28. y = x + 5

29. y = x - 4

30. y = -x - 6

31. y = -2x + 2

32. y = 3x - 4

33. x + y = 1

34. x - y = 6

ANALYZE TABLES Determine whether each relation or equation is linear. Justify your answer. 35.

Real-World Link Walking is the top sports activity among Americans over the age of 7. Source: Statistical Abstract of the United States

x

y

-1

36.

1

-1

-1

0

0

0

-1

1

1

1

-1

4

2

-1

-2

-1

0

0

1

2

2

4

2

39. y = x2

40. y = 5

GEOMETRY For Exercises 41–44, use the following information. The formula for the perimeter of a square with sides s units long is P = 4s. 41. Find three ordered pairs that satisfy this condition. 42. Draw the graph that contains these points. 43. Why do negative values of s make no sense in the context of the situation? 44. Does this equation represent a function? Explain.

368 Chapter 7 Functions and Graphing Duomo/CORBIS

y

y

38. 3x + y = 20

37.

x

x

s s

s s

EXTRA

PRACTICE


H.O.T. Problems

45. MUSIC Aisha has $50 to spend on music. Single songs cost $1 to download and entire CDs cost $10. Find an equation to represent the number of single songs x and the number of CDs y Aisha can buy with $50. Then, find three ordered pairs that satisfy this condition. 46. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would graph a linear equation. 47. CHALLENGE Compare and contrast the functions shown in the tables. (Hint: Compare the change in values for each column.) 48. OPEN ENDED Write and graph a linear equation that has (-2, 4) as a solution.

x

y

x

y

-1

-2

-1

1

0

0

0

0

1

2

1

1

2

4

2

4

49. NUMBER SENSE Explain why a linear function has infinitely many solutions. 50.

Writing in Math How can linear equations represent a function? Include in your answer a description of four ways that you can represent a function, and an example of a linear equation that could be used to determine the cost of x pounds of bananas that are $0.49 per pound.

Miles Driven d

51. The graph describes the distance d Brock can drive his car on g gallons of gasoline. How many gallons of gas will he need to drive 280 miles? A 14 gal 80 B 16 gal 60 C 17 gal 40 D 19 gal

52. Monica has $440 to pay a painter to paint her bedroom. The painter charges $55 per hour. The equation y = 440 - 55x represents the amount of money left after x number of hours worked by the painter. What does the solution (7, 55) represent? F Monica has $7 left after 55 hours of painting.

20

0

1

2 3 Gallons g

4

G Monica has $55 left after 7 hours of painting. H The job is completed after 7 hours. J The job is completed after 55 hours.

Determine whether each relation is a function. Explain. (Lesson 7-1) 53. {(0, 6), (-3, 9), (4, 9), (-2, 1)} 54. (-0.1, 5), (0, 10), (-0.1, -5) 55. VOLUNTEERING In a survey of high school students, 28% said they volunteered at least 2 hours a week. In a class of 32 high school students, how many would you predict volunteer at least 2 hours a week? (Lesson 6-10) 56. SHOPPING Two cans of soup cost $0.88. One can costs $0.20 more than the other. How much would 5 cans of each type of soup cost? (Lesson 1-1)

PREREQUISITE SKILL Evaluate each expression. (Lesson 1-2) 18 - 10 57. _ 8-4

16 - 7 58. _ 1-3

46 - 22 59. _ 2005 - 2001

31 - 25 60. _ 46 - 21

Lesson 7-2 Representing Linear Functions

369

Language of Functions Equations that are functions can be written in a form called function notation as shown below. equation

function notation

y = 4x + 10

f(x) = 4x + 10 Read f(x) as f of x.

So, f (x) is simply another name for y. Letters other than f are also used for names of functions. For example, g(x) = 2x and h(x) = -x + 6 are also written in function notation. domain

range

In a function, x represents the domain values, and f(x) represents the range values.

f(x)

=

4 x + 10

f(3) represents the element in the range that corresponds to the element 3 in the domain. To find f(3), substitute 3 for x in the function and simplify.

Read f(3) as f of 3.

f (x) = 4x + 10 f (3) = 4(3) + 10 f (3) = 12 + 10 or 22

Write the function. Replace x with 3. Simplify.

So, the function value of f for x = 3 is 22.

Reading to Learn 1. RESEARCH Use the Internet or a dictionary to find the everyday meaning of the word function. Write a sentence describing how the everyday meaning relates to the mathematical meaning. 2. Write your own rule for remembering how the domain and the range are represented using function notation. 3. Copy and complete the table below. x

f(x) = 3x + 5

0

f(0) = 3(0) + 5

f(x)

1 2 3

4. If f(x) = 4x - 1, find each value. a. f(2)

b. f(-3)

c. f

_21

5. Find the value of x if f(x) = -2x + 5 and the value of f(x) is -7. 370 Chapter 7 Functions and Graphing

7-3

Rate of Change

Main Ideas • Solve problems involving rates of change.

New Vocabulary rate of change

The graph shows the changes in height and distance of a small airplane during 30 minutes of flight.

À«>i *ÃÌ Ónää

a. Between which two consecutive points did the vertical position of the airplane increase the most? decrease the most? How do you know?

6iÀÌV> *ÃÌ vÌ®

• Find rates of change.

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#

%

! "

£Èää

&

$

£Óää nää {ää ä

b. What is happening to the airplane between points A and B?

Óää Èää £äää £{ää £nää ÀâÌ> *ÃÌ vÌ®

c. Find the ratio of the vertical change to the horizontal change for each section of the graph. Which section is the steepest?

Rate of Change A rate of change is a rate that describes how one quantity changes in relation to another quantity. The rate of change of the vertical position of the airplane to the horizontal position from point B to point C is shown below. change in vertical position 2400 - 2000 ___ =_ 800 - 400

change in horizontal position

400 or 1 ft in vertical position for every 1 ft =_ 400

in horizontal position

TECHNOLOGY The table shows the growth of subscribers to satellite radio. Find the rate of change from 2003 to 2005. 7.7 - 1.8 rate of change = _ 2005 - 2003

= 2.95

← change in subscribers ← change in time Simplify.

Year 2003 2004 2005

Total Subscribers (millions) 1.8 4.5 7.7

Source: www.govtech.net

So, the rate of change from 2003 to 2005 was an increase of about 2.95 million people per year.

1. TECHNOLOGY Find the rate of change from 2004 to 2005 in the table above. Personal Tutor at pre-alg.com Lesson 7-3 Rate of Change

371

Rates of change can be positive or negative. This corresponds to an increase or decrease in the y-value between the two data points. When a quantity does not change over time, it is said to have a zero rate of change.

EXAMPLE

Compare Rates of Change

GEOMETRY The table shows how the perimeters of an equilateral triangle and a square change as side lengths increase. Compare the rates of change.

Perimeter y Triangle Square 0 0 6 8 12 16

Side Length x 0 2 4

change in y change in x 6 or 3 For each side length increase of 2, =_ the perimeter increases by 6. 2 change in y square rate of change = _ change in x For each side length increase of 2, _ = 8 or 4 the perimeter increases by 8. 2

triangle rate of change = _

y

28 24 20 16 12 8 4

Perimeter (in.)

The perimeter of square increases at a faster rate than the perimeter of a triangle. A steeper line on the graph indicates a greater rate of change for the square.

square

triangle

0

1 2 3 4 5 6 7 Side Length (in.)

x

2. GEOMETRY The perimeter of a regular hexagon changes as its side lengths increase by 1 inch. Compare this rate of change with the rates of change for the triangle and the square described above.

Real-World Link The temperature of the air is about 3˚F cooler for every 1000 feet increase in altitude. Source: hot-air-balloons.com

Negative Rate of Change EARTH SCIENCE The data points on the graph show the relationship between altitude and temperature. Find the rate of change.

/i«iÀ>ÌÕÀi LÛi -i> iÛi Y

change in temperature = __

Broken Lines In Example 3, there are no data points between the points that represent temperature. So, a broken line was used to help you easily see trends in the data.

change in altitude 4.2°C - 24°C Temperature goes from = __ 3 km - 0 km 24°C to 4.2°C. Altitude goes from 0 km to 3 km. -19.8°C =_ Simplify. 3 km = -6.6°C/km Express as a unit rate.

Ón Ó{ Óä

£È £Ó n

{ ä

£

X

Ó Î { x È Ç ÌÌÕì ®

So, the rate of change is -6.6°C/km, or a decrease of 6.6°C per 1-kilometer increase in altitude.

372 Chapter 7 Functions and Graphing David Keaton/CORBIS

/i«iÀ>ÌÕÀi Â ®

rate of change

3. Bracy received $200 cash for her birthday. The table shows the amount y remaining after x weeks. Find the rate of change. Interpret its meaning.

Weeks

Amount ($)

x

y

2

$160

4

$120

6

$80

Rates of Change Rate of Change

positive

zero

negative

Real-Life Meaning

increase

no change

decrease

Y

Y

Y

SLANTS UPWARD

SLANTS DOWNWARD HORIZONTAL LINE

Graph X

"

1.

35 30 25 20 15 10 5 0

Example 2 (p. 372)

"

X

Find the rate of change for each linear function. Amount of Water (gal)

(pp. 371–373)

2.

y

1 2 3 4 5 6 7 8 Time (min)

x

Time (h)

Wage ($)

x

y

0

0

1

12

2

24

3

36

3. AGE The graph shows the median ages that men got married in different years. Compare the rates of change of the median age between 1999 and 2001 and between 2001 and 2003. iÌÌ} >ÀÀi` ÓÇ°Ó ÓÇ°£ }i Þi>ÀÃ®

Examples 1, 3

X

"

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3OURCE "UREAU OF THE #ENSUS

Lesson 7-3 Rate of Change

373


Find the rate of change for each linear function. 4.

36 24 12

0

6.

5.

y

1 2 3 Number of Feet

Time (min)

Temperature (˚C)

HELP

Number of Inches

HOMEWORK

x

Temperature (°F)

y

28 24 20 16 12 8 4 0

7.

x

400 1200 2000 Altitude (m)

Time (h)

Distance (mi) y

x

y

x

0

58

0.0

0

1

56

0.5

25

2

54

1.5

75

3

52

3.0

150

TELEVISION The graph shows the percent of households that had cable television in the United States. 8. Find the rate of change in the percent of households from 2001 to 2003. 9. Find the rate of change in the percent of households from 1999 to 2001.

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È°ä

Èn°È Èn°Ó

ÈÇ°n ÈÇ°{

£ Óää£ 9i>À

ÓääÎ

ä £Ç

3OURCE .IELSEN -EDIA 2ESEARCH

10. ANALYZE TABLES The table shows late fees for DVDs and video games at a video store. Compare the rates of change.

EXTRA PRACTICE See pages 777, 800. Self-Check Quiz at pre-alg.com


Days Late x

DVDs

Video Games

0

0

0

2

$3

$4

4

$6

5

$7.50

$8 $10

>vÀ> `ÀÃ £xä

£Óx ÕLiÀ

11. ANALYZE GRAPHS The graph shows the populations of California condors in the wild and in captivity. Write several sentences that describe how the populations have changed since 1965. Include the rate of change for several key intervals.

Late Fee y

£ää Çx

7ILD

xä Óx

#APTIVITY

ä ½Èx ½Çä ½Çx ½nä ½nx ½ä ½x ½ää ½äx 9i>À

3OURCE 53 &ISH AND 7ILDLIFE 3ERVICE

H.O.T. Problems

12. CHALLENGE Describe the rate of change for a graph that is a horizontal line and a graph that is a vertical line. 13. OPEN ENDED Describe a real-world relationship between two quantities that involves a positive rate of change. 14.

Writing in Math Use the data about airplane flight on page 371 to explain how rate of change affects the graph of the airplane’s position.

15. The graph shows the population of Kendall. In which decade was there the greatest population change? £ä]äää ]äää

16. The table shows a relationship between time and altitude of a hot-air balloon. Which is the best estimate for the rate of change for the balloon from 1 to 5 seconds?

*«Õ>Ì

n]äää

Time (s) 1 2 3 4 5

Ç]äää È]äää x]äää {]äää ä

£Èä

£Çä

£nä 9i>À

£ä

Altitude (ft) 6.3 14.5 22.7 30.9 39.1

Óäää

A 1960–1970

C 1980–1990

B 1970–1980

D 1990–2000

F 7.6 ft/s

H 8.2 ft/s

G 7.8 ft/s

J

8.8 ft/s

Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2) 17. y = 2x + 5

18. y = -3x

19. x + y = 7

Determine whether each relation is a function. (Lesson 7-1) 20. {(2, 12), (4, -5), (-3, -4), (11, 0)}

21. {(-4.2, 17), (-4.3, 16), (-4.3, 15), (-4.3, 14)}

Solve each problem by using the percent equation. (Lesson 6-8) 22. 10 is what percent of 50?

23. Find 95% of 256.

24. RAIN A raindrop falls from the sky at about 17 miles per hour. How many feet per second is this? Round to the nearest foot per second. (Lesson 6-1) Find each product. Write in simplest form. (Lesson 5-3) 8 25. 7 _

( 21 )

10 14 26. -_ · -_ 15

28

14 2 27. _ · 3_ 15

7

5 1 28. -1_ · 3_ 4

9

PREREQUISITE SKILL Rewrite y = kx by replacing k with each given value. (Lesson 1-3) 29. k = 5

30. k = -2

31. k = 0.25

1 32. k = _ 3

Lesson 7-3 Rate of Change

375

7-4

Constant Rate of Change and Direct Variation

Main Ideas The graph shows the relationship between time and distance of a car traveling 55 miles per hour.

Travel Time

a. Choose any two points on the graph and find the rate of change.

• Solve problems involving direct variation.

b. Repeat Part a with a different pair of points. What is the rate of change?

New Vocabulary linear relationship constant rate of change direct variation constant of variation

Distance (mi)

• Identify proportional and nonproportional relationships by finding a constant rate of change.

350 300 250 200 150 100 50 0

y

(4, 220) (3, 165) (2, 110) (1, 55) 1 2 3 4 5 6 Time (h)

x

c. MAKE A CONJECTURE What is the rate of change between any two points on the line.

Constant Rates of Change The graph above is a straight line. Relationships that have straight-line graphs are called linear relationships. Notice in the graph above that as the time in hours increases by 1, the distance in miles increases by 55. +1

+1

Time (h)

0

1

Distance (mi)

0

55

+1

2

+1

3

4

110 165 220

Rate of Change change in distance 55 __ = _ or 55 miles per hour 1 change in time

+ 55 + 55 + 55 + 55

The rate of change between any two data points in a linear relationship is the same or constant. Another way of describing this is to say that a linear relationship has a constant rate of change. Constant Rate of Change


Not a Constant Rate of Change

Y

Y

"

Constant Rate of Change

X

"

Y

X

"

X

EXAMPLE

Use a Graph to Find a Constant Rate of Change

TECHNOLOGY An Internet advertisement contains a circular icon that decreases in size until it disappears. Find the constant rate of change for the radius in the graph shown. Describe what the rate means.

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Choose any two points on the line and find the rate of change between them. We will use the points at (2, 5) and (6, 4). (2, 5) → 2 seconds, radius 5 centimeters (6, 4) → 6 seconds, radius 4 centimeters change in radius rate of change = __ change in time 4 cm - 5 cm = __ 6s-2s 1 cm = -_ 4s

= -0.25 cm/s

Ç È x { Î Ó £ £

ä

← centimeters ← seconds

Ó Î { x È Ç X ÕLiÀ v -iV`Ã

The radius goes from 5 cm to 4 cm. The time goes from 2 s to 6 s. Simplify. Express this as a unit rate.

The rate of change -0.25 cm/s means that the radius of the circle is decreasing at a rate of 0.25 centimeter per second.

*iÀiÌiÀ v -µÕ>Ài V®

1. TECHNOLOGY Another icon on the advertisement described above is a square that increases in size. Find the constant rate of change for the perimeter of the square in the graph shown. Describe what the rate means.

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£ Ó Î { x X ÕLiÀ v -iV`Ã

Some linear relationships are also proportional. That is, the ratio of each non-zero y-value compared to the corresponding x-value is the same. Look Back To review proportional relationships, see Lesson 6-2.

Number of People x

1

2

3

4

Cost of Parking y

4

8

12

16

cost of parking y 8 16 4 12 __ →_ =4 _ =4 _ =4 _ =4 2 3 1 4 number of people x

The ratios are equal, so the linear relationship is proportional. Number of People x

1

2

3

4

Cost of Tickets y

13

22

31

40

cost of tickets y 31 40 13 22 1 _ __ →_ = 13 _ = 11 _ = 10_ = 10 2 3 3 4 1 number of people x

The ratios are not equal, so the linear relationship is nonproportional. Extra Examples at pre-alg.com

Lesson 7-4 Constant Rate of Change and Direct Variation

377

EXAMPLE

Use Graphs to Identify Proportional Linear Relationships

POOLS The height of the water as a pool is being filled is recorded in the table. Determine if there is a proportional linear relationship between the height of the water and the time.

} 1« > * i}Ì °®

£ä

To determine if the quantities are proportional, height y find _ for points on the graph.

Real-World Link The Johnson Space Center in Houston, Texas, has a 6.2 million gallon pool used to train astronauts for space flight. It is 202 feet long, 102 feet wide, and 40 feet deep. Source: www.jsc.nasa.gov

time x 10 _5 = 2.5 _ = 2.5 2 4

n È { Ó ä

15 20 _ = 2.5 _ = 2.5 6 8

x £ä £x Óä Óx /i °®

Since the ratios are the same, the height of the water is proportional to the time.

2. PLANTS Determine whether the relationship between time and distance of a car on page 376 is a proportional relationship. Personal Tutor at pre-alg.com

Direct Variation A special type of linear equation that describes constant rate of change is called a direct variation. The graph of a direct variation always passes through the origin and represents a proportional linear relationship. Direct Variation Words

Directly Proportional Since k is a constant rate of change in a direct variation, we can say the following. • y varies directly with x. • y is directly proportional to x.

A direct variation is a relationship in which the ratio of y to x is a constant, k. We say y varies directly with x.

Symbols

y = kx, where k ≠ 0

Example

y = 2x

Model

y y 2x

x

O

In the equation y = kx, k is called the constant of variation or constant of y proportionality. The direct variation y = kx can be written as k = _ x . In this form, you can see that the ratio of y to x is the same for any corresponding values of y and x. In other words, x and y vary in such a way that they have a constant ratio, k.

Use Direct Variation to Solve Problems TECHNOLOGY The time it takes to burn amounts of information on a CD is given in the table. a. Write an equation that relates the amount of information and the time it takes. Step 1 Find the value of k using the equation y = kx. Choose any point in the table. Then solve for k. y = kx

Direct variation

10 = k(2.5) Replace y with 10 and x with 2.5. 4=k

378 Chapter 7 Functions and Graphing NASA/JSC


Amount of Information (megabytes)

Time (s)

Rate of changes (MB/s)

x

y

y k=_ x

2.5

10

4

15

4

10

3.75

40

4

25

100

4

Step 2 Use k to write an equation. y = kx Direct variation y = 4x Replace k with 4.

b. Predict how long it will take to fill a 700 megabyte CD with information. y = 4x

Write the direct variation equation.

y = 4(700) Replace x with 700. y = 2800

Multiply.

It will take 2800 seconds or about 46 minutes and 40 seconds to fill a 700 megabyte CD with information.

««iÃ

UNIT COST The graph shows the cost of apples. 3A. Write an equation that relates cost and weight. 3B. Predict how much 5 pounds of apples would cost.

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fÎ

Y Ó] f£°Èä

fÓ f£

ä°x] fä°{ä

fä

£°Óx] f£

X

£ Ó 7i}Ì L®

Î

Proportional Linear Relationships Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change.

Words

B

Graph

a _ is constant and

Symbols

b

change in b _ is constant. change in a

"

Find the constant rate of change for each linear function and interpret its meaning. 1.

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2.

-i} vvii

Y

Y Ç /Ì> ->iÃ f®

(p. 377)

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Example 1

A

£ Ó Î { x È Ç ÕLiÀ v 7iiÃ

X

È x { Î Ó £ ä

£

Ó Î { x È Ç X ÕLiÀ v *Õ`Ã


379

Example 2 (p. 378)

Determine whether a proportional linear relationship exists between the two quantities shown in each of the functions indicated. Explain your reasoning. 3. Exercise 1

(p. 379)

HOMEWORK

HELP


ÕÃÌâi` ÛÌ>ÌÃ

8.

iÀV> À«>iÃ

{ä

xäää

Îx

{äää

Îä

ÃÌ f®


PHYSICAL SCIENCE The length of a spring stretches directly with the amount of weight attached to it. When a 25-gram weight is attached, a spring stretches 8 centimeters. 5. Write a direct variation equation relating the weight x and the amount of stretch y. 6. Estimate the stretch of the spring when it has a 60-gram weight attached.

ÃÌ>Vi ®

Example 3

4. Exercise 2

Óx Óä

Îäää Óäää £äää

£x ä

£ä

£

Ó

x

Î { x /i ®

È

Ç

n

x £ä £x Óä Óx Îä Îx {ä

ä

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9.

vÌ} 7i}ÌÃ

10.

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Îx Îä Óx Óä £x £ä x ä

Èä xä {ä Îä Óä £ä

£ Ó Î { x È Ç X ÕLiÀ v À 7i}ÌÃ

ä

£ä Óä Îä {ä xä Èä Çä X /ViÌÃ -`

Determine whether a proportional linear relationship exists between the two quantities shown in each of the functions indicated. Explain your reasoning. 11. Exercise 7 EXTRA

PRACTICE


12. Exercise 8

13. Exercise 9

14. Exercise 10

FOOD COSTS The cost of cheese varies directly with the number of pounds bought. Suppose 2 pounds cost $8.40. 15. Write an equation that could be used to find the unit cost of cheese. 16. Find the cost of 3.5 pounds of cheese.


CONVERSIONS The number of centimeters in a measure varies directly as the number of inches. 17. Write an equation that could be used to convert inches to centimeters. 18. How many inches is 16.51 centimeters?

H.O.T Problems

19. OPEN ENDED Graph a line that shows a 2-unit increase in y for every 1-unit increase in x. State the rate of change. 20. REASONING Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. A linear relationship that has a constant rate of change is a proportional relationship. 21.

Writing in Math Write about two quantities in real life that have a proportional linear relationship. Describe how you could change the situation to make the relationship between the quantities nonproportional.

22. Which is a true statement about the graph below? A There is not a constant rate of change.

i>} >À«iÌÃ f£Èä /Ì> ÃÌ

f£{ä

B The two quantities are not proportional.

f£Óä f£ää

C A proportional linear relationship exists.

fnä fÈä ä

D The total cost varies directly with the number of rooms.

£ Ó Î { ÕLiÀ v ,Ã

23. ENTERTAINMENT Admission to a water park is $36 for 3 people, $48 for 4 people, and $60 for 5 people. What is the rate of change, and what does the rate mean in this situation? (Lesson 7-3) Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2) 1 25. y = _ x

24. y = 5 - 3x

26. x - y = 10

2

27. SPACE The table shows how many stars a person can see in the night sky. How many stars can be seen with a small telescope? (Lesson 4-1)

Unaided eye in urban area

3 · 102 stars

Unaided eye in rural area

2 · 103 stars

With binoculars

3 · 104 stars

With small telescope

2 · 106 stars

Source: Kids Discover

PREREQUISITE SKILL Subtract. (Lesson 2-3) 28. -11 - 13

29. 15 - 31

30. -26 - (-26)

31. 9 - (-16)


381

CH

APTER

7


Determine whether each relation is a function. Explain. (Lesson 7-1) 1. {(0, 5), (1, 2), (1, -3), (2, 4)} 2. {(-6, 3.5), (-3, 4.0), (0, 4.5), (3, 5.0)} x

9

11

13

17

21

y

7

3

-1

-5

-7

4.

->Û}Ã

>>Vi f®

3.

12. SAVINGS The graph shows Felisa’s and Julian’s savings accounts several weeks after they were opened. Compare the rates of change. (Lesson 7-3)

Y

£ää ä nä Çä Èä xä {ä Îä Óä £ä ä

Y iÃ>

Õ>

£

Ó

5. MULTIPLE CHOICE The relation {(2, 11), (-9, 8), (14, 1), (5, 5)} is NOT a function when which ordered pair is added to the set? (Lesson 7-1)

A (8, -9)

C (0, 0)

B (6, 11)

D (2, 18)

Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2)

X

,iÌ} 6ì >iÃ

7. y = 5x - 1

fÓx

Graph each equation by plotting ordered pairs. (Lesson 7-2) 8. 3x - y = 7

x

PART TIME JOB For Exercises 13 and 14, use the following information. Ivy’s income varies directly with the number of hours she works. When Ivy works 4 hours she earns $44. (Lesson 7-4) 13. Write an equation that relates hours worked x and amount of pay y. 14. Predict the amount earned after 20 hours of work. 15. Find the constant rate of change for the linear function shown below and interpret its meaning. (Lesson 7-4)

9. x = 2

10. INSECTS The average flea can jump 150 times its own length. This can be represented by the equation y = 150x, where x is a flea’s 1 -inch long length. How far can a flea that is 16 jump? (Lesson 7-2) 11. TRAVEL Find the rate of change for the linear function. (Lesson 7-3) Time (h)

x

0

0.5

1.5

3

Distance (mi)

y

0

30

90

180


ÕÌ ,i>}

6. y = 8

{

7iiÃ

X

•

Î

fÓä f£x f£ä fx

ä

£

Ó

Î

{

x

ÕLiÀ v >iÃ

16. MULTIPLE CHOICE Which is a true statement about the graph shown in Exercise 15? (Lesson 7-4) F A proportional linear relationship exists. G The two quantities are not proportional. H There is not a constant rate of change. J The amount remaining varies directly with the number of games.

EXPLORE

7-5

Algebra Lab

It’s All Downhill The steepness, or slope, of a hill can be described by a ratio. vertical change height slope = __ horizontal change length

hill

vertical change

horizontal change

• Use posterboard or a wooden board, tape, and three or more books to make a “hill.”

y x

• Measure the height y and 1 1 inch or _ inch. Record the length x of the hill to the nearest _ 2 4 measurements in a table like the one below. Hill

Height y (in.)

Length x Car Distance (in.) (in.)

y

Slope x

1 2 3

ACTIVITY Step 1 Place a toy car at the top of the hill and let it roll down. Measure the distance from the bottom of the ramp to the back of the car when it stops. Record the distance in the table. Step 2

For the second hill, increase the height by adding one or two more books. Roll the car down and measure the distance it rolls. Record the dimensions of the hill and the distance in the table.

Step 3 Take away two or three books so that hill 3 has the least height. Roll the car down and measure the distance it rolls. Record the dimensions of the hill and the distance in the table. Step 4 Find the slopes of hills 1, 2, and 3 and record the values in the table.

ANALYZE THE RESULTS 1. How did the slope change when the height increased and the length decreased? 2. How did the slope change when the height decreased and the length increased? 3. MAKE A CONJECTURE On which hill would a toy car roll the farthest— 18 4 or _ ? Explain by describing the relationship a hill with slope _ 25 18 between slope and distance traveled. 4. Make a fourth hill. Find its slope and predict the distance a toy car will go when it rolls down the hill. Test your prediction by rolling a car down the hill. Explore 7-5 Algebra Lab: It’s All Downhill

383

7-5

Slope

Main Idea • Find the slope of a line.

New Vocabulary slope

Some roller coasters can make you feel heavier than a shuttle astronaut feels on liftoff. This is because the speed and steepness of the hills increase the effects of gravity. a. Write the rate of change comparing the height of the roller coaster to the length of the drop as a fraction in simplest form.

56 ft

b. Find the rate of change of a hill that has the same length but is 14 feet higher than the hill above. Is this hill steeper or less steep than the original?

42 ft

Slope Slope describes the steepness of a line. It is the ratio of the rise, or the vertical change, to the run, or the horizontal change. rise slope = _ run

← vertical change ← horizontal change

Note that the slope is the same for any two points on a straight line. It represents a constant rate of change.

Use Rise and Run to Find Slope ROADS Find the slope of a road that rises 25 feet for every horizontal change of 80 feet. rise slope = _ run 25 ft =_ 80 ft 5 =_ 16

Write the formula. rise = 25 ft, run = 80 ft

25 ft

Simplify. 80 ft

5 The slope of the road is _ or 0.3125. 16

1. RAMPS What is the slope of a wooden wheelchair ramp that rises 2 inches for every horizontal change of 24 inches? 384 Chapter 7 Functions and Graphing Tony Freeman/PhotoEdit

EXAMPLE

Use a Graph to Find Slope

Find the slope of each line. a.

b.

ÀÕ Î

Y

{] Ó®

È] n®

"

X

ÀÃi È

ÀÃi { Î] {®

x] {® ÀÕ

"

rise _4 slope = _ run =

rise -6 2 _ slope = _ or -_ run = 9

3

2A.

2B.

Y

Ó] Ó®

3

Y "

X

ä] ä®

" X Ó] {® x] Î®

Slope The slope m of a line passing through points at (x1, y1) and (x2, y2) is the ratio of the difference in y-coordinates to the corresponding difference in x-coordinates.

Words

Model

y (x 1, y 1) (x 2, y 2) O

y2 - y1

Symbols m = _ x - x , where x2 ≠ x1 2

x

1

Choosing Points • Any two points on a line can be chosen as (x1, y1) and (x2, y2). • The coordinates of both points must be used in the same order. Check: In Example 3a, let (x1, y1) = (5, 3) and let (x2, y2) = (2, 2), then find the slope.

EXAMPLE

Positive and Negative Slopes


b.

y (2, 2)

x

y -y

2 1 m=_ x -x

3-2 m=_ 5-2 _ m= 1 3

1

x

O

(5, 3)

O

2

y (⫺2, 1)

Definition of slope (x1, y1) = (2, 2), (x2, y2) = (5, 3)

(0, ⫺3)

y -y

2 1 m=_ x -x 2

1

Definition of slope

(x1, y1) = (-2, 1), 0 - (-2) (x2, y2) = (0, -3) -4 m=_ or -2 2

-3 - 1 m=_

Find the slope of the line that passes through each pair of points. 3B. C(1, -5), D(8, 3) 3A. A(-4, 3), B(1, 2) Extra Examples at pre-alg.com

Lesson 7-5 Slope

385

EXAMPLE

Zero and Undefined Slopes


b.

y

y (⫺5, 3)

(⫺1, 1)

(3, 1) (⫺5, 0) x

O

y -y

2 1 m=_ x -x 2

1

1-1 m=_

3 - (-1) _ m = 0 or 0 4

O

y -y

x

Definition of slope

2 1 m=_ x -x

(x1, y1) = (-1, 1), (x2, y2) = (3, 1)

0-3 (x1, y1) = (-5, 3), m =_ -5 - (-5) (x2, y2) = (-5, 0)

2

1

-3 m=_

Definition of slope

Division by 0 is undefined.

0

The slope is undefined.

Find the slope of the line that passes through each pair of points. 4B. G(2, 4), H(2, -1) 4A. E(-1, 7), F(5, 7)

Compare Slopes There are two major hills on a hiking trail. Hill 1 rises 6 feet vertically for every 42-foot run. Hill 2 rises 10 feet vertically for every 98-foot run. Which statement is true? Make a Drawing Whenever possible, make a drawing that displays the given information. Then use the drawing to estimate the answer.

A Hill 1 is steeper than Hill 2.

C Both hills have the same steepness.

B Hill 2 is steeper than Hill 1.

D You cannot find which hill is steeper.

Read the Test Item To compare the steepness of the hills, find the slopes. Solve the Test Item Hill 1

Hill 2

rise slope = _ run

rise slope = _ run

6 ft =_

rise = 6 ft, run = 42 ft 42 ft 1 =_ or about 0.14 7

10 ft =_

rise = 10 ft, run = 98 ft 98 ft 5 =_ or about 0.10 49

0.14 > 0.10, so the first hill is steeper than the second. The answer is A.

5. A home builder has four models with roofs having the dimensions in the table. Which roof is the steepest? F roof A G roof B H roof C J roof D



Roof A B C D

Length (ft) Height (ft) 15 5 16 4 20 10 21 14

Example 1 (p. 384)

1. CARPENTRY In a stairway, the slope of the handrail is the ratio of the riser to the tread. If the tread is 12 inches long and the riser is 8 inches long, find the slope.

handrail

tread riser

Example 2 (p. 385)

Find the slope of each line. 2.

3.

y

y (⫺1, 2)

(⫺3, 0) O

x

(1, 2)

x

O

(0, ⫺2)

Examples 3, 4

Find the slope of the line that passes through each pair of points.

(pp. 385–386)

4. A(3, 4), B(4, 6)

Example 5

7. MULTIPLE CHOICE Which bike ramp is the steepest? A 1 C 3 B 2 D 4

(p. 386)

HOMEWORK

HELP

For See Exercises Examples 8, 9 1 10–13 2 14–19 3, 4 33, 34 5

5. J(-8, 0), K(-8, 10)

6. X(-7, 0), Y(-1, -5) Bike Ramp 1 2 3 4

Length (ft) 8 4 3 4

Height (ft) 6 10 5 8

8. SKIING Find the slope of a snowboarding beginner hill that decreases 24 feet vertically for every 30-foot horizontal increase. FT

9. HOME REPAIR The bottom of a ladder is placed 4 feet away from a house. It reaches a height of 16 feet on the side of the house. What is the slope of the ladder?

FT


11.

y

y

(1, 3) (⫺2, 0) O

x

x

O (⫺1, ⫺1) (3, ⫺4)

Lesson 7-5 Slope

387

Find the slope of each line. Y

12.

13.

Y

Î] Ó®

È] £®

X

"

X

" Ó] £®

Î] Î®

Find the slope of the line that passes through each pair of points. 14. A(1, -3), B(5, 4) 17. J(-3, 6), K(-5, 9)

15. Y(4, -3), Z(5, -2) 18. N(2, 6), P(-1, 6)

16. S(-9, -4), T(-9, 8) 19. D(5, -1), E(-3, 4)

20. ROLLER COASTERS The first hill of the Texas Giant at Six Flags over Texas drops 137 feet vertically over a horizontal distance of about 103 feet. The Magnum XL-200 at Cedar Point in Ohio drops 195 feet vertically over a horizontal distance of about 113 feet. Which coaster has the steeper slope?

The Kingda Ka roller coaster in Jackson, New Jersey, is the tallest roller coaster in the world, standing at 456 feet. Source: rcdb.com

ANALYZE GRAPHS For Exercises 21–23, use the graph. 21. Which section of the graph shows the smallest increase in attendance? Describe the slope. 22. What happened to the attendance at the amusement park from 2002–2003? Describe the slope of this part of the graph. 23. The attendance for 1997 was 300 million. Describe the slope of a line connecting the data points from 1997 to 1998.

ÕÃiiÌ *>À 6ÃÌÀÃ ÌÌi`>Vi Ã®

Real-World Link

ÎÎä ÎÓx ÎÓä Î£x Î£ä Îäx Îää ä ½n ½ ½ää ½ä£ ½äÓ ½äÎ ½ä{ 9i>À

3OURCE )NTERNATIONAL !SSOCIATION OF !MUSEMENT 0ARKS AND !TTRACTIONS

Find the slope of the line that passes through each pair of points. 1 _ 1 , 5 1 , X 2_ ,6 25. W 3_

(

24. F(0, 1.6), G(0.5, 2.1) EXTRA

PRACTICE


H.O.T. Problems

2

4

) (

2

)

26. ANALYZE TABLES What is the slope of the line represented by the data in the table? 27. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the slopes of two lines or segments on a graph.

x -1 0 1 2

y -6 -8 -10 -12

1 28. OPEN ENDED Draw a line whose slope is -_ . 4

29. FIND THE ERROR Mike and Chloe are finding the slope of the line that passes through Q(-2, 8) and R(11, 7). Who is correct? Explain your reasoning. Mike 8-7 m=_ -2 - 11

Chloe 7-8 m= _ 11 - 2

30. CHALLENGE The graph of a line goes through the origin (0, 0) and C(a, b). State the slope of this line and explain how it relates to the coordinates of point C. 388 Chapter 7 Functions and Graphing Courtesy of Six Flags Theme Parks

31. REASONING Determine whether the following sentence is true or false. If true, provide an example. If false, provide a counterexample. As the constant of variation increases in a direct variation, the slope of the graph becomes steeper. 32.

Writing in Math Explain how slope is used to describe roller coasters. Include a description of slope and an explanation of how changes in rise or run affect the steepness of a roller coaster.

iÃ /À>Ûii`

{n 33. The graph {ä shows the ÎÓ distance traveled Ó{ by Ebony and £È Rocco during the n first five hours of a seven-hour ä long bicycle ride. Find the true statement.

LÞ ,VV

34. The table shows attendance figures for the Atlanta Falcons football team for three years. Find the slope of a line connecting the years 2001 and 2003 on the graph. Year 2003 2002 2001

£ Ó Î { x È ÕLiÀ v ÕÀÃ

A Rocco’s speed was 6 mph.

Attendance 563,676 550,974 451,333

Source: atlantafalcons.com

B Ebony’s speed was 16 mph. C Rocco traveled a total of 30 miles.

F 12,702

H 99,641

G 56,171.5

J 112,343

D Ebony’s traveled a total of 40 miles.

ÀViÃ

ÀVÕviÀiVi V®

35. Determine whether a proportional linear relationship exists between the two quantities shown in the graph. Explain your reasoning. (Lesson 7-4)

Îä Óä £x £ä x ä

36. FRUIT The table at the right shows the cost of pears and oranges. Compare the rates of change. (Lesson 7-3)

Solve each equation. (Lesson 3-2) 37. -123 = x - 183 38. -205 + t = -118

PREREQUISITE SKILL Solve each equation for y. (Lesson 7-2) 40. x + y = 6

41. 3x + y = 1

42. -x + 5y = 10

Y

Óx

£

Ó Î { x X ,>`ÕÃ V®

Cost y Weight (lb) x

Pears

Oranges

0

$0

$0

2

$2.40

$2.20

5

$6

$5.50

39. -350 + z = 125

y 43. _ - 7x = -5 2

Lesson 7-5 Slope

389

EXTEND

7-5


Slope and Rate of Change

In this activity, you will investigate the relationship between slope and rate of change.

• Attach the force sensor to the graphing calculator. Place the sensor in a ring stand as shown. • Make a small hole in the bottom of a paper cup. Straighten a paper clip and use it to create a handle to hang the cup on the force sensor. Place another cup on the floor below. • Set the device to collect data 100 times at intervals of 0.1 second.

ACTIVITY Step 1 Hold your finger over the hole in the cup. Fill the cup with water. Step 2 Begin collecting data as you begin to allow the water to drain. Step 3 Make the hole in the cup larger. Then repeat Steps 1 and 2 for a second trial.

ANALYZE THE RESULTS 1. Use the calculator to create a graph of the data for Trial 1. The graph will show the weight of the cup y as a function of time x. Describe the graph. 2. Create the graph for Trial 2. Compare the steepness of the two graphs. Which has a greater slope? 3. What happens as the time increases? 4. Did the cup empty at a faster rate in Trial 1 or Trial 2? Explain. 5. Describe the relationship between slope and the rate at which the cup was emptied. 6. MAKE A CONJECTURE What would a graph look like if you emptied a cup using a hole half the size of the original hole? twice the size of the second hole? Explain. 7. Water is emptied at a constant rate from containers shaped like the ones shown below. Draw a graph of the water level in each of the containers as a function of time. a.

b.

390 Chapter 7 Functions and Graphing Horizons Companies

c.

7-6

Slope-Intercept Form

Main Ideas • Determine slopes and y-intercepts of lines. • Graph linear equations using the slope and y-intercept.

New Vocabulary y-intercept slope-intercept form

A landscaping company charges a $20 fee to mow a lawn plus $8 per hour. The equation y = 8x + 20 represents this situation where x is the number of hours it takes to mow the lawn and y is the total cost of mowing the lawn.

Number of Hours, x 1 2 3

Total Cost, y

a. Copy and complete the table to find the total cost of mowing the lawn. b. Use the table to graph the equation. In which quadrant does the graph lie? Explain. c. Find the y-coordinate of the point where the graph crosses the y-axis and the slope of the line. How are they related to the equation?

Slope and y-Intercept The

y

y-intercept is the y-coordinate of a point where the graph crosses the y-axis. An equation with a y-intercept that is not 0 represents a nonproportional relationship. The equation y = 2x + 1 is written in the form y = mx + b, where m is the slope and b is the y-intercept. This is called slope-intercept form.

Reading Math Different Forms Both equations below are written in slope-intercept form. y = x + (-2) y=x-2

y 2x 1 x

O

slope 2 y-intercept 1

y = mx + b slope

EXAMPLE

y-intercept

Find the Slope and y-Intercept

3 State the slope and the y-intercept of the graph of y = _ x - 7. 5

3 y=_ x-7

Write the original equation.

3 y = _x + (-7)

Write the equation in the form y = mx + b.

↑ ↑ y = mx + b

3 m=_ , b = -7

5 5

5

3 The slope of the graph is _ , and the y-intercept is -7. 5

State the slope and the y-intercept of the graph of each equation. 1A. y = 8x + 6 1B. y = x - 3 Extra Examples at pre-alg.com

Lesson 7-6 Slope-Intercept Form

391

EXAMPLE BrainPOP® pre-alg.com

Write an Equation in Slope-Intercept Form

State the slope and the y-intercept of the graph of 5x + y = 3. 5x + y = 3

Write the original equation.

5x + y - 5x = 3 - 5x

Subtract 5x from each side.

y = -5x + 3 Simplify and write in slope-intercept form. The slope of the graph is -5, and the y-intercept is 3.

State the slope and the y-intercept of the graph of each equation. 1 2A. -9x + y = -5 2B. y - 6 = _x 2

Graph Equations You can use the slope-intercept form of an equation to graph a line.

EXAMPLE

Graph an Equation

1 Graph y = -_ x - 4 using the slope and y-intercept. 2

Step 1

1 Find the slope and y-intercept. m = -_

Step 2

Graph the y-intercept point at (0, -4).

Step 3

2

b = -4 y

-1 1 Write the slope -_ as _ . Use it to locate a 2 2 second point on the line. -1 ← change in y: down 1 unit m=_ 2

← change in x: right 2 units

down 1 unit

Another point on the line is at (2, -5). Step 4

x

O

(0, 4) (2, 5) right 2 units

Draw a line through the two points.

1 3. Graph y = _ x + 1 using the slope and y-intercept. 3


a. Graph the equation. First, find the slope and the y-intercept. slope = 12 y-intercept = 24 Plot the point at (0, 24). Then go up 12 and right 1. Connect these points.

ÃÌ f®

Real-World Career Business Owner Business owners must understand the factors that affect cost and profit. Graphs are a useful way for them to display this information.

BUSINESS A T-shirt company charges a design fee of $24 for a pattern and then sells the shirts for $12 each. The total cost y can be represented by y = 12x + 24, where x represents the number of T-shirts. n{ ÇÓ Èä {n ÎÈ Ó{ £Ó

Y

b. Describe what the y-intercept and the X ä £ Ó Î { x È Ç slope represent. ÕLiÀ v /-ÀÌÃ The y-intercept 24 represents the design fee. The slope 12 represents the cost per T-shirt, which is the rate of change. Since it is not reasonable for the number of T-shirts or the cost to be negative, the graph is in Quadrant I only.

392 Chapter 7 Functions and Graphing Cris Haigh/Getty Images

4. WRITING Tim has written 30 pages of a novel. He plans to write 12 pages per week until he is finished. The total number of pages written y can be represented by y = 12x + 30, where x represents the number of weeks. Graph the equation. Describe what the y-intercept and the slope represent. Personal Tutor at pre-alg.com

Examples 1, 2 (pp. 391–392)

Example 3 (p. 392)

State the slope and the y-intercept of the graph of each equation. 1. y = x + 8

Example 4

HOMEWORK

HELP

For See Exercises Examples 9–14 1, 2 15–23 3 24–27 4

3. x + 3y = 6

Graph each equation using the slope and y-intercept. 1 4. y = _ x+1 4

(pp. 392–393)

2. x + y = 2

5. 3x + y = 2

6. x - 2y = 4

BUSINESS Mrs. Allison charges $25 for a basic cake that serves 12 people. A larger cake costs an additional $1.50 per serving. The total cost can be given by y = 1.5x + 25, where x represents the number of additional slices. 7. Graph the equation. 8. Explain what the y-intercept and the slope represent.

State the slope and the y-intercept of the graph of each equation. 9. y = x + 2 12. 2x + y = -3

10. y = 2x - 4

11. x + y = -3

13. 5x + 4y = 20

14. y = 4

Graph each equation using the slope and y-intercept. 15. y = x + 5

16. y = -x + 6

17. y = 2x - 3

3 18. y = _ x+2 4

19. x + y = -3

20. x + y = 0

21. -2x + y = -1

22. 5x + y = -3

1 23. y + _ x=1 5

AUTOMOBILES For Exercises 24 and 25, use the following information. To replace a set of brakes, an auto mechanic charges $40 for parts plus $50 per hour. The total cost y can be given by y = 50x + 40 for x hours. 24. Graph the equation using the slope and y-intercept. 25. State the slope and y-intercept of the graph of the equation and describe what they represent.

EXTRA

PRACTICE


HANG GLIDING For Exercises 26–28, use the following information. The altitude in feet y of a hang glider who is slowly landing can be given by y = 300 - 50x, where x represents the time in minutes. 26. Graph the equation using the slope and y-intercept. 27. State the slope and y-intercept of the graph of the equation and describe what they represent. 28. The x-intercept is the x-coordinate of a point where a graph crosses the x-axis. Name the x-intercept and describe what it represents. Graph each equation using the slope and y-intercept. 29. x - 3y = -6

30. 2x + 3y = 12

31. y = -3

Lesson 7-6 Slope-Intercept Form

393

H.O.T. Problems

32. OPEN ENDED Draw the graph of a line that has a y-intercept but no x-intercept. What is the slope of the line? 33. FIND THE ERROR Carlotta and Alex are finding the slope and y-intercept of x + 2y = 8. Who is correct? Explain your reasoning. Alex slope = -_1 2 y-intercept = 4

Carlotta slope = 2 y-intercept = 8

34. CHALLENGE What is the x-intercept of the graph of y = mx + b? Explain how you know. 35.

Writing in Math

How can knowing the slope and y-intercept help you graph an equation? Include an explanation of how to write an equation for a line if you know the slope and y-intercept.

36. Which best represents the graph of y = 3x - 1? A

Î Ó £

Y

ÎÓ£ " £ Ó Î

B

Î Ó £

Y Î Ó £

C

ÎÓ£ " £ Ó Î

£ Ó ÎX

Y

D

Î Ó £ ÎÓ£ " £ Ó Î

ÎÓ£ "£ Ó Î X £ Ó Î

£ Ó ÎX

Y

£ Ó ÎX

37. CARS The cost of gas varies directly with the number of gallons bought. Marty bought 18 gallons of gas for $49.50. Write an equation that could be used to find the unit cost of a gallon of gas. Then find the unit cost. (Lesson 7-5) 38. BIRDSEED Find the constant rate of change for the linear function in the table at the right and interpret its meaning. (Lesson 7-4)

PREREQUISITE SKILL Simplify. (Lesson 1-2) 39. 2(18) - 1

40. (-2 - 4) ÷ 10


41. -1(6) + 8

Amount of Birdseed (lb) x 4 8 12

42. 5 - 8(-3)

Total Cost ($) y 11.20 22.40 33.60

EXTEND

7-6


The Family of Linear Graphs

A graphing calculator is a valuable tool when investigating characteristics of linear functions. Before graphing, you must create a viewing window that shows both the x- and y-intercepts of the graph of a function. You can use the standard viewing window [-10, 10] scl: 1 by [-10, 10] scl: 1 or set your own minimum and maximum values for the axes and the scale factor by using the WINDOW option.

The tick marks on the x scale and on the y scale are 1 unit apart.

[-10, 10] scl: 1 by [-10, 10] scl: 1

You can use a TI-83/84 Plus graphing The x-axis goes calculator to enter several functions and from -10 to 10. graph them at the same time on the same screen. This is useful when studying a family of functions. The family of linear functions has the parent function y = x.

The y-axis goes from -10 to 10.

ACTIVITY 1 Graph y = 3x - 2 and y = 3x + 4 in the standard viewing window and describe how the graphs are related. Step 1 Graph y = 3x + 4 in the standard viewing window. • Clear any existing equations from the Y= list. KEYSTROKES:

y ⫽ 3x ⫹ 4

CLEAR

• Enter the equation and graph. KEYSTROKES:

3 X,T,␪,n

4 ZOOM 6

Step 2 Graph y = 3x - 2. • Enter the function y = 3x – 2 as Y2 with y = 3x + 4 already existing as Y1. KEYSTROKES:

3 X,T,␪,n

2

• Graph both functions in the standard viewing window. KEYSTROKES:

ZOOM 6

The first function graphed is Y1 or y = 3x + 4. The second function graphed is Y2 or y = 3x - 2. Press TRACE . Move along each function using the right and left arrow keys. Move from one function to another using the up and down arrow keys. The graphs have the same slope, 3, but different y-intercepts at 4 and -2. Extend 7-6 Graphing Calculator Lab: The Family of Linear Graphs

395

EXERCISES Graph y = 2x - 5, y = 2x - 1, and y = 2x + 7. 1. Compare and contrast the graphs. 2. How does adding or subtracting a constant c from a linear function affect its graph? 3. Write an equation of a line whose graph is parallel to y = 3x - 5, but is shifted up 7 units. 4. Write an equation of the line that is parallel to y = 3x - 5 and passes through the origin. 5. Four functions with a slope of 1 are graphed in the standard viewing window, as shown at the right. Write an equation for each, beginning with the left most graph.

[-10, 10] scl:1 by [-10, 10] scl:1

_

_

Clear all functions from the Y= menu and graph y = 1 x, y = 3 x, y = x, and 3 4 y = 4x in the standard viewing window. 6. How does the steepness of a line change as the coefficient for x increases? 7. Without graphing, determine whether the graph of y = 0.4x or the graph of y = 1.4x has a steeper slope. Explain. Clear all functions from the Y= menu and graph y = -4x and y = 4x. 8. How are these two graphs different? 9. How does the sign of the coefficient of x affect the slope of a line? 10. Describe the similarities and differences between the graph of y = 2x - 3 and the graph of each equation listed below. a. y = 2x + 3 b. y = -2x – 3 c. y = 0.5x + 3 11. Write an equation of a line whose graph lies between the graphs of y = -3x and y = -6x. For Exercises 12–14, use the following information. A garden center charges $75 per cubic yard for topsoil. The delivery fee is $25. 12. Describe the change in the graph of the situation if the delivery fee is changed to $35. 13. How does the graph of the situation change if the price of a cubic yard of topsoil is increased to $80? 14. What are the prices of a cubic yard of topsoil and delivery if the graph has slope 70 and y-intercept 40? 396 Chapter 7 Functions and Graphing

7-7

Writing Linear Equations

Main Idea • Write equations given the slope and y-intercept, a graph, a table, or two points.

You can determine the approximate outside temperature by counting the chirps of crickets, as shown in the table.

Number of Chirps in 15 Seconds

Temperature (°F)

0

40

5

45

10

50

15

55

20

60

a. Graph the ordered pairs (chirps, temperature). Draw a line through the points. b. Find the slope and the y-intercept of the line. What do these values represent?

c. Write an equation in the form y = mx + b for the line. Then translate the equation into a sentence.

Write Equations There are many different methods for writing linear equations. If you know the slope and y-intercept, you can write the equation of a line by substituting these values in y = mx + b.

EXAMPLE

Write Equations From Slope and y-Intercept

Write an equation in slope-intercept form for each line. a. slope = 4, y-intercept = -8 y = mx + b

b. slope = 0, y-intercept = 5 y = mx + b

Slope-intercept form

y = 4x + (-8) Substitute

y = 0x + 5

Substitute

y = 4x - 8

y=5

Simplify.


Simplify.

1 1A. slope = -_ , y-intercept = 0 2

EXAMPLE

1 1B. slope = 2, y-intercept = -_ 3

Write an Equation From a Graph y

Write an equation in slope-intercept form for the line graphed. Check Equation To check, choose another point on the line and substitute its coordinates for x and y in the equation.

The y-intercept is 1. From (0, 1), you can go down 3 units and right 1 unit to another point -3 , or -3. on the line. So, the slope is _ y = mx + b

1 Slope = intercept form

y = -3x + 1 Replace m with -3 and b with 1.

y -intercept

down 3 units

O

x

right 1 unit

Lesson 7-7 Writing Linear Equations

397

2. Write an equation in slope-intercept form for the line graphed.

Y

X

"

EXAMPLE

Write an Equation to Make a Prediction

EARTH SCIENCE On a summer day, the temperature at altitude 0 meters, or sea level, is 30°C. The temperature decreases 2°C for every 305 meters increase in altitude. Predict the temperature for an altitude of 2000 meters.

Use a Table Translate the words into a table of values to help clarify the meaning of the slope. For every increase of 305 meters in altitude, the temperature decreases by 2°C. Alt. (m)

Explore You know the rate of change of temperature to altitude (slope) and the temperature at sea level (y-intercept). Make a table of ordered pairs.

30

305

28

610

26

0

30

305

28

610

26

Plan

Write an equation to show the relationship between altitude x and temperature y. Then, substitute the altitude of 2000 meters into the equation to find the temperature.

Solve

Step 1

Temp. (°C)

0

Altitude, Temperature y (°C), x

Step 2 Find the y-intercept b.

Find the slope m. change in y m=_ change in x -2 = 305

≈ -0.007

← ← ← decrease of -2°C ← increase of 305 m

change in temperature __ change in altitude

Simplify.

Step 3

(x, y) = (altitude, temperature) = (0, b) When the altitude is 0, or sea level, the temperature is 30°C. So, the y-intercept is 30.

Write the equation. y = mx + b Slope-intercept form ≈ -0.007x + 30 Replace m with -0.007 and b with 30. Step 4 Substitute the altitude of 2000 meters. Write the equation. y = -0.007x + 30 = -0.007(2000) + 30 Replace x with 2000. ≈ 17 Simplify.

So, at an altitude of 2000 meters, the temperature is about 17°C. Check

As you go up, the temperature drops about 2°C for every 300 meters. Since 2000 ÷ 300 is about 7, at 2000 meters the temperature will drop about 7 × 2°C or 14°C. If the temperature is 30°C at 0 meters, the temperature at 2000 meters is about 30 - 14 or about 16°C. So the answer is reasonable.



3. PIANO LESSONS The cost of 7 half-hour piano lessons is $151. The cost of 11 half-hour lessons is $223. Write a linear equation that shows the cost y for x half-hour lessons. Then use the equation to find the cost of 3 half-hour lessons. Personal Tutor at pre-alg.com

You can also write an equation for a line if you know the coordinates of two points on a line.

EXAMPLE

Write an Equation Given Two Points

Write an equation for the line that passes through (-2, 5) and (2, 1). Step 1

Find the slope m. y2 - y1 m=_ x -x 2

Definition of slope

1

5-1 =_ or -1 (x1, y1) = (-2, 5), (x2, y2) = (2, 1) -2 - 2

Step 2

y = mx + b 5 = -1(-2) + b 3=b

Check Equation To check, substitute the coordinates of the other point into the equation.

Find the y-intercept b. Use the slope and the coordinates of either point.

Step 3

Slope-intercept form Replace (x, y) with (-2, 5) and m with -1. Simplify.

Substitute the slope and y-intercept. y = mx + b Slope-intercept form y = -1x + 3 Replace m with -1 and b with 3. y = -x + 3 Simplify.

y = -x + 3 (1) -(2) + 3 1 -2 + 3 1=1

4. Write an equation for the line that passes through (5, 1) and (8, -2).

EXAMPLE

Write an Equation From a Table

Use the table of values to write an equation in slope-intercept form. Step 1 Find the slope m. Use the coordinates of any two points. y2 - y1 m=_ x -x 2

Definition of slope

1

-2 - 6 4 m=_ or -_ (x1, y1) = (-5, 6), (x2, y2) = (5, -2)

Alternate Strategy

Step 2

y = mx + b

Slope-intercept form Replace (x, y) with (-5, 6) and m with -45. Simplify.

x

y

-5

6

2=b

0

2

y-intercept = 2

-5

6

5

-2

10

-6

15

-10

Find the y-intercept b. Use the slope and the coordinates of any point. 6 = -45(-5) + b

Step 3

y

5

5 - (-5)

If a table includes the y-intercept, simply use this value and the slope to write an equation.

x

Substitute the slope and y-intercept. y = mx + b


y = -45 x + 2

4 Replace m with -_ and b with 2. 5

Lesson 7-7 Writing Linear Equations

399

5. Write an equation in slope-intercept form to represent the table of values shown below.

Example 1 (p. 397)

-6 -3

3

y

-1

0

2

Write an equation in slope-intercept form for each line. 1. slope = 1, y-intercept = 1

2. slope = 0, y-intercept = -7

3.

4.

2

Example 2

x

y

y

(pp. 397–398) x

O x O

Example 3 (pp. 398–399)

Example 4 (p. 399)

PICNICS It costs $50 plus $10 per hour to rent a park pavilion. 5. Write an equation in slope-intercept form that shows the cost y for renting the pavilion for x hours. 6. Find the cost of renting the pavilion for 8 hours. Write an equation in slope-intercept form for the line passing through each pair of points. 7. (2, 2) and (4, 3)

Example 5 (pp. 399–400)

HOMEWORK

HELP

For See Exercises Examples 10–15 1 16, 17 3 18–23 2 24–29 4 30, 31 5

8. (3, -4) and (-1, 4)

9. Write an equation in slope-intercept form to represent the table of values.

x

-4

0

4

8

y

-4

-1

2

5

Write an equation in slope-intercept form for each line. 10. slope = 2, y-intercept = 6

11. slope = -4, y-intercept = 1

12. slope = 0, y-intercept = 5


1 14. slope = -_ , y-intercept = 8 3

2 15. slope = _ , y-intercept = 0 5

SOUND For Exercises 16 and 17, use the table that shows the distance that a rip current travels through the ocean. 16. Write an equation in slope-intercept form to represent the data in the table. Describe what the slope means. 17. Estimate how far the rip current travels in one minute.


Times (s)

Distance (ft)

x

y

0

0

1

2.4

2

4.8

3

7.2

Write an equation in slope-intercept form for each line. 18.

19.

y

20.

y

x

O

x

22.

y

23.

y

x

O

Coyotes communicate by using different barks and howls. Because of the way in which sound travels, a coyote is usually not in the area from which the sound seems to be coming. Source: livingdesert.org

EXTRA

PRACTIICE

y

x

O

Real-World Link

x

O

O

21.

y

x

O

Write an equation in slope-intercept form for the line passing through each pair of points. 24. (-2, -1) and (1, 2)

25. (-4, 3) and (4, -1)

26. (0, 0) and (-1, 1)

27. (4, 2) and (-8, -16)

28. (8, 7) and (-9, 7)

29. (5, -6) and (3, 2)

Write an equation in slope-intercept form for each table of values. 30.

x

-1

0

1

2

y

-7

-3

1

5

31.

x

-3

-1

1

3

y

7

5

3

1

SOUND For Exercises 32 and 33, use the table that shows the distance that sound travels through dry air at 0°C. 32. Write an equation in slope-intercept form to represent the data in the table. Describe what the slope means. 33. Estimate the number of miles that sound travels through dry air in one minute.

Times (s)

Distance (ft)

x

y

0

0

1

1088

2

2176

3

3264


H.O.T. Problems

34. COMPUTERS A computer repair company charges a fee and an hourly charge. After two hours the repair bill is $110, and after three hours it is $150. How much would it cost for 1.5 hours of work? 35. OPEN ENDED Choose a slope and y-intercept. Write an equation and then graph the line. 36. SELECT A TOOL Mr. Awan has budgeted $860 to have his dining room painted. The estimated cost for materials is $100. The painter charges $35 per hour and estimates that the work will take about 20 hours to complete. Which of the following tools might Mr. Awan use to determine whether he has budgeted enough money to paint the dining room? Justify your choice. Then use the tool to solve the problem. draw a model

paper/pencil

calculator

Lesson 7-7 Writing Linear Equations Gail Shumway/Getty Images

401

CHALLENGE A CD player has a pre-sale price of $c. Kim buys it at a 30% discount and pays 6% sales tax. After a few months, she sells it for $d, which was 50% of what she paid originally. 37. Express d as a function of c. 38. How much did Kim sell it for if the pre-sale price was $50? 39.

Writing in Math Explain how you can model data with a linear equation and how to find the y-intercept and slope by using a table.

41. The graphs show how much a video store pays for three different movies. Which equation is NOT represented by one of the graphs? fä fnä

f£ää

fÇä

fä

fÈä

fnä

fxä

ÃÌ

ÕÌ Ài> "ÜiÃ

40. Lorena borrowed $100 from her father and plans to pay him back at a rate of $10 per week. The graph shows the amount Lorena owes her father. Find the equation that represents this relationship.

fÇä

-OVIE " -OVIE !

f{ä

fÈä

fÎä

fxä

fÓä

f{ä

f£ä

fÎä

fä

-OVIE #

ä

fÓä

£

Ó Î { x È Ç ÕLiÀ v ÛiÃ

f£ä

n

F y = 10x

fä ä

£

Ó

Î { x È Ç n ÕLiÀ v 7iiÃ

£ä

G y = 25x

A y = 10x - 100

C y = 100 - 10x

H y = 50x

B y = 10x - 90

D y = -100 - 10x

J y = 60x

State the slope and the y-intercept for the graph of each equation. (Lesson 7-6) 42. y = 6x + 7

43. y = -x + 4

44. -3x + y = -2

45. CONSTRUCTION Find the slope of the road that rises 48 feet for every 144 feet measured horizontally. (Lesson 7-5) Write each number in scientific notation. (Lesson 4-7) 46. 345,000

47. 1,680,000

48. 0.00072

49. 0.001

PREREQUISITE SKILL (Lesson 1-7) 50. State whether a scatter plot containing the following set of points would show a positive, negative, or no relationship. 402 Chapter 7 Functions and Graphing

x

0

2

4

5

4

3

6

y

15

20

36

44

32

30

50

7-8

Prediction Equations

• Draw lines of fit for sets of data. • Use lines of fit to make predictions about data.

New Vocabulary line of fit

The scatter plot shows the number of years people in the United States are expected to live, according to the year they were born. a. Use the line drawn through the points to predict the life expectancy of a person born in 2020. b. What are some limitations in using a line to predict life expectancy?

Life Expectancy (yr)

Main Ideas

85 80 75 70 65 60 55 50 45 40 0

y

1920 1940 1960 1980 2000 Year

x


Lines of Fit When real-life data are collected, the points graphed usually do not form a straight line, but may approximate a linear relationship. A line of fit can be used to show such a relationship. A line of fit is a line that is very close to most of the data points.

EXAMPLE

Make Predictions from a Line of Fit

MONEY The table shows the changes in the number of college-bound students taking the ACT.

Labeling In Example 1, the x-axis could have been labeled “Years Since 1991” to simplify the graph and the prediction equation.

£Óää ££ää £äää ää

Number (thousands)

1991 1995 1998 2000 2002 2004

796 945 995 1065 1116 1171

Source: ACT, Inc.

nää ½ä

½ää 9i>À

½£ä

b. Use the line of fit to predict the number of students taking the ACT in 2015. Extend the line so that you can find the y-value for an x-value of 2015. The y-value for 2015 is about 1480. So, the number of students taking the ACT is approximately 1480.

ÕLiÀ v -ÌÕìÌÃ ÌÕÃ>`Ã®

ÕLiÀ v -ÌÕìÌÃ ÌÕÃ>`Ã®

a. Make a scatter plot and draw a line of fit for the data.

Year

£xää £{ää £Îää £Óää ££ää £äää ää nää ä ½ä

½ää ½£ä 9i>À

Lesson 7-8 Prediction Equations

½Óä

403

1. ENTERTAINMENT Make a scatter plot of the percent of U.S. households that have a digital video camera and draw a line of fit. Predict the percent of U.S. households with a digital video camera in 2015. Year

1999

2000

2001

2002

2003

2004

2005

% of U.S. Households

3%

7%

10%

14%

17%

19%

21%

Source: Parks Associates


Prediction Equations You can also make predictions from the equation of a line of fit.

EXAMPLE

Make Predictions from an Equation

SWIMMING The scatter plot shows the winning Olympic times in the women’s 800-meter freestyle event from 1968 through 2004.

There may be general trends in sets of data. However, not every data point may follow the trend exactly.

y 560

a. Write an equation in slope-intercept form for the line of fit that is drawn. Step 1 First, select two points on the line and find the slope. We have chosen (1980, 525) and (1992, 500). Notice that they are not original data points. y2 - y1 m=_ x -x 2

1

Time (s)

Trends

Women’s 800-Meter Freestyle Event

(1980, 525)

540 520 500

(1992, 500) 0 ’68

Definition of slope

525 - 500 =_ 1980 - 1992

(x1, y1) = (1992, 500), (x2, y2) = (1980, 525)

≈ -2.1

Simplify.

’76

’84 Year

’92

’00


Step 2 Next, find the y-intercept. y = mx + b


525 = -2.1(1980) + b Replace (x, y) with (1980, 525) and m with -2.1. 4683 ≈ b

Simplify.

Step 3 Write the equation.

Lines of fit can help make predictions about recreational activities. Visit pre-alg.com to continue work on your project.

y = mx + b


y = -2.1x + 4683

Replace m with -2.1 and b with 4683.

b. Predict the winning time in the women’s 800-meter freestyle event in the year 2012. y = -2.1x + 4683

Write the equation of the line of fit.

= -2.1(2012) + 4683

Replace x with 2012.

= 457.8

Simplify.

A prediction for the winning time in the year 2012 is approximately 457.8 seconds or 7 minutes, 37.8 seconds.



x

Time (s)

’68 ’72 ’76 ’80 ’84 ’88 ’92 ’96 ’00 ’04

2. SWIMMING Write an equation in slope-intercept form for the line of fit that is drawn. Predict the winning time in the men’s 100-meter butterfly in 2012.

56 55 54 53 52 51

Year Source: The World Almanac

Example 1 (p. 403)

NEWSPAPERS For Exercises 1 and 2, use the table that shows the number of Sunday newspapers in the U.S. 1. Make a scatter plot and draw a line of fit. 2. Use the line of fit you drew in Exercise 1 to predict the number of Sunday newspapers in the U.S. in 2010.

Year 1998 1999 2000 2001 2002 2003

Sunday Newspapers 898 905 917 913 913 917

Example 2 (p. 404)

SPENDING For Exercises 3 and 4, use the line of fit drawn that shows the billions of dollars spent by travelers in the United States. 3. Write an equation in slope-intercept form for the line of fit. 4. Use the equation to predict how much money travelers will spend in 2008.

Amount ($ billions)

Source: Statistical Abstract of the U.S.

600 500 400 300 200 100

(4, 479) (6, 502.5)

1 2 3 4 5 6 7 Years Since 1997

0

Source: Travel Industry Association of America

HOMEWORK

HELP

For See Exercises Examples 5, 6, 10, 11 1 7–9, 12–15 2

ENTERTAINMENT For Exercises 5 and 6, use the table that shows the number of movie tickets sold in the United States. Year

1998

1999

2000

2001

2002

2003

2004

Tickets Sold (millions)

1481

1465

1421

1487

1578

1523

1507

Source: boxofficemojo.com

5. Make a scatter plot and draw a line of fit. 6. Use the line of fit to predict movie attendance in 2010. PRESSURE For Exercises 7–9, use the table that shows the approximate barometric pressure at various altitudes. 7. Make a scatter plot of the data and draw a line of fit. 8. Write an equation for the line of fit you drew in Exercise 7. Use it to estimate the barometric pressure at 60,000 feet. Is the estimation reasonable? Explain. 9. Do you think that a line is the best model for this data? Explain.

Altitude (ft) 0 5000 10,000 20,000 30,000 40,000 50,000

Barometric Pressure (in. mercury) 30 25 21 14 9 6 3

Source: New York Public Library Science Desk Reference


405

POLE VAULTING For Exercises 10 and 11, use the table that shows the men’s winning Olympic pole vault heights to the nearest inch. 10. Make a scatter plot and draw a line of fit. 11. Use the line of fit to predict the winning pole vault height in the 2008 Olympics. Year

1976

1980

1984

1988

1992

1996

2000

2004

Height (in)

217

228

226

232

228

233

232

234


EARTH SCIENCE For Exercises 12–15, use the table that shows the latitude and the average temperature in July for five cities in the United States. Real-World Link In 1964, thirteen competitors broke or equaled the previous Olympic pole vault record a total of 36 times. This was due to the new fiberglass pole.

City

Source: Chance

Latitude ( N)

Average July High Temperature ( F)

Chicago, IL

41

73

Dallas, TX

32

85

Denver, CO

39

74

New York, NY

40

77

Duluth, MN

46

66

Fresno, CA

37

97


EXTRA

PRACTIICE


H.O.T. Problems

12. Make a scatter plot of the data and draw a line of fit. 13. Describe the relationship between latitude and temperature shown by the graph. 14. Write an equation for the line of fit you drew in Exercise 12. 15. Use the equation in Exercise 14 to estimate the average July temperature for a location with latitude 50° north. Round to the nearest degree Fahrenheit. 16. OPEN ENDED Make a scatter plot with at least ten points that appear to be somewhat linear. Draw two different lines that could approximate the data. 17. CHALLENGE The table at the right shows the percent of public schools in the United States with Internet access. Suppose you use (Year, Percent of Schools) to write a linear equation describing the data. Then you use (Years Since 1995, Percent of Schools) to write an equation. Is the slope or y-intercept of the graphs of the equations the same? Explain.

Year

Years Since 1995

Percent of Schools

1995

0

50

1997

2

78

1999

4

95

2001

6

99

2003

8

100

Source: National Center for Education Statistics

18.

Writing in Math Use the information about life expectancy on page 403 to explain how a line can be used to predict life expectancy for future generations. Include a description of a line of fit and an explanation of how lines can represent sets of data that are not exactly linear.

406 Chapter 7 Functions and Graphing Michael Steele/Getty Images

A The scoring average of the points leader increased over time. B The scoring average of the points leader decreased over time.

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19. The scatter plot at the right shows the scoring average of the WNBA points leader from 1997 through 2004. Which statement best describes the relationship on the scatter plot?

ÓÓ°ä Ó£°x Ó£°ä Óä°x Óä°ä

C The scoring average of the points leader remained the same over time. D The scoring average of the points leader could not be determined over time.

ÓÓ°x

ä

¼Ç ¼n ¼ ¼ää ¼ä£ ¼äÓ ¼äÎ ¼ä{ 9i>À

-ÕÀVi\ ÜL>°V

Write an equation in slope-intercept form for each line. (Lesson 7-7) 20. slope = 3, y-intercept = 5

21. slope = -2, y-intercept = 2

22.

23.

y

y x

O

O

x

Graph each equation using the slope and y-intercept. (Lesson 7-6) 24. y = x - 2

1 26. y = _ x

25. y = -x + 3

2

27. ENTERTAINMENT In 2005, a movie actor was to pay 10 percent commission to his management company for work negotiated on his behalf. The commission amount totaled $660,000. How much money did he earn from his movies? (Lesson 6-8) Solve each proportion. (Lesson 6-2) 16 a 28. _ =_ 3

5 15 29. _ =_ x 10

24

n 2 30. _ =_ 16

36

8 12 and y = _ . Write in simplest form. (Lesson 5-3) 31. Evaluate xy if x = _ 9

30

1 GEOGRAPHY Africa makes up _ of all the land on Earth. Use the 5

table to find the fraction of Earth’s land that is made up by other continents. Write each fraction in simplest form. (Lesson 5-2)

Continent

Decimal Portion of Earth’s Land

Antarctica

0.095

32. Antarctica

33. Asia

Asia

0.295

34. Europe

35. North America

Europe

0.07

North America

0.16

Source: Incredible Comparisons


407

CH

APTER

7



Key Vocabulary X

Be sure the following Key Concepts are noted in your Foldable.

Key Concepts Functions

/

s

W

(Lesson 7-1)

• In a function, each member in the domain is paired with exactly one member in the range.

Representing Linear Functions

(Lesson 7-2)

• A solution of a linear equation is an ordered pair that makes the equation true. • A linear equation can be represented by a set of ordered pairs, a table of values, or a graph.

Rate of Change and Slope

(Lessons 7-3, 7-4, and 7-5)

• A change in one quantity in relation to another quantity is called the rate of change. • When a quantity increases over time, it has a positive rate of change. When a quantity decreases over time, it has a negative rate of change. When a quantity does not change over time, it has a zero rate of change. • Linear relationships have constant rates of change. • Two quantities a and b have a proportional

change in b a linear relationship if _ is constant and _

is constant.

b

constant of variation (p. 378) constant rate of change (p. 376) direct variation (p. 378) family of functions (p. 395) function (p. 359) line of fit (p. 403) linear equation (p. 365) linear relationship (p. 376) rate of change (p. 371) slope (p. 384) slope-intercept form (p. 391) vertical line test (p. 360) y-intercept (p. 391)

change in a

• Slope can be used to describe rates of change. • Slope is the ratio of the rise, or the vertical change, to the run, or the horizontal change.

Writing and Predicting Linear Equations (Lessons 7-6, 7-7, and 7-8)

Vocabular y Check Choose the term that best matches each statement or phrase. Choose from the list above. 1. a relation in which each member of the domain is paired with exactly one member of the range 2. a value that describes the steepness of a line 3. can be drawn through data points to approximate a linear relationship 4. a description of how one quantity changes in relation to another quantity 5. a graph of this is a straight line 6. a linear equation that describes rate of change

• In the slope-intercept form y = mx + b, m is the slope and b is the y-intercept.

7. a way to determine whether a relation is a function

• You can write a linear equation by using the slope and y-intercept, two points on a line, a graph, a table, or a verbal description.

8. the rate of change between any two data points is the same

• A line of fit is used to approximate data.


9. k in the equation y = kx 10. an equation written in the form y = mx + b



Functions

(pp. 359–363)

Determine whether each relation is a function. Explain. 11. {(1, 12), (-4, 3), (6, 36), (10, 6)}

Example 1 Determine whether {(-9, 2), (1, 5), (1, 10)} is a function. Explain. $OMAIN X

2ANGE Y

£

Ó x £ä

12. {(11.8, -9), (10.4, -2), (11.8, 3.8)} 13. {(0, 0), (2, 2), (3, 3), (4, 4)} 14. {(-0.5, 1.2), (3, 1.2), (2, 36)} GASOLINE Use the table that shows the cost of gas in different years.

Year 2002 2003 2004

Cost $1.36 $1.59 $1.82


15. Is the relation a function? Explain. 16. Describe how the cost of gas is related to the year.

This relation is not a function because 1 in the domain is paired with two range values, 5 and 10. Example 2 The table shows the number of Alternative Fuel Vehicles (AFVs). Do these data represent a function? Explain.

Year 2001 2002 2003

AFVs 623,043 895,984 930,538

Source: eia.doe.gov

This relation is a function because during each year, there is only one value of AFVs.

7–2

Linear Equations in Two Variables

(pp. 365–369)

Graph each equation by plotting ordered pairs.

Example 3 Graph y = -x + 2 by plotting ordered pairs.

17. y = x + 4

18. y = x - 2

19. y = -x

20. y = 2x

The ordered pair solutions are: (0, 2), (1, 1), (2, 0), (3, -1).

21. CANDY A regular fruit smoothie x costs $1.50, and a large fruit smoothie y costs $3. Find two solutions of 1.5x + 3y = 12 to determine how many of each type of fruit smoothie Lisa can buy with $12.

Then plot and connect the points. y (0, 2) (1, 1) y x 2 (2, 0) x O (3, 1)


409

CH

A PT ER

7 7–3


Rate of Change

(pp. 371–375)

Find the rate of change for each function. 22. Time (s) Distance (m) x 0 1 2

y 0 8 16

7–4

5

4

4

8

3

change in time

0 min - 4 min 1 ft or _ 1 ft/min = -4 min -4 1 The rate of change is _ ft/min, or a -4 1 decrease of foot per minute. 4

Constant Rate of Change and Direct Variation Time (min) x 0 20 40 60

Distance (mi) y 0 3 6 9

25. FRUIT The cost of peaches varies directly with the number of pounds bought. If 3 pounds of peaches cost $4.50, find the cost of 5.5 pounds.

Slope

0

5 ft - 4 ft = __

y 45 43 41

24. Find the constant rate of change for the linear function and interpret its meaning.

7–5

Time Water (min) Level (ft)

change in water level rate of change = __

23. Time (h) Temperature (° F) x 1 2 3

Example 4 The table shows the relationship between time and water level of a pool. Find the rate of change.

(pp. 376–381)

Example 5 Find the rate of change in population from 2000 to 2004 for El Paso, Texas. y2 - y1 rate of change = _ x -x 2

x

Population (1000s) y

2000 2004

564 592

Year

1

Definition of slope

592 - 564 Substitute. =_ 2004 - 2000

=7 Simplify. So, the rate of change in population was 7 thousand people per year.

(pp. 384–389)

Find the slope of the line that passes through each pair of points.

Example 6 Find the slope of the line that passes through A(0, 6) and B(4, -2).

26. J(3, 4), K(4, 5)

y2 - y1 m=_ x -x

Definition of slope

-2 - 6 m=_

(x1, y1) = (0, 6) (x2, y2) = (4, -2)

-8 or -2 m=_

The slope is -2.

27. C(2, 8), D(6, 7)

2

28. ANIMALS A lizard is crawling up a hill that rises 5 feet for every horizontal change of 30 feet. Find the slope.


1

4-0 4



7–6

Slope-Intercept Form

(pp. 391–394)

Graph each equation using the slope and y-intercept. 29. y = -x + 4 30. y = -2x + 1

Example 7 State the slope and yintercept of the graph of y = -2x + 3.

31. y = 1x - 2 3

Writing Linear Equations

y = mx + b

33. BIRDS The altitude in feet y of an albatross who is slowly landing can be given by y = 400 - 100x, where x represents the time in minutes. State the slope and y-intercept of the graph of the equation and describe what they represent.

7–7

32. x + y = -5

y = -2x + 3

The slope of the graph is -2, and the y-intercept is 3.

(pp. 397–402)

Write an equation in slope-intercept form for each line. 34. slope = -1, y-intercept = 3 35. slope = 6, y-intercept = -3 Write an equation in slope-intercept form for the line passing through each pair of points. 36. (3, 7), (4, 4) 37. (1, 5), (2, 8) BIRTHDAYS For Exercises 38 and 39, use the following information. It costs $100 plus $30 per hour to rent a movie theater for a birthday party. 38. Write an equation in slope-intercept form that shows the cost y for renting the theater for x hours. 39. Find the cost of renting the theater for 4 hours.

Example 8 Write an equation in slope-intercept form for the line that passes through (5, 9) and (2, 0). Step 1 Find the slope m. y2 - y1 m= _ x -x 2

1

9-0 m=_ or 3 5-2

Step 2

Find the y-intercept b. Use the slope and the coordinates of either point. y = mx + b 9 = 3(5) + b -6 = b

Step 3

Definition of slope (x1, y1) = (5, 9) (x2, y2) = (2, 0)

Slope-intercept form Substitute. Simplify.

Substitute the slope and y-intercept. y = mx + b y = 3x + (-6) y = 3x - 6

Slope-intercept form Substitute. Simplify.


411

CH

A PT ER

7 7–8


Prediction Equations

(pp. 403–407)

ART The table shows the attendance for an annual art festival. Year 2002 2003 2004 2005

Attendance 2500 2650 2910 3050

Example 9 Make a scatter plot and draw a line of fit for the table showing the attendance at home games for the first four games of a high school football season. Game 1 2 3 4

40. Make a scatter plot and draw a line of fit.

HOUSING The table shows the changes in the median price of existing homes. Year

Median Price ($ thousands)

1991

97.1

1995

110.5

1998

128.4

2000

139.0

2002

158.1

2003

170.0

Draw a line that fits the data. ÌÌi`>Vi Õ`Ài`Ã®

41. Use the line of fit to predict art festival attendance in 2010.

£È £x £{ £Î £Ó ££ ä

Source: National Association of REALTORS

42. Make a scatter plot and draw a line of fit for the data. 43. Use the line of fit to predict the median price for an existing home for the year 2015.

Attendance 1100 1200 1300 1500

£

Ó Î { >i

x

Use the line of fit to predict the attendance for the seventh home game. Extend the line so that you can find the y-value for an x-value of 7. The y-value for 7 is about 19. So, a prediction for the attendance at the seventh home game is approximately 1900 people. Óä

44. Use the line of fit to predict the value of y when x = 7. y

ÌÌi`>Vi Õ`Ài`Ã®

£ £n £Ç £È £x £{ £Î £Ó ££ O

x ä


£

Ó

Î

{ x >i

È

Ç

n

CH

A PT ER

7

Practice Test

Determine whether each relation is a function. Explain.

1. {(-3, 4), (2, 9), (4, -1), (-3, 6)} 2. {(1, 2), (4, -6), (-3, 5), (6, 2)} 3. {(7, 0), (9, 3), (11, 1), (13, 0)}

F y = 3x + 10

Graph each equation by plotting ordered pairs.

4. y = 2x + 1

5. 3x + y = 4

6. MULTIPLE CHOICE Find the rate of change for the linear function represented in the table. Hours Worked Money Earned ($)

1 5.50

2 11.00

3 16.50

16. MULTIPLE CHOICE Victor works at a barber shop. He gets paid $10 an hour plus $3 for every hair cut he performs. Which equation represents Victor’s hourly earnings?

4 22.00

A increase $6.50/h

G y = 10x + 3 H y = 3x - 10 J

y = 10x - 3

RECYCLING For Exercises 17 and 18 use the graph and the information below. Ramiro collected 150 pounds of cans to recycle. He plans to collect an additional 30 pounds each week. The graph shows the amount of cans he plans to collect. {ää

B increase $5.50/h

Îxä Îää

D decrease $6.50/h

Óxä

7. JOBS Determine whether a proportional linear relationship exists between the two quantities in Exercises 6. Explain your reasoning.

>Ã L®

C decrease $5.50/h

£xä £ää

Find the slope of the line that passes through each pair of points. 10. A(2, 5), B(4, 11) 12. F(8, 5), G(7, 9)

11. C(-4, 5), D(6, -3) 13. H(11, 6), J(9, -1)

State the slope and y-intercept of the graph of each equation. Then graph each equation using the slope and y-intercept. 2 x-4 14. y = _ 3

15. 2x + 4y = 12


ä] £xä®

xä ä

FUND-RAISING The total profit for a school varies directly with the number of potted plants sold. Suppose the school earns $57.60 if 12 plants are sold. 8. Write an equation that could be used to find the profit per plant sold. 9. Find the total profit if 65 plants are sold.

Î] Ó{ä®

Óää

£

Ó Î { ÕLiÀ v 7iiÃ

x

17. Find the equation of the line. 18. What does the slope of the line represent? GARDENING For Exercises 19 and 20, use the table and the information below. The full-grown height of a tomato plant and the number of tomatoes it bears are recorded for five tomato plants. 19. Make a scatter plot of the Height Number of data and draw a line of fit. (in.) Tomatoes 27 12 20. Use the line of fit to predict 33 18 the number of tomatoes a 43-inch tomato plant will 19 9 bear. 40 16 31


15

413

CH

A PT ER


7


Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The graph of the line y = 2x + 2 is shown on the coordinate grid below. n Ç È x { Î Ó £ n ÇÈ x{ÎÓ£" £ Ó Î { x È Ç n

Y

Question 3 When answering GRIDDABLE questions, first fill in the answer on the top row. Then pencil in exactly one bubble under each number or symbol.

£ Ó Î { x È Ç nX

Which table of ordered pairs contains only points on this line? A x C y x y

B

2

2

2

3

1

0

1

1

0

2

0

1

1

4

1

3

x

y

1

3. GRIDDABLE A bus traveled 209 miles at an average speed of 70 miles per hour. About how many hours did it take for the bus to reach its destination? Round your answer to the nearest half hour.

D

x

y

0

1

1

0

3

0

1

1

6

1

3

2

9

2

5

2. Mario has 125 coins in his collection. He plans to add another 5 coins each week until he has doubled the amount in his collection. Which equation can be used to determine w, the number of weeks it will take to double the size of the coin collection? H 5w + 125w = 250 F 5w + 125 = 125 G 5w + 125 = 250 J 2(5w + 125) = 250 414 Chapter 7 Functions and Graphing

4. Chris, Candace, Jamil, and Lydia ate breakfast at a restaurant. The total amount of the bill, including tax and tip was $46.60. 1 of the bill, Chris paid $10, Candace paid _ 4 Jamil paid 20% of the bill, and Lydia paid the rest. Who paid the greatest amount? A Chris B Candace C Jamil D Lydia 5. Mr. Williams wants to purchase some blank CDs. He compared the prices from four different office supplies stores. Which store’s prices are based on a constant unit price? H Number Total F Number Total of CDs 10

Cost $3.65

of CDs 10

Cost $2.50

20

$6.65

20

$5.00

30

$9.65

30

$7.50

$12.65

40

$10.00

Number of CDs 10

Total Cost $3.00

40

G Number

J

of CDs 10

Total Cost $3.50

20

$7.00

20

$5.50

30

$10.00

30

$7.50

40

$12.00

40

$10.00



6. The Saturn V rocket that took Apollo astronauts to the moon weighed 6,526,000 pounds at lift-off. Write its weight in scientific notation. A 6.526 × 10-7 C 6.526 × 106 B 6.526 × 10-6 D 6.526 × 107

9. Which of the following statements is true? F 0.4 > 40% H 40% > 0.04 G 0.04 = 40% J 40% ≤ 0.04 10. The cost, c, of hiring a plumber can be found using the equation c = 75 + 40h, where h is the number of hours the plumber worked. For how many hours was the plumber hired if he charged $215? C 3.75 h A 3h B 3.5 h D 4.25 h

7. Mary-Ann saved $56 when she purchased a television on clearance at an electronics store. If the sale price was 20% off the regular price, what was the regular price? F $250 H $275 G $260 J $280

11. The expression 2(n + 4) describes a pattern of numbers. What is the tenth term of the sequence? F 2 H 25 G 15 J 28

8. The graph shows the shipping charges per order, based on the number of items shipped in the order. Which statement best describes this graph?

Pre-AP Record your answers on a sheet of paper. Show your work.

Shipping Charge ($)

y 18 16 14 12 10 8 6 4 2 0

12. Krishnan is considering three plans for cellular phone service. The plans each offer the same services for different monthly fees and different costs per minute. Plan

Monthly Fee

X

$0

x

1 2 3 4 Number of Items

A As the number of items increases, the shipping charge decreases.

Cost per Minute $0.24

Y

$15.95

$0.08

Z

$25.95

$0.04

a. For each plan, write an equation that shows the total monthly cost c for m minutes of calls. b. What is the cost of each plan if Krishnan uses 100 minutes per month? c. Which plan costs the least if Krishnan uses 100 minutes per month? d. Which plan costs the least if Krishnan uses 300 minutes per month?

B As the number of items increases, the shipping charge increases. C As the number of items decreases, the shipping charge increases. D There is no relationship between the number of items shipped and the shipping charge.


1

2

3

4

5

6

7

8

9

10

11

12

Go to Lesson...

7-2

7-7

3-8

5-2

6-1

4-7

6-8

7-1

6-5

1-5

3-7

7-6

Chapters 7 Standardized Test Practice

415


8 •


•

Make connections among various representations of a numerical relationship.

•


Key Vocabulary null or empty set (p. 426) identity (p. 426) inequality (p. 430)

Real-World Link Capacity You can use an inequality to express the maximum number of people that can be held in the Radio City Music Hall in New York City.

Equations and Inequalities Make this Foldable to help you organize notes on equations and 1 inequalities. Begin with a plain sheet of 8_” by 11” paper. 2

1 Fold in half lengthwise.

2 Fold in thirds and then fold each third in half.

3 Open. Cut one side

4 Label each tab with a

along the folds to make tabs.

lesson number as shown.

416 Chapter 8 Equations and Inequalities CORBIS SYGMA



Option 1


Solve each equation. Check your solution.

Example 1

(Lesson 3-3)

Solve _r + 6 = 5.

1. 2x + 5 = 13

2. 4n - 3 = 5

d 3. 16 = 8 + _

c 4. _ + 3 = -9

5. -18 = 4b + 10

h 6. -9 - _ =5

3

-4

4

7. SALES Suppose a computer costs $600. If Tony pays a down payment of $150 and plans to pay the balance in 6 equal installments, how much would each installment be? (Lesson 3-3)

Find each sum or difference. (Lessons 2-2 and 2-3)

8. -28 + (-16)

4

_r + 6 = 5 4

_r + 6 - 6 = 5 - 6 4

_r = -1

4 r 4 _ = 4(-1) 4

r = -4

Write the equation. Subtract 6 from each side. Simplify. Undo division. Multiply each side by 4. Simplify.

Example 2

Find -30 - (-42). 9. 17 + (-25)

10. -13 + 24

11. 36 + (-18)

12. 31 - 48

13. -16 - 7

14. 4 - (-12)

15. -23 - (-29)

-30 - (-42) = -30 + 42 To subtract -42, add 42.

= 12

Simplify.

16. STOCKS The stock market fell 507.99 points on October 19, 1987. If the stock market began the day at a 2246.73, what was its value at the end of the day? (Lesson 2-3)

Find each product or quotient. (Lessons 2-4 and 2-5)

Example 3

Find 6 × (-15).

17. -6(8)

18. -3 · 5

19. -6(-25)

20. 2(-4)(-9)

21. 64 ÷ (-32)

22. -15 ÷ 3

23. -12 ÷ (-3)

24. 24 ÷ (-2)

6 × (-15) = -90 The factors have different signs so the product is negative.

25. CHEMISTRY A solution cooled at a rate of 6˚F every 5 minutes. What was the change 1 hour? (Lesson 2-5) in temperature after _ 2


417

EXPLORE

8-1

Algebra Lab

Equations with Variables on Each Side In Chapter 3, you used algebra tiles and an equation mat to solve equations in which the variable was on only one side of the equation. You can use algebra tiles and an equation mat to solve equations with variables on each side of the equation.

ACTIVITY 1 The following example shows how to solve x + 3 = 2x + 1 using algebra tiles. Step 1

Model the equation. 1

x

1

x

x 1

1 x3

Step 2

Remove the same number of x-tiles from each side of the mat until there is an x-tile by itself on one side.

2x 1

1

x

1

x

x 1

1 x x3

2x x 1

Variables Keep in mind that, unlike the numerical tiles, the size of the x-tile does not correspond to the number that it represents. Remember that the x-tile represents an unknown number, and so even though the x-tile appears to be only slightly larger than the 1-tile, x could actually be much larger, or even smaller, than 1.

Step 3

Remove the same number of positive tiles from each side of the mat until the x-tile is by itself on one side.

1

1

x 1

1

2

x

There are two positive tiles on the left side of the mat and one x-tile on the right side. Therefore, x = 2. Since 2 + 3 = 2(2) + 1, the solution is correct.

ANALYZE THE RESULTS Use algebra tiles to model and solve each equation. 1. 2x + 3 = x + 5

2. 3x + 4 = 2x + 8

3. 3x = x + 6

4. 6 + x = 4x

5. 2x - 4 = x - 6

6. 5x - 1 = 4x - 5

7. Which property of equality allows you to remove a positive tile from each side of the mat? 8. Explain why you can remove an x-tile from each side of the mat. 418 Chapter 8 Equations and Inequalities

Some equations are solved by using zero pairs. Remember, you may add or subtract a zero pair from either side of an equation mat without changing its value. The following example shows how to solve 2x + 1 = x - 5.

Review Vocabulary Zero Pair A pair of numbers that, when added together, equal zero. Example: 3 + -3 = 0

ACTIVITY 2 Step 1 Model the equation.

1 x

x

x

1 2x 1

Step 2

Remove the same number of x-tiles from each side of the mat until there is an x-tile by itself on one side.

Step 4

Remove the zero pair from the left side. There are 6 negative tiles on the right side of the mat.

1

1

1 x

x

2x x 1

It is not possible to remove the same number of 1-tiles from each side of the mat. Add 1 negative tile to the left side to make a zero pair. Add 1 negative tile to the right side of the mat.

1

x 5

x

1

Step 3

1

1

1

1

1

x x 5

1 1 1

x

1 1

1

x 1 (1)

1 x

1 1

5 (1)

1 1

1

x

1 1 1 1

6

Therefore, x = -6. Since 2(-6) + 1 = -6 - 5, the solution is correct.

ANALYZE THE RESULTS Use algebra tiles to model and solve each equation. 9. 2x + 3 = x - 5

10. 3x - 2 = x + 6

11. x - 1 = 3x + 7

12. x + 6 = 2x - 3

13. 2x + 4 = 3x - 2

14. 4x - 1 = 2x + 5

15. Does it matter whether you remove x-tiles or 1-tiles first? Explain. 16. Explain how you could use models to solve -2x + 5 = -x - 2. Explore 8-1 Algebra Lab: Equations with Variables on Each Side

419

8-1

Solving Equations with Variables on Each Side

Main Idea • Solve equations with variables on each side.

Each bag on the balance contains the same number of blocks. (Assume that the paper bag weighs nothing.) a. The two sides balance. Without looking in a bag, how can you determine the number of blocks in each bag? b. Explain why your method works. c. Suppose x represents the number of blocks in the bag. Write an equation that is modeled by the balance. d. Explain how you could solve the equation.

Equations with Variables on Each Side To solve equations with Look Back To review Addition and Subtraction Properties of Equality, see Lesson 3-3.

variables on each side, use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side. Then solve the equation.

EXAMPLE

Equations with Variables on Each Side

Solve 2x + 3 = 3x. Check your solution. 2x + 3 = 3x 2x - 2x + 3 = 3x - 2x 3=x Subtract 2x from the left side of the equation to isolate the variable.

Write the equation. Subtract 2x from each side. Simplify.

Subtract 2x from the right side of the equation to keep it balanced.

To check your solution, replace x with 3 in the original equation. CHECK

2x + 3 = 3 x 2(3) + 3 3(3) 6+39 9=9

Write the equation. Replace x with 3. Simplify. The statement is true.

The solution is 3.

1. Solve 7x = 5x + 4. Check your solution. 420 Chapter 8 Equations and Inequalities


EXAMPLE BrainPOP® at pre-alg.com

Equations with Variables on Each Side

Solve each equation. Check your solution. a. 5x + 4 = 3x - 2 Write the equation. 5x + 4 = 3x - 2 5x - 3x + 4 = 3x - 3x - 2 Subtract 3x from each side. 2x + 4 = -2 Simplify. 2x + 4 - 4 = -2 - 4 Subtract 4 from each side. 2x = -6 Simplify. x = -3 Check your solution.

b. 2.4 + a = 2.5a - 4.5 Write the equation. 2.4 + a = 2.5a - 4.5 2.4 + a - a = 2.5a - a - 4.5 Subtract a from each side. 2.4 = 1.5a - 4.5 Simplify. 2.4 + 4.5 = 1.5a - 4.5 + 4.5 Add 4.5 to each side. 6.9 = 1.5a Simplify.

6.9 1.5a _ =_


4.6 = a

Check your solution.

1.5

1.5

2A. 2x + 3 = 3x - 2

2B. 3.2 + 0.3x = 0.2x + 1.4

RENTALS Under Plan A, an annual membership costs $30 plus $1.50 for each DVD rental. Under Plan B, the annual membership costs $12 plus $3 for each DVD rental. What number of DVD rentals results in the same yearly cost? Let v represent the number of videos rented. $30 plus $1.50 for each video Real-World Link In 1980, only 1% of American households owned a VCR. Today more than 86% own either a VCR or DVD player. Source: Statistical Abstract of the United States

equals

$12 plus $3 for each video

30 + 1.50v = 12 + 3v 30 + 1.50v = 12 + 3v Write an equation. 30 + 1.5v - 1.5v = 12 + 3v - 1.5v Subtract 1.5v from each side. 30 = 12 + 1.5v Simplify. 30 - 12 = 12 - 12 + 1.5v Subtract 12 from each side. 18 = 1.5v Simplify. 18 1.5v _ =_ 1.5

1.5

12 = v

Divide each side by 1.5. Simplify.

The yearly cost is the same for 12 rentals.

3. CRUISES Red Bird Cruises charges $85 per day plus a one-time fee of $75 for taxes and gratuities. King Cruises charges $100 per day plus a fee of $30. For what number of days do the cruise companies charge the same? Personal Tutor at pre-alg.com Lesson 8-1 Solving Equations with Variables on Each Side Jose Luis Pelaez, Inc./CORBIS

421

Solve each equation. Check your solution. Example 1 (p. 420)

Example 2 (p. 421)

Example 3 (p. 421)

HOMEWORK

HELP


1. 4x - 8 = 5x

2. 4x + 9 = 7x

3. 12x = 2x + 40

4. 6a = 26 + 4a

5. 4x - 1 = 3x + 2

6. 4k + 24 = 6k - 10

7. 7.2 - 3c = 2c - 2

8. 3 - 3.7b = 10.3b + 10

9. CAR RENTAL Suppose you can rent a car from ABC Auto for either $25 a day plus $0.45 a mile or for $40 a day plus $0.25 a mile. What number of miles results in the same cost for one day?

Solve each equation. Check your solution. 10. 2x + 3 = x

11. n - 14 = 3n

12. 8 - 2c = 2c

13. q - 2 = -q + 1

14. 13y - 18 = -5y + 36

15. -s + 4 = 7s - 3

16. 7d - 13 = 3d + 7

17. 2f - 6 = 7f + 24

18. 12n - 23.2 = -14n + 28.8

19. 3.1w + 5 = 0.8 + w

Define a variable and write an equation to find each number. Then solve. 20. Twice a number is 220 less than six times the number. What is the number? 21. Fourteen less than three times a number equals the number. Find the number. 22. GEOGRAPHY South Carolina’s coastline is 358 kilometers longer than twice the coastline of North Carolina. It is also 842 kilometers longer than the coastline of North Carolina. Find the lengths of the coastlines of South Carolina and North Carolina. 23. MUSIC DOWNLOADS Denzel is comparing Web sites for downloading music. One charges a $5 membership fee plus $0.50 per track. Another charges $1.00 per track, but has no monthly fee. How many songs would Denzel have to buy for him to spend the same amount at both Web sites? Solve each equation. Check your solution.

EXTRA

PRACTICE


H.O.T. Problems

24. 12 + 1.5a = 3a

25. 12.6 - x = 2x

26. 2b + 6.2 = 13.2 - 8b

27. 3c + 4.5 = 7.2 - 6c

28. 12.4y + 14 = 6y - 2

29. 4.3n - 1.6 = 2.3n + 5.2

30. 0.4x = 2x + 1.2

1 1 b + 8 =_ b-4 31. _ 3

2

32. CELLULAR PHONES One cellular phone carrier charges $29.75 a month plus $0.15 a minute for international calls. Another carrier charges $19.95 a month and $0.29 a minute for international calls. For how many minutes is the cost of the plans the same? 33. NUMBER SENSE Three times the quantity y + 7 is equal to four times the quantity y - 2. What value of y makes the sentence true?

422 Chapter 8 Equations and Inequalities

34. OPEN ENDED Write an example of an equation with variables on each side. State the steps you would use to isolate the variable.

The trends in attendance at various sporting events can be represented by equations. Visit pre-alg.com.

35. CHALLENGE An empty bucket is put under two faucets. If one faucet is turned on alone, the bucket fills in 6 minutes. If the other faucet is turned on alone, the bucket fills in 4 minutes. If both are turned on, how many seconds will it take to fill the bucket? 36.

Writing in Math

Explain how solving equations with variables on each side is like solving equations with variables on just one side. Include examples of both types of equations and an explanation of how they are alike and how they are different.

9 37. The formula F = _ C + 32 is used to 5 find the Fahrenheit temperature when a Celsius temperature is known. For what value are the Celsius and Fahrenheit temperatures the same?

A -72°

39. Olivia’s manager gave her a choice as to how she wants to be paid.

C 0°

B -40° D 32° 38. Two weeks ago the sewing club had 1 less than 3 times their average attendance. Last week they had 3 more than their average attendance. If the attendance for both weeks were equal, what is the average attendance of the sewing club? F 1

H 3

G 2

J 4

Pay per Hour

Pay for Each Dollar of Appliance Sales

Plan 1

$3

15¢

Plan 2

$4

10¢

Which equation shows what Olivia’s sales need to be in one hour to earn the same amount under either plan? A 3 + 0.15s = 4 + 0.10s B 3s + 0.15 = 4s + 0.10 C 3 + 0.10s = 4 + 0.15s D 3(s + 0.15) = 4(s + 0.10)

40. Find the true statement. (Lesson 7-8) • A line of fit is close to most of the data points. • A line of fit describes the exact coordinates of each point in the data set. • A line of fit always has a positive slope. 41. What equation represents the table of values? (Lesson 7-7)

x

-4

-8

-12

-16

y

6

8

10

12

PREREQUISITE SKILL Use the Distributive Property to rewrite each expression as an equivalent algebraic expression. (Lesson 3-1) 42. 4(x - 8)

43. 2(1.2c + 14)

1 44. _ (n - 9) 2

Lesson 8-1 Solving Equations with Variables on Each Side

423

8-2

Solving Equations with Grouping Symbols

Main Ideas • Solve equations that involve grouping symbols. • Identify equations that have no solution or an infinite number of solutions.

New Vocabulary null or empty set identity

Josh starts walking toward the park at a rate of 2 mph. One hour later, his sister Maria starts on the same path, riding her bike at 10 mph. The table shows expressions for the distance Maria and Josh have traveled after a given time.

Rate (mph)

a. What does t represent?

Time (hours)

Distance (miles)

Josh

2

t

2t

Maria

10

t-1

10(t - 1)

b. Why is Maria’s time shown as t - 1? c. Write an equation that represents the time when Maria catches up to Josh. (Hint: They will have traveled the same distance.)

Solve Equations with Grouping Symbols To find how many hours it takes Maria to catch up to Josh, you can solve the equation 2t = 10(t - 1). First, use the Distributive Property to remove the grouping symbols.

EXAMPLE

Solve Equations with Parentheses

a. Solve the equation 2t = 10(t - 1). Check your solution. 2t = 10(t - 1)

Write the equation.

2t = 10(t) - 10(1)

Use the Distributive Property.

2t = 10t - 10

Simplify.

2t - 10t = 10t - 10t - 10

Review Vocabulary Dimensional Analysis The process of including units of measurement when computing (Lesson 5-3)

Subtract 10t from each side.

-8t = -10

Simplify.

-10 -8t _ =_


-8 5 1 t = _ or 1_ 4 4

-8

CHECK

Simplify.

Use dimensional analysis. 2 miles _ 1 · 5 hour or 2 _ miles. Josh traveled _ 2

4

hour

Maria traveled one hour less than Josh. She traveled 10 miles _ 1 _ · 1 hour or 2 _ miles. hour

4

2

1 hour, or 15 minutes. Therefore, Maria caught up to Josh in _ 4



b. Solve 5(a - 4) = 3(a + 1.5).

Alternative Method You can also solve the equation by subtracting 3a from each side first, then adding 20 to each side.

5(a - 4) = 3(a + 1.5) 5a - 20 = 3a + 4.5 5a - 20 + 20 = 3a + 4.5 + 20 5a = 3a + 24.5 5a - 3a = 3a - 3a + 24.5 2a = 24.5 24.5 2a _ =_

Use the Distributive Property. Add 20 to each side. Simplify. Subtract 3a from each side. Simplify. Divide each side by 2.

2

2

Write the equation.

a = 12.25

Simplify.

Solve each equation. Check your solution. 1A. 3x = 4(x + 2) 1B. -0.2(3c + 15) = 3(0.8c - 8)

Sometimes a geometric figure is described in terms of only one of its dimensions. To find the dimensions, you may have to solve an equation that contains grouping symbols.

EXAMPLE

Use an Equation to Solve a Problem

GEOMETRY The perimeter of a rectangle is 46 inches. Find the dimensions if the length is 5 inches greater than twice the width. Words Variable Equation

Review Vocabulary Perimeter The distance around a geometric figure; Example: The perimeter of a square with sides that are 5 inches long is 20 inches. (Lesson 3-8)

2 times width

+

2 times length

= perimeter w

Let = the width. Let 2w + 5 = the length. 2w

+

2w + 2(2w + 5) = 46 2w + 4w + 10 = 46 6w + 10 = 46 6w + 10 - 10 = 46 - 10 6w = 36 w=6

2(2w + 5)

=

46

2w + 5

Write the equation. Use the Distributive Property. Simplify. Subtract 10 from each side. Simplify. Mentally divide each side by 6.

Evaluate 2w + 5 to find the length. 2(6) + 5 = 12 + 5 or 17

Replace w with 6.

The width is 6 inches. The length is 17 inches.

2. RECYCLING Sofia recycled 3 pounds less than 3 times the amount that James recycled. If they recycled a total of 53 pounds, how many pounds did each person recycle? Personal Tutor at pre-alg.com Lesson 8-2 Solving Equations with Grouping Symbols

425

No Solution or All Numbers as Solutions Some equations have no solution. That is, no value of the variable results in a true sentence. When this occurs, the set of solutions for the equation contains no elements. A set that contains no elements is called the null or empty set, shown by the symbol ∅ or {}.

EXAMPLE

No Solution

1 1 Solve 3x + _ = 3x - _ .


3 2 1 1 Write the equation. 3x + _ = 3x - _ 3 2 1 1 3x - 3x + _ = 3x - 3x - _ Subtract 3x from each side. 3 2 _1 = -_1 Simplify. 3 2 1 1 The sentence _ = -_ is never true. So, the solution set is ∅. 3 2

3. Solve 6x + 4 = 2(3x - 5). Check your solution. An equation that is true for every value of the variable is called an identity.

EXAMPLE

All Numbers as Solutions

Solve 2(2x - 1) + 6 = 4x + 4. 2(2x - 1) + 6 = 4x + 4 Write the equation. 4x - 2 + 6 = 4x + 4 Use the Distributive Property. 4x + 4 = 4x + 4 Simplify. 4x + 4 - 4 = 4x + 4 - 4 Subtract 4 from each side. 4x = 4x Simplify. x=x Mentally divide each side by 4. The sentence x = x is always true. The solution set is all numbers.

4. Solve 20f + (-8f - 15) = 3(4f - 5). Check your solution.

Example 1 (pp. 424–425)

Example 2 (p. 425)


Solve each equation. Check your solution. 1. 3(g - 3) = 6

2. 4(x + 1) = 28

3. 2(a - 2) = 3(a - 5)

4. 16(z + 3) = 4(z + 9)

5. 5(2c + 7) = 80

6. 6(3d + 5) = 75

7. GEOMETRY The perimeter of a rectangle is 20 feet. The width is 4 feet less than the length. Find the dimensions of the rectangle. Solve each equation. Check your solution. 8. 12 - h = -h + 3 10. 3(2g + 4) = 6(g + 2)


9. 3n + 4 = 3(n + 2) 11. 4(f + 3) + 5 = 17 + 4f

HOMEWORK

HELP

For See Exercises Examples 12–19 1 20, 21 3 22, 23 4 24, 25 2

Solve each equation. Check your solution. 12. 2(d + 6) = 3d - 1

13. 6n - 18 = 4(n + 2.1)

14. 3(a - 3) = 2(a + 4)

15. 3(s + 22) = 4(s + 12)

16. 4(x - 2) = 3(1.5 + x)

17. 3(a - 1) = 4(a - 1.5)

18. 2(3.5n + 6) = 2.5n - 2

19. 4.2x - 9 = 3(1.2x + 4)

20. 2(x - 5) = 4x - 2(x + 5) + 1

21. (3x + 2) + (-x + 5) = 2x - 7

22. 8y - 5 = 5(y - 1) + 3y

23. 10z + 4 = 2(5z + 8) - 12

24. GEOMETRY The perimeter of a rectangle is 32 feet. Find the dimensions of the rectangle if the length is 4 feet longer than three times the width. Then find the area of the rectangle. 25. BASKETBALL Camilla has three times as many points as Lynn. Lynn has five more points than Kim. Camilla, Lynn, and Kim combined have twice as many points as Jasmine. If Jasmine has 25 points, how many points does each of the other three girls have? Find the dimensions of each rectangle. The perimeter is given. 26. P = 460 ft

27. P = 440 yd

28. P = 11 m w

w

w

2w - 2 3w - 60

w + 30

Solve each equation. Check your solution. 1 29. _ (2n - 5) = 4n - 1

1 30. y - 2 = _ (y + 6)

1 (24b + 60) 31. -3(4b - 10) = _ 2

3 1 32. _ a+4=_ (3a + 16) 4 4

33. 0.4d = 2d + 1.24

a-6 a-2 =_ 34. _

2

3

12

35. GEOMETRY The triangle and the rectangle have the same perimeter. Find the dimensions of each figure. Then find the perimeter of each figure.

4

x -3 x

x +2

x +1

x +1

EXTRA

PRACTICE


H.O.T. Problems

36. DECORATING A gallon of paint covers about 350 square feet. A painter estimates the area to paint by multiplying the combined wall lengths by the height and subtracting 15 square feet for each window or door. Suppose a rectangular room measures 15 feet long by 12 feet wide. The room is 9 feet high and has two windows and two doors. How many gallons of paint are needed to paint the room using two coats of paint? 37. OPEN ENDED Give an example of an equation that has no solution and an equation that is an identity. 38. CHALLENGE An apple costs the same as 2 oranges. Together, an orange and a banana cost 10¢ more than an apple. Two oranges cost 15¢ more than a banana. What is the cost for one of each fruit? Lesson 8-2 Solving Equations with Grouping Symbols

427

39. SELECT A TOOL/TECHNIQUE Jamie has two spools with an equal length of plastic fencing that she is going to use to fence a rectangular and a triangular section of grass. The length of the rectangle will be 40 feet greater than the width, and the length of each side of the triangle will be 45 feet longer than the width of the rectangle. What technique(s) could be used to find the lengths of the sides? Justify your response and use your technique(s) to solve the problem. draw a model

use paper/pencil

use a calculator

40. NUMBER SENSE Three times the sum of three consecutive integers x, x + 1, and x + 2, is 72. What are the integers? 41.

Writing in Math Why is the Distributive Property important for solving equations? Include in your answer a definition of the Distributive Property and a description of its use in solving equations.

42. Wes leaves downtown driving 55 miles per hour. Emma follows 1 hour later, driving 60 miles per hour. Which equation can be used to determine how long it is after Wes leaves that Emma will catch up? A 55x = 60x - 1

C 55x = 60x

B 60x = 55(x - 1)

D 55x = 60(x - 1)

43. Find the value of x so that the polygons have the same perimeter. F 3 G 6 H 8

X { XÓ

X Î

X { X x

J 12

ALGEBRA Solve each equation. Check your solution. (Lesson 8-1) 44. 4x = 2x + 5

45. 3x + 5 = 7 - 2x

46. 1.5x + 9 = 3x - 3

47. HOUSING The table shows the median price of existing homes. Make a scatter plot and draw a line of fit for the data. Use the line of fit to predict the median price for an existing home in 2010. (Lesson 7-8) Year

1991

1995

1998

2000

2002

2003

Median Price (thousands)

97.1

110.5

128.4

139.0

158.1

170.0

Source: National Association of REALTORS

Express each number in scientific notation. (Lesson 4-7)

48. 4,500,000

49. -37,000

50. 0.000498

PREREQUISITE SKILL Evaluate each expression. (Lesson 1-3) 52. 2t + 8, t = -3 53. b + 11, b = -15 428 Chapter 8 Equations and Inequalities

51. -0.00203

54. 4a, a = -6

Meanings of at Most and at Least The phrases at most and at least are used in mathematics. In order to use them correctly, you need to understand their meanings. Phrase

Meaning

Mathematical Symbol

at most

• no more than • less than or equal to

≤

at least

• no less than • greater than or equal to

≥

Here is an example of one common use of each phrase, its meaning, and a mathematical expression for the situation. Verbal Expression You can spend at most $20. Meaning You can spend $20 or any amount less than $20. Mathematical Expression s ≤ 20, where s represents the amount you spend.

Verbal Expression Meaning

A person must be at least 18 to vote. A person who is 18 years old or any age older than 18 may vote. Mathematical Expression a ≥ 18, where a represents age.

Notice that the word or is part of the meaning in each case.

Reading to Learn 1. Write your own rule for remembering the meanings of at most and at least. For each expression, write the meaning. Then write a mathematical expression using ≤ or ≥. 2. You need to earn at least $50 to help pay for a class trip. 3. The sum of two numbers is at most 6. 4. You want to drive at least 250 miles each day. 5. You want to hike 4 hours each day at most. 6. There are no more than 25 apples in the basket. 7. It will take at least 5 hours to finish this project. Reading Math Meanings of at Most and at Least Bob Daemmrich/The Image Works

429

8-3

Inequalities

Main Ideas • Write inequalities. • Graph inequalities.

New Vocabulary

Children under 6 eat free.

Speed Limit

Must be over 40 inches tall to ride.

inequality

35

a. Name three ages of children who can eat free at the restaurant. Does a child who is 6 years old eat free? b. Name three heights of children who can ride the ride at the amusement park. Can a child who is 40 inches tall ride? c. Name three speeds that are legal. Is a driver who is traveling at 35 mph driving at a legal speed?

Write Inequalities A mathematical sentence that contains , ≤, or ≥ is called an inequality.

EXAMPLE

Write Inequalities

Write an inequality for each sentence. a. Your age is less than 6 years. Words Variable Inequality

Your age

is less than

6 years.

Let a represent your age. a

6

35

1A. Your height is greater than or equal to 40 inches. 1B. Your speed is less than or equal to 35 miles per hour. 430 Chapter 8 Equations and Inequalities


The table below shows some common verbal phrases and the corresponding mathematical inequalities.

Reading Math Inequalities Notice that ≤ and ≥ combine the symbol < or > with part of the symbol for equals, =.

Inequalities

• is less than • is fewer than

≤

• is greater than • is more than • exceeds

≥

• is less than or equal to • is no more than • is at most

• is greater than or equal to • is no less than • is at least

NUTRITION A food can be labeled low fat if it has no more than 3 grams of fat per serving. Write an inequality to describe low-fat foods. Words

Grams of fat per serving is no more than 3.

Variable

Let = number of grams of fat per serving.

Inequality

f

≤

3

The inequality is f ≤ 3.

2. DRIVER’S EDUCATION A student must have at least 10 hours of instructorassisted driving time. Write an inequality to describe this situation.

Inequalities with variables are open sentences. When the variable in an open sentence is replaced with a number, the inequality may be true or false.

EXAMPLE

Determine Truth of an Inequality

For the given value, state whether each inequality is true or false. a. s - 7 < 5, s = 14

Reading Math Inequality Symbols ≮ means is not less than.

s-7 12 is false, the equation 12 = 12 is true. Therefore, this sentence is true.

3A. 3 + x ≤ 12, x = 6

3B. y - 7 < 10, y = 17

Personal Tutor at pre-alg.com Lesson 8-3 Inequalities

431

Graph Inequalities Inequalities can be graphed on a number line. The graph helps you visualize the values that make the inequality true.

EXAMPLE

Graph Inequalities

Graph each inequality on a number line. a. x > 4

b. x ≥ 4

Inequalities When inequalities are graphed, an open dot means the number is not included and a closed dot means it is.

2

3

4

5

6

2

The open circle means the number 4 is not included in the graph.

4

5

6

The closed circle means the number 4 is included in the graph.

c. x < 4 2

3

d. x ≤ 4 3

4

5

6

2

4A. x < 5

4B. x ≥ -2

4C. x > 0

EXAMPLE

Write an Inequality

3

4

5

6

4D. x ≤ 2

Write the inequality for the graph. 4

5

6

7

8

9 10 11 12 13 14

An open circle is on 10, so the point 10 is not included in the graph. The arrow points to the right, so the graph includes all numbers greater than 10. The inequality is x > 10.

5A.

5B. x { Î Ó £ ä £ Ó Î { x

Example 1 (p. 430)

Example 2 (p. 431)

Example 3 (p. 431)

-7 -6 -5 -4 -3 -2 -1 0 1 2 3

Write an inequality for each sentence. 1. Lacrosse practice will be no more than 45 minutes. 2. Mario is more than 60 inches tall. 3. SOCCER More than 8000 fans attended the Wizards’ opening soccer game at Arrowhead Stadium in Kansas City, Missouri. Write an inequality to describe the attendance. For the given value, state whether the inequality is true or false. 4. n + 4 > 6, n = 12


5. 34 ≤ 4r, r = 8

Example 4 (p. 432)

Example 5 (p. 432)

Graph each inequality on a number line. 6. n > 3

HELP

For See Exercises Examples 12–15

1

16, 17

2

18–23

3

24–35

4

36–39

5

8. x < 7

9. d ≥ -6

Write the inequality for each graph. 10.

11. 7

HOMEWORK

7. y ≤ 14

8

-22

9 10 11 12 13 14 15

-20

-18

-16

-14

Write an inequality for each sentence. 12. The elevators in an office building have been approved for a maximum load of 3600 pounds. 13. Kyle’s earnings were no more than $60. 14. The race time of 86 minutes was greater than the winner’s time. 15. After a withdrawal, a savings account is now less than $500. ANALYZE TABLES For Exercises 16 and 17 use the table that shows the average amount of time students ages 14 to 18 spend on homework per week. 16. Inali spends at least an hour more than the average Average Hours Group time spent by boys on homework each week. Write per week an inequality for Inali’s homework time. Male 5.4 17. Anna usually spends no more than the average Female 6.8 time spent by girls on homework each week. Write Source: Horatio Alger Association an inequality to represent Anna’s homework time. For the given value, state whether each inequality is true or false. 18. 18 − x > 4, x = 12

19. 14 + n < 23, n = 8

20. 5k > 35, k = 7

14 < 7, c = 2 21. _ c

y 22. _ ≥ 2, y = 9 3

23. 16 ≤ 3d, d = 8

Graph each inequality on a number line. 24. a > 4

25. x > 6

26. d ≤ 5

27. w ≤ 8

28. n < 11

29. x < 5

30. t ≥ 9

31. b ≥ 8

32. x > -4

33. n ≥ -3

34. x ≤ -5

35. x < -2

Write the inequality for each graph. 36.

37. -10 -9 -8 -7 -6 -5 -4 -3 -2

38.

PRACTICE


0

1

39. -2 -1

EXTRA

-7 -6 -5 -4 -3 -2 -1

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

40. SPORTS There are more than 32,150 high school girls basketball and track programs in the United States. If there are 15,089 girls track programs, write and solve an inequality to determine the number of girls basketball programs. 41. RESEARCH Use the Internet or another source to find the state or national spending limits on certain government branches, organizations, or projects. Write an inequality to express one or more of these limits. Lesson 8-3 Inequalities

433

H.O.T. Problems

42. NUMBER SENSE Provide a counterexample to the statement, “All numbers less than 0 are negative integers.” 43. OPEN ENDED Write four examples of inequalities, each using one of the symbols , ≤, and ≥. Tell the meaning of each inequality. 44. CHALLENGE Graph the solutions for each compound inequality. a. y < -2 or y > 3 (Hint: In a sentence, or means either part is true.) b. y ≥ 0 and y ≤ 5 (Hint: In a sentence, and means both parts must be true.) 45.

Writing in Math How can inequalities help you describe relationships? Illustrate your answer with a real-world example that uses an inequality symbol and an explanation of the relationships described by the inequality.

46. Young adults are not allowed to vote in elections before their 18th birthday. Which graph represents the age of people who are allowed to vote? A £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ

B £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ

C £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ

47. Which inequality represents the graph below? £ä n Ç È x {

F x ≥ -8 G x ≤ -8 H x > -8 J x < -8

D £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ

ALGEBRA Solve each equation. Check your solution. (Lesson 8-2) 48. 2(3 + x) = 14

49. 63 = 9(2y – 3)

50. 3(n – 1) = 1.5(n + 2)

51. ALGEBRA Four times a number minus 6 is equal to the sum of 3 times the number and 2. Define a variable and write an equation to find the number. (Lesson 8-1) Find each quotient. Write in simplest form. (Lesson 5-4) 3 1 1 7 1 2 52. _ ÷_ 53. _ ÷ _ 54. _ ÷_ 3 3 5 2 2 4 7 1 _ 55. MOWING Keri has gallon of gasoline left. Her mower uses _ gallon to cut 8 6 an average yard. How many average yards can she mow? (Lesson 5-4)

PREREQUISITE SKILL Solve each equation. (Lesson 3-3) 56. x + 19 = 32

57. a + 7 = -3

58. 26 + c = 19

59. 44 – c = 26

60. y – 9.7 = 10.1

61. r – 1.6 = -0.6


8-4

Solving Inequalities by Adding or Subtracting

Main Idea • Solve inequalities by using the Addition and Subtraction Properties of Inequality.

The paper bag on the balance may contain some blocks. The scale models an inequality because the two sides are not equal. The side with the bag and 2 blocks weighs less than the side with 5 blocks. So, the inequality is x + 2 < 5.

x+2 b, then a + c > b + c and a - c > b - c. 2. if a < b, then a + c < b + c and a - c < b - c.

Inequalities When you add or subtract any number from each side of an inequality, the inequality symbol remains the same.

Examples

23

2+33-4

5 -1

These properties are also true for a ≥ b and a ≤ b.

EXAMPLE

Solve an Inequality Using Subtraction

Solve x + 3 > 10. Check your solution. x + 3 > 10

Write the inequality.

x + 3 - 3 > 10 - 3 Subtract 3 from each side. x>7

Simplify.

(continued on the next page)

Lesson 8-4 Solving Inequalities by Adding or Subtracting

435

To check your solution, try any number greater than 7. CHECK x + 3 > 10

8 + 3 10

Checking Solutions Try a number less than 7 to show that it is not a solution.

Write the inequality. Replace x with 8.

11 > 10 This statement is true. Any number greater than 7 will make the statement true. Therefore, the solution is x > 7.

1. Solve z + 4 > 3. Check your solution.

EXAMPLE

Solve an Inequality Using Addition

Solve -6 ≥ n - 5. Check your solution. -6 ≥ n - 5


-6 + 5 ≥ n - 5 + 5 Add 5 to each side. -1 ≥ n

Simplify.

The solution is -1 ≥ n or n ≤ -1.

2. Solve -3 ≥ g - 7. Check your solution.

EXAMPLE

Graph Solutions of Inequalities

1 Solve a + _ < 2. Graph the solution on a number line. 2

1 a+_ 4 9. x + 3.75 ≤ 5

Example 4 (p. 437)

10. 7 > z + 2 3

11. SAVINGS Chris is saving money to buy a stereo. He has $62.50, but his goal is to save at least $100. What is the least amount Chris still needs to save to reach his goal?

Extra Examples at pre-alg.com Courtesy Ohio Expo Center

8. x - 6 ≤ 4


437

HOMEWORK

HELP

For See Exercises Examples 12–23 1, 2 24–32 3 33, 34 4

Solve each inequality. Check your answer. 12. p + 7 < 9

13. t + 6 > -3

14. -13 ≥ 9 + b

15. 16 > -11 + k

16. 3 ≥ -2 + y

17. 25 < n + (-12)

18. r - 5 ≤ 2

19. a - 6 < 13

20. j - 8 ≤ -12

21. -8 > h - 1

22. 22 > w - (-16)

23. -30 ≤ d + (-5)

Solve each inequality. Then graph the solution on a number line. 24. n + 4 < 9

25. t + 7 > 12

26. p + (-5) > -3

27. -3 + z > 2

28. -13 ≥ x - 8

29. -32 ≥ a + (-5)

1 30. 3 ≤ _ +a

2 31. 4 ≥ s - _

3 32. -_ <w-1

2

3

4

33. TRANSPORTATION A certain minivan has a maximum carrying capacity of 1100 pounds. If the luggage weighs 120 pounds, what is the maximum weight allowable for passengers? 34. MARINE BIOLOGY Manatees can weigh up to 1000 pounds and are generally no more than 10 feet long. Suppose a manatee is currently 6.25 feet long. Write and solve an inequality to find how much longer the manatee could grow. HURRICANES For Exercises 35–37, use the diagram below. Types of Storms Tropical Storm

Depression 39

Hurricane 74

Wind Speed of Storm (mph)

35. A hurricane has winds that are at least 74 miles per hour. Suppose a tropical storm has winds that are 42 miles per hour. Write and solve an inequality to find how much the winds must increase so the storm is a hurricane. 36. Tropical storm Alpha has winds of 50 miles per hour. Write and solve an inequality to find how much the winds need to decrease so that the storm is downgraded to a depression. 37. A major storm has wind speeds that are at least 110 miles per hour. Write and solve an inequality that describes how much greater these wind speeds are than a hurricane with the slowest winds. Solve each inequality. Check your solution. EXTRA

PRACTIICE

38. 1 + y ≤ 2.4

39. 2.9 < c + 7

40. f - 4 ≥ 1.4

41. z - 2 > -3.8

3 1 < 2_ 42. b - _ 2 4

2 1 43. g - 1_ > 2_


3

6

44. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would need to solve an inequality using addition or subtraction.


H.O.T. Problems

45. OPEN ENDED Write an inequality for the solution graphed below. 16

18

20

22

24

46. FIND THE ERROR Dylan and Jada are using the statement x minus three is greater than or equal to 15 to find values of x. Who is correct? Explain. Jada x - 3 = 15 x - 3 + 3 = 15 + 3 x = 18

Dylan x - 3 ≥ 15 x - 3 + 3 ≥ 15 + 3 x ≥ 18

47. CHALLENGE Is it always, sometimes, or never true that x - 1 < x? Explain. 48.

Writing in Math

How is solving an inequality similar to solving an

equation?

49. Trevor has $25 to spend on a T-shirt and shorts for gym class. The shorts cost $14. Based on the inequality 14 + t ≤ 25, where t represents the cost of the T-shirt, what is the most Trevor can spend on the T-shirt?

50. The length of the rectangle is greater than its width. Which inequality represents the possible values of x? X x® V £Ó V

A $9 B $10.99 C $11

F x ≤ 17

H x ≥ 17

G x < 17

J x > 17

D $11.50

ALGEBRA For the given value, state whether each inequality is true or false. (Lesson 8-3) 51. x - 5 > 4, x = 9

52. 9 + a ≤ 3, a = -7

d 53. _ ≥ 8, d = 4 2

54. GEOMETRY The perimeter of a rectangle is 24 centimeters. Find the dimensions if the length is 3 more than twice the width. (Lesson 8-2) ALGEBRA Use the Distributive Property to rewrite each expression as an equivalent algebraic expression. (Lesson 3-1) 55. 4(2 + 8) 56. -2(n + 6) 57. 5(x - 3.5) 58. (9 - d)(-3c)

PREREQUISITE SKILL Solve each equation. (Lesson 3-4) 59. -7x = 14

60. -3y = -27

d 61. _ = -6 -3

c 62. _ = 12 -4


439

CH

APTER

8



Graph each inequality on a number line.

(Lesson 8-1)

(Lesson 8-3)

1. 6y + 42 = 4y

12. x < -3

13. y ≥ 5

2. 12x - 19 = 3x + 8

4 14. _ >d 5

15. f < 11.2

3. 7m - 12 = 2.5m + 2 Define a variable and write an equation to find each number. Then solve. (Lesson 8-1) 4. Twice a number is 150 less than 5 times the number. What is the number? 1 5. One fourth of a number plus 3 is _ that 2 number minus 1. What is the number? 6. TESTS Bobby’s score is 5 less than twice Allan’s score. It is also 45 points greater than Allan’s score. What score did the two boys receive? (Lesson 8-1) 7. MULTIPLE CHOICE An online computer game community has two membership plans. The first plan gives you unlimited play time for $40 a month. The second plan charges a monthly access fee of $4.25 plus $2.75 for each hour you play. After how many hours do the two plans cost the same amount? (Lesson 8-1) A 6.6

B 9.0

C 11.2

D 13.0

Solve each equation. Check your solution. (Lesson 8-2)

8. 8(p - 4) = 2(2p + 1) 9. 0.2x - 1.4 = 15.82 - 0.5x 10. b + 2(b + 5) = 3(b - 1) + 13 11. MULTIPLE CHOICE Which of the following 17 ≤ y? graphs represents the inequality -_ (Lesson 8-3)

5

F x {Î Ó £ ä £ Ó Î { x

G £ä ££ £Ó £Î £{ £x £È £Ç £n £ Óä

18. FITNESS The table shows a gym class’s average results for boys and girls participating in the long jump. Gender

Male Female

x {Î Ó £ ä £ Ó Î { x x {Î Ó £ ä £ Ó Î { x


Distance

17 feet 5 inches 14 feet 3 inches

Cheyenne could jump no farther than 12 inches more than the average distance for males. Write an inequality that gives the possible distances that Cheyenne could jump. (Lesson 8-3) Solve each inequality. Check your solution. (Lesson 8-4) 19. d + 10 ≥ 12 20. c - (-5) < 24 21. 5 < g - 21 22. -32 ≤ 17 + j 23. k - 3 > 7 24. 7 ≤ m + 1 25. MULTIPLE CHOICE Shanté has $50 to spend on a back-to-school outfit. The blouse she wants is $17. Based on the inequality 17 + s ≤ 50 where s is the cost of a skirt, what is the most that Shanté can spend on a skirt? (Lesson 8-4)

A $17 B $33

H J

Write an inequality for each sentence. (Lesson 8-3) 16. More than 35,000 people attended a concert in Toronto. 17. Toby wants to spend no more than 3 hours working on his model car.

C $43 D $50

8-5

Solving Inequalities by Multiplying or Dividing

Main Ideas • Solve inequalities by multiplying or dividing by a positive number. • Solve inequalities by multiplying or dividing by a negative number.

An astronaut in a space suit weighs about 300 pounds on Earth, but only 50 pounds on the Moon because of weaker gravity. weight on Moon

weight on Earth

300

50

>

Weight of Astronaut (lb)

Location Earth

300

Moon

50

Pluto

67

Mars

113

Neptune

407

Jupiter

796

If the astronaut and space suit each weighed half as much, would the inequality still be true? That is, would the astronaut’s weight still be greater on Earth? a. Divide each side of the inequality 300 > 50 by 2. Is the inequality still true? Explain by using an inequality. b. Would the weight of 5 astronauts be greater on Pluto or on Earth? Explain by using an inequality.

Multiply or Divide by a Positive Number The application above demonstrates how you can solve inequalities by using the Multiplication and Division Properties of Inequalities.

Multiplication and Division Properties

Positive Number

Words

When you multiply or divide each side of an inequality by the same positive number, the inequality remains true.

Symbols

For all numbers a, b, and c, where c > 0,

_ 1. if a > b, then ac > bc and _ c > c. a

The inequality c > 0 means that c is a positive number.

b

_ 2. if a < b, then ac < bc and _ c < c. a

Examples

2 -9

4(2) < 4(6)

3 -9 _ >_

8 < 24

1 > -3

3

b

3

These properties are also true for a ≥ b and a ≤ b. Lesson 8-5 Solving Inequalities by Multiplying or Dividing

441

EXAMPLE

Multiply or Divide by a Positive Number

Solve each inequality. Check your solution. a. 8x ≤ 40 8x ≤ 40


8x _ _ ≤ 40


8

8

x≤5

Simplify.

The solution is x ≤ 5. You can check this solution by substituting 5 or a number less than 5 into the inequality. d b. _ >7 2

_d > 7

2 d 2_ > 2(7) 2

d > 14

Write the inequality. Multiply each side by 2. Simplify.

The solution is d > 14. You can check this solution by substituting a number greater than 14 into the inequality. f 1B. _ < -5

1A. 3x > -15

4

Ling earns $8 per hour. Which inequality can be used to find how many hours he must work in a week to earn at least $120? A 8x < 120

B 8x ≤ 120

C 8x > 120

D 8x ≥ 120

Read the Test Item Key Words Before taking a standardized test, review the meanings of phrases like at least and at most.

You are to write an inequality to represent a real-world problem. Solve the Test Item Words Variable

Amount earned per hour

times number of hours is at least the amount earned each week.

Let x represent the number of hours worked.

Inequality 8

·

x

120

≥

The answer is D.

3 2. It takes Alfonzo _ hour to mow a lawn. Which inequality can be used to 4 find the number of lawns he can mow if he works 15 hours per week? 3 3 3 3 F _ x ≤ 15 G _ x ≥ 15 H _ x > 15 J _ x < 45 4

4



4

4

Multiply or Divide by a Negative Number What happens when each side of an inequality is multiplied or divided by a negative number? Multiply each side by -1.

Graph 3 and 4 on a number line.

x {Î Ó £ ä £ Ó Î { x

x {Î Ó £ ä £ Ó Î { x

Since 3 is to the left of 4, 3 < 4.

Since -3 is to the right of -4, -3 > -4.

Notice that the numbers being compared switched positions as a result of being multiplied by a negative number. In other words, their order reversed. These and other examples suggest the following properties.

Multiplication and Division Properties Negative Number The inequality c < 0 means that c is a negative number.

Words

When you multiply or divide each side of an inequality by the same negative number, the inequality symbol must be reversed for the inequality to remain true.

Symbols

For all numbers a, b, and c, where c < 0,

_ 1. if a > b, then ac < bc and _ c < c. a

b

_ 2. if a < b, then ac > bc and _ c > c. a

7>1

Examples

b

-4 < 16

16 -4 -2(7) < -2(1) Reverse the symbols. _ > _ -4

-4

1 > -4

-14 < -2

These properties are also true for a ≥ b and a ≤ b.

EXAMPLE

Multiply or Divide by a Negative Number

Solve each inequality and check your solution. Then graph the solution on a number line. x a. _ ≤4

b. -7x > -56

-3

x _ ≤4

-3 x -3 _ ≥ -3(4) -3

x ≥ -12

Write the inequality. Multiply each side by -3 and reverse the symbol. Check this result.

⫺14

y 4

3A. - _ < 3


⫺12

⫺10

⫺8

-56 -7x _ 63 2

5 3 5. _ ≤_ y 7 4

3 1 6. _y ≤ _

(p. 443)

B 13

C 14

HELP

For See Exercises Examples 11–18 1 19, 20, 2 42, 44 21–28 3

D 20

Solve each inequality. Check your solution. Then graph the solution on a number line. 8. -4t > -20

HOMEWORK

4

7. MULTIPLE CHOICE Koto delivers pizzas on weekends. Her average tip is $1.50 for each pizza that she delivers. How many pizzas must she deliver to earn at least $20 in tips? A 10

Example 3

24

9. -8z ≤ -24

2 g 10. 18 > -_ 3

Solve each inequality. Check your solution. 11. 13a ≥ -26

12. -15 ≤ 5b

13. 144 < 12d

14. 15 ≥ 3t

p 15. _ > 5 6

h 16. 7 ≥ _ 14

17. 3m < 33

18. 8z ≤ -24

19. SOCCER Tomás wants to spend less than $100 for a new soccer ball and shoes. The ball costs $24. Write and solve an inequality that gives the amount that Tomás can spend on shoes. 20. ARCADE Montel spends $0.75 every time he plays his favorite video game. Montel has $10. Write and solve an inequality that shows how many times Montel can play the video game. Solve each inequality. Check your solution. Then graph the solution on a number line. 21. -8 ≤ -4w

22. -6a > -78

23. -25t ≤ 400

24. 18 > -2g

y 25. -_ ≥ 2.4 4

n 26. _ ≥ -0.8 -5

x 27. 6 > _ -7

r 28. _ < -2 -2

29. SWIMMING Andrea swims 40 meters per minute, and she wants to swim at least 2000 meters this morning. Write and solve an inequality to find how long she should swim. EXTRA

PRACTIICE


H.O.T. Problems

Solve each inequality. Check your solution. Then graph the solution on a number line. y -0.3

c 30. -5 ≥ -_

31. -19 > _

1 32. -_ x ≥ -9

1 33. -36 < -_ b

y 34. _ < -7 -3

k 35. _ _ -0.4

m 37. _ ≤ 1.2 -7

4.5

3

2

38. OPEN ENDED Write an inequality that can be solved using the Division Property of Inequality, where the inequality symbol is not reversed. 39. CHALLENGE The product of an integer and -7 is less than -84. Find the least integer that meets this condition.

444 Chapter 8 Equation and Inequalities

40. FIND THE ERROR Brittany and Tamika each solved -45 ≥ 9k. Who is correct? Explain your reasoning. Brittany -45 ≥ 9k

Tamika -45 ≥ 9k

-45 9k 9 ≤ 9

-45 ≥ 9k 9 9

-5 ≥ k

-5 ≤ k

41.

Writing in Math

Use the information on page 441 to explain how inequalities can be used in studying space. Illustrate your answer with inequalities that compare the weight of two astronauts on Mars and on the Moon.

42. The solutions for which inequality are represented by the following graph?

44. Which number is NOT a possible length of the rectangle if the area is less than 36 square inches?

Óä £ £n £Ç £È £x £{

x A _ ≤5

-3 x ≥5 B _ -3

x in.

x C _ -5 3

4 in.

43. GRIDDABLE Isabel is putting water into a 20-gallon fish tank using a 2-quart pitcher. How many pitchers of water will she need to fill the tank?

F 6

H 8

G 7

J

9

ALGEBRA Solve each inequality. Check your solution. (Lessons 8-4) 45. -4 + x > 23

46. c + 18 ≤ -2

47. 6 > n - 10

48. CRAFTS It takes Carolyn two hours to complete a cross-stitch pattern. Carolyn can spend no more than fourteen hours cross-stitching. Write an inequality that represents this situation and use it to determine whether Carolyn can complete 8 cross-stitch patterns. (Lesson 8-3) Find each product. Write in simplest form. (Lesson 5-3) 1 3 49. _ · _ 8

4

7

ab 4 52. _ · _

5 1 51. 2_ · -_

3 _ 50. -_ ·5 9

2

2

6

bc

PREREQUISITE SKILL Solve each equation. (Lesson 3-5) 53. 2x + 3 = 9

54. 5a - 6 = 14

55. 3n - 8 = -26

56. _t + 5 = 2 3

57. _c - 1 = 4

d 58. _ + 3 = 19

4

2

Lesson 8-5 Solving Inequalities by Multiplying or Dividing

445

8-6

Solving Multi-Step Inequalities BrainPOP

Main Idea • Solve inequalities that involve more than one operation.

pre-alg.com

More than 10 million Americans are “frequent runners.“ A rule of thumb for training is that you will generally have enough endurance to finish a race that is up to 3 times your average daily distance. a. Write an inequality that represents the relationship between daily average distance and possible race lengths. b. Your average daily run is 2 kilometers. Write and solve an inequality that represents the amount that you need to increase your daily run by to have enough endurance for a 12-kilometer race.

Inequalities with More than One Operation An inequality may involve more than one operation. To solve the inequality, work backward to undo the operations, just as you did in solving multi-step equations.

EXAMPLE

Solve a Two-Step Inequality

Solve 6x + 15 > 9 and check your solution. Graph the solution on a number line.

Common Misconception Do not reverse the inequality sign just because there is a negative sign in the inequality. Only reverse the sign when you multiply or divide by a negative number.

6x + 15 > 9 6x + 15 - 15 > 9 - 15 6x > -6 x > -1 CHECK

Write the inequality. Subtract 15 from each side. Simplify. Mentally divide each side by 6.

6x + 15 > 9 Write the inequality. 6(0) + 15 > 9 Replace x with a number greater than -1. Try 0. 0 + 15 > 9 Simplify. 15 > 9 The solution checks.

Graph the solution, x > -1. 5

4

3

2

1

0

1

2

3

4

5

1. Solve 8y -2 ≤ 14 and check your solution. Graph the solution on a number line. 446 Chapter 8 Equations and Inequalities Bob Thomas/Getty Images

EXAMPLE

Reverse the Inequality Symbol

Solve 10 - 3a ≤ 25 + 2a and check your solution. Graph the solution on a number line. 10 - 3a ≤ 25 + 2a


10 - 3a - 2a ≤ 25 + 2a - 2a Subtract 2a from each side. 10 - 5a ≤ 25

Simplify.

10 - 10 - 5a ≤ 25 - 10 Inequalities Remember that you must reverse the inequality symbol if you multiply or divide each side of an inequality by a negative number.


-5a ≤ 15

Simplify.

15 -5a _ ≥_

Divide each side by -5 and change ≤ to ≥.

-5

-5

a ≥ -3

Simplify.

Check your solution by substituting a number greater than -3. Graph the solution, a ≥ -3. 5 4 3 2 1 0

1

2

3

4 5

2. Solve 5b + 8 ≥ 7b + 2 and check your solution. Graph the solution on a number line.

When inequalities contain grouping symbols, you can use the Distributive Property to begin simplifying the inequality.

RUNNING Refer to the application at the beginning of the lesson. Tammy wants to be able to run at least the standard marathon distance of 26.2 miles. If the length of her current daily runs is about 4 miles, how many miles should she increase her daily run by to meet her goal? Words Variable Inequality

Real-World Link One of the most popular races in America is the Chicago marathon. In 2004, about 33,000 people completed the 26.2-mile race.

3

times

4 miles

plus

amount of increase

is greater than or equal to

desired distance.

Let d = the amount of increase. 3

·

(4

d)

+

3(4 + d) ≥ 26.2


12 + 3d ≥ 26.2

Multiply.

≥

3d ≥ 14.2


14.2 3d ≥ _ 3 3


26.2

d ≥ 4.7 3

Source: chicagomarathon.com

In order to have enough endurance to run a marathon, Tammy should increase the distance of her average daily run by at least 4.73 miles. Extra Examples at pre-alg.com Matthew Stockman/Getty Images

Lesson 8-6 Solving Multi-Step Inequalities

447

3. BUSINESS Banks estimate the value of a business to determine loans and insurance. The formula for the value of a coffee shop is 40% of its annual sales plus the value of its inventory. The value of Holmes Coffee is at least $150,000. Write and solve an inequality to find the annual sales at Holmes Coffee if its inventory is $26,000. Personal Tutor at pre-alg.com

Solve each inequality and check your solution. Then graph the solution on a number line. Example 1 (p. 446)

Example 2 (p. 447)

Example 3 (p. 447)

HOMEWORK

HELP


1. 3x + 4 ≤ 31

2. 12a - 4 > 20

3. 2n + 5 > 11 – n

4. y + 1 ≥ 4y + 4

5. 16 - 2c < 14

6. 18 ≤ 12 - 2n

7. -3(b - 1) > 18

8. -2(k + 1) ≥ 16

9. MONEY A company pays Dante’s Web site for advertising on the site. The Web site earns $10 per month plus $0.05 each time a visitor to the site clicks on the advertisement. What is the least number of clicks he needs to make $45 per month or more from this advertiser?

Solve each inequality and check your solution. Then graph the solution on a number line. 10. 2x + 8 > 24

11. 6q + 4 ≤ 28

12. 3y - 1 ≤ 5

13. 9t - 5 ≤ -14

14. 3 + 4c > -13

15. 9 + 2p ≤ 15

16. 4 - 3k ≤ 19

17. 16 - 4n > 20

18. -3b + 4 < -2

19. -5a - 8 > 12

20. 2(n + 3) < -4

21. 2(d + 1) > 16

For Exercises 22 and 23, write and solve an inequality. 22. SALES You earn $2 for every magazine subscription you sell plus a salary of $10 each week. How many subscriptions do you need to sell each week to earn at least $40 each week? 23. HIKING You hike along the Appalachian Trail at 3 miles per hour. You stop for one hour for lunch. You want to walk at least 18 miles. How many hours should you expect to spend on the trail? Solve each inequality and check your solution. Then graph the solution on a number line. 24. 3x - 2 > 10 - x

25. c - 1 < 3c + 5

26. 2 + 0.3y ≥ 11

27. 0.5a - 1.4 ≤ 2.1

1 (6 - c) > 5 28. _ 2

m 29. _ +9≥5 2

30. Four times a number less 6 is greater than two times the same number plus 8. For what number or numbers is this true? 448 Chapter 8 Equations and Inequalities

31. One half of the sum of a number and 6 is less than 25. For what numbers is this true? 32. REAL ESTATE A new real estate agent receives a monthly salary of $1500 plus a 3.5% commission on every home sold. For what amount of monthly sales will the agent earn at least $5000? 33. REPAIRS Carl is having a mechanic fix his car. The mechanic said that the job was going to cost at least $375 for parts and labor. If the cost of the parts was $150, and the mechanic charges $60 an hour, how many hours is the mechanic planning on working on the car?

Real-World Career Real Estate Agent Real estate agents help people buying and selling a home. All states require prospective agents to pass a written test, which usually contains a section on mathematics.


EXTRA

PRACTICE


H.O.T. Problems

34. SCHOOL Nate has scores of 85, 91, 89, and 93 on four tests. What is the least number of points he can get on the fifth test to have an average of at least 90? 35. FUND-RAISERS The booster club at Jefferson High School sells football programs for $1 each. The costs to make the programs are $60 for page layout plus $0.20 for printing each program. If they print 400 programs, how many programs must the Club sell to make at least $200 profit? 36. CAR RENTAL The costs for renting a car from Able Car Rental and from Baker Car Rental are shown in the table. For what mileage does Baker have the better deal? Use the inequality 30 + 0.05x > 20 + 0.10x. Explain why this inequality works.

Rental Car Costs Cost per Day

Cost per Mile

Able

$30

$0.05

Baker

$20

$0.10

37. CELL PHONE SERVICES While reviewing prepay phone plans, Miko found that FoneCom charges a $5.35 monthly fee plus $0.10 per minute. Miko currently has BestPhone service at $10 per month plus $0.05 per minute. Miko figures that her monthly bill would be more with FoneCom. For how many minutes per month does she use the phone? 38. TRAVEL Tim is taking the train to Seattle to visit his grandparents. He was given $5.00 to spend on snacks and reading material. Granola bars cost $0.75 each and a newspaper is $1.25. If Tim buys a newspaper, how many granola bars can he get? 39. OPEN ENDED Write a multi-step inequality that can be solved by first adding 3 to each side. 40. CHALLENGE Assume that k is an integer. Solve the inequality 10 - 2|k| > 4. 41. FIND THE ERROR Jerome and Ryan are solving 2(2y + 3) > y + 1. Who is correct? Explain your reasoning. Jerome 2(2y + 3) > y + 1 4y + 6 > y + 1

42.

Ryan 2(2y + 3) > y + 1 4y + 3 > y + 1

Writing in Math Use the information about running found on page 446 to explain how multi-step inequalities are used in running. Lesson 8-6 Solving Multi-Step Inequalities

Doug Martin

449

43. Sandra’s scores on the first five science tests are shown in the table. Which inequality represents the score she must receive on the sixth test to have an average score of more than 88? A s ≥ 86

44. An art teacher wants to buy at least 2 canvases for each student in her painting class. If there are 30 students in the class and if canvases cost $16 per package, what other information is needed to find the amount the teacher should budget for canvases?

Test

Score

B s ≤ 88

1

85

C s < 88

2

84

F The number of art projects planned for the course

D s > 86

3

90

G The cost of paints and brushes

4

95

5

88

H The number of canvases in each package J The total budget available for art supplies

ALGEBRA Solve each inequality. Check your solution. (Lessons 8-4 and 8-5) q 3

45. 6x < -27

46. -5m ≥ -15

47. 8 > _

n 48. _ ≤ -11

49. -9 + k > 20

50. 22 ≤ -15 + y

51. 12 + z ≤ 8

52. 14 ≥ 7 + a

-4

53. SCHOOL If 12 of the 20 students in a class are boys, what percent are boys? (Lesson 6-5) 1 54. Write _ as a percent. (Lesson 6-5) 200

Express each ratio as a unit rate. (Lesson 6-1) 55. $5 for 2 loaves of bread 57. 24 meters in 4 seconds

56. 200 miles on 12 gallons 58. 9 monthly issues for $11.25

GEOMETRY Find the missing dimension in each rectangle. (Lesson 3-8) 59.

18.4 ft

ᐉ

60.

w 5.1 m

Area = 30.6 m2

Perimeter = 49.6 ft

Math and Recreation Just for Fun It is time to complete your project. Use the information and data you have gathered about

recreational activities to prepare a Web page or poster. Be sure to include a scatter plot and a prediction for each activity. Cross-Curricular Project at pre-alg.com


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Key Vocabulary


Key Concepts Solving Equations


• Use the Addition or Subtraction Property of Equality to isolate the variables on one side of an equation.

identity (p. 426) inequality (p. 430) null or empty set (p. 426)

Vocabulary Check Determine whether each statement is true or false. If false, replace the underlined word or phrase to make a true statement. 1. When an equation has no solution, the solution set is the null set.

• Use the Distributive Property to remove the grouping symbols.

2. The inequality n + 8 - 8 ≥ 14 - 8 demonstrates the Subtraction Property of Inequality.

Solving Inequalities

3. An equation that is true for every value of the variable is called an inequality.

(Lessons 8-3 to 8-6)

• An inequality is a mathematical sentence that contains , ≤, or ≥.

4. The inequality x (4) < 7(4) demonstrates 4 the Division Property of Inequality.

• Solving an inequality means finding values for the variable that make the inequality true.

5. A mathematical sentence that contains , ≤, or ≥ is called an empty set.

• When you multiply or divide each side of an inequality by a positive number, the inequality symbol remains the same.

6. When the final result in solving an equation is 5 = -8, the solution set is the null set.

• When you multiply or divide each side of an inequality by a negative number, the inequality symbol must be reversed.

7. The symbol ≥ means is less than or equal to.

• To solve an inequality that involves more than one operation, work backward to undo the operations.

Vocabulary Review at pre.alg.com


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Solving Equations with Variables on Each Side

(pp. 420–423)


Example 1 Solve 7x = 3x - 12.

8. 2a + 9 = 5a

7x = 3x - 12 Write the equation. 7x - 3x = 3x - 3x - 12 Subtract 3x from each

9. x - 4 = 3x

10. 3y - 8 = y

11. 19t = 26 + 6t

12. 12 + 1.5x = 9x

13. 5b - 1 = 2.5b - 4

side.

4x = -12

Simplify.

4x -12 _ =_


4

14. CONCERTS An outdoor concert venue is planning on increasing the number of 1 for next year. This will concerts by 14 increase their number of concerts by 3. How many concerts will they host this year?

4

x = -3

Simplify.

15. An online DVD rental club has two membership plans as shown. In how many months would the total cost of the two plans be the same?

A

Membership Fee $20

Cost Per Month $5

B

$30

$3

Plan

8–2

Solving Equations with Grouping Symbols

(pp. 424–428)

Solve each equation. Check your solution. 16. 4(k + 1) = 16 17. 2(n - 5) = 8 18. 11 + 2q = 2(q + 4) 3 1 19. _ (t + 8) = _ t 2

4

20. 4(x + 2.5) = 3(7 + x) 21. 3(x + 1) - 5 = 3x - 2 22. GEOMETRY The perimeter of a rectangle is 84 meters. Find the dimensions of the rectangle if the length is 3 meters less than twice the width.


Example 2 Solve 2(x + 3) = 15. 2(x + 3) = 15 2x + 6 = 15 2x + 6 - 6 = 15 - 6 2x = 9 9 2x _ =_ 2 2

x = 4.5

Write the equation. Use the Distributive Property. Subtract 6 from each side. Simplify. Divide each side by 2. Simplify.



8–3

Inequalities

(pp. 430–434)

For the given value, state whether each inequality is true or false.

Example 3 State whether n + 11 < 14 is true or false for n = 5.

23. x + 4 > 9, x = 12 25. 6r > 30, r = 5

n + 11 < 14 Write the inequality. 5 + 11 < 14 Replace n with 5. 16 ≮ 14 Simplify.

26. 15 ≤ 5n, n = 8

The sentence is false.

24. 12 - t < 5, t = 3

27. 3n + 1 ≥ 14, n = 7 28. 23 ≤ _c + 2, c = 10 4

29. CAMPING When camping, Stephán and his friends usually use at least 3 logs for fire each night. Write an inequality that represents this situation. 30. DIVING In a diving competition, the diver in first place has a total score of 345.4. Ming has scored 68.2, 68.9, 67.5, and 71.7 for her first four dives and has one more dive remaining. Write an inequality to show the score x that Ming must receive on her fifth dive in order to overtake the diver in first place.

8–4

Solving Inequalities by Adding or Subtracting

(pp. 435–439)

Solve each inequality. Then graph the solution on a number line.

Example 4 Solve x - 7 ≤ 3. Then graph the solution on a number line.

31. b - 9 ≥ 8

32. 15 > 3 + n

33. x + 4.8 ≤ 2

34. r + 5.7 ≤ 6.1

1 35. t + _ 19 47. 5n + 4 ≤ 24 48. 6 ≥ _r + 1 7

t 49. _ + 15 < 21 -2

50. 3(a + 8.4) > 30 1 51. _ + 2b < 13 + 5b 4

52. SALES A car sales associate receives a monthly salary of $1700 a month plus 8% commission on every car sold. For what amount of monthly sales will the sales associate earn at least $4200?


Example 6 Solve 4t + 7 < -5. Write the inequality. 4t + 7 < -5 4t + 7 - 7 < -5 - 7 Subtract 7 from each side. 4t < -12 Simplify. t < -3 Mentally divide each side by 4.

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Practice Test


Write an inequality for each graph.

1. 7x - 3 = 10x 2. p - 9 = 4p

14.

3. 2(6 - 5d) = -8

15.

4. 4(a + 3) = 20

16.

⫺2 ⫺6

5. 2.3n - 8 = 1.2n + 3

0 ⫺4

2 ⫺2

4

0

6

2

4

£ ä £ Ó Î { x È Ç n

3 5 6. _ y-5=_ y-3 8 8

7. 6 + 2(x - 4) = 2(x - 1) 1 8. _ (9b + 1) = b - 1 3

9. MULTIPLE CHOICE For a project, a class is divided into two groups and each group has to make a video. Group A’s video is 20 seconds less than twice the length of Group B’s video. Group A’s video is also 255 seconds longer than Group B’s video. Which equation represents this information? A 2a + 20 = b + 255

17. SALES The Cookie Factory has a fixed cost of $300 per month plus $0.45 for each cookie sold. Each cookie sells for $0.95. How many cookies must be sold during one month for the profit to be at least $100? 18. MULTIPLE CHOICE Danny earns $8.50 per hour working at a movie theater. Which inequality can be used to find how many hours he must work each week to earn at least $100 a week? F

B 2b - 20 = b + 255

8.50h < 100

H 8.50h ≤ 100

G 8.50h > 100

C 20b - 255 = b + 20

J

8.50h ≥ 100

D a + 255 = b - 20 For Exercises 10–12, define a variable and write an equation to find each number. Then solve. 10. Eight more than three times a number equals four less than the number. 11. The product of a number and five is twelve more than the number. 12. GEOMETRY The perimeter of the rectangle is 22 feet. Find the dimensions of the rectangle. w 2w + 3.5

13. SHOPPING The cost of purchasing four shirts is at least $120. Write an inequality to describe this situation.


Solve each inequality and check your solution. Then graph the solution on a number line. 19. -4 ≥ p - 2

20. 3x ≥ 15

21. -42 < -0.6x

22. c - 3 ≤ 4c + 9

23. 7(3 - 2b) ≥ 5b + 2

1 1 24. _ (a + 4) > _ (a - 8) 2

4

25. MULTIPLE CHOICE The Lapeer Nature Club wants to raise at least $4000 for conservation. They have been given a $150 dollar donation and are selling canvas bags for $55 each to raise the rest of the money. Which inequality describes how many bags they need to sell in order to reach this goal? A x ≥ 35 B x ≤ 35 C x ≤ 70 D x ≥ 70


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Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The cost, c, of renting a moving truck can be found using the equation c = 20.75 + 31.50d, where d is the number of days you rent the truck. What would be the total cost of renting a truck for 4 days? C $158.50 A $126.00 B $146.75 D $172.25

4. A sequence of numbers was generated using the rule 3n - 1, where n represents a number’s position in the sequence. Which sequence fits this rule? A 1, 3, 5, 7, 9, … B 1, 4, 7, 10, 13, … C 2, 4, 6, 8, 10, … D 2, 5, 8, 11, 14, … 3 4 5. Which fraction is between _ and _ ? 2 F _

4

5

3

2. A music store surveyed 100 of its customers about their preferred styles of music. The results of the survey are shown in the survey.

5 G _ 7 19 H _ 25

7 J _ 8

Favorite Style of Music Style Country

Frequency 25

Rock

38

Jazz

18

Classical

12

Other

7

If the store only uses these data to order new CDs, what conclusion can be drawn from the data? F More than half of each order should be country and rock CDs. G More than half of each order should be rock CDs. H Only country, rock, and jazz CDs should be ordered. J About a fourth of each order should be classical music CDs.

3. GRIDDABLE Find the next term in the pattern below. 1, 3, 7, 13, 21, 31, … 456 Chapter 8 Equations and Inequalities

6. GRIDDABLE Colleen is using a punch recipe that calls for 12 ounces of fruit juice for every 40 ounces of lemon-lime soda. If she uses 60 ounces of lemon-lime soda, how many ounces of fruit juice will she need? 7. Robert, Isabelle, Michael, and Katrina are going to a football game. The total cost of the tickets is $231.75. Robert paid $60, Isabelle 1 of paid 20% of the total cost, Michael paid _ 4 the total cost, and Katrina paid the rest. Who paid the greatest amount? A Robert B Isabelle C Michael D Katrina 8. A couch is on sale for 20% off the regular price of $480. How much money is discounted off the regular price? F $384 G $362 H $96 J $84 Standardized Test Practice at pre-alg.com


9. Kelly’s deck has an area of 660 square feet.

12. The cost, c, of renting a car can be found using the equation c = 50 + 0.10m, where m is the number of miles you drive the car. What would be the total cost of renting a car and driving it 200 miles? H $50 F $20 G $0.10 J $70

FT

What is the length of the deck if the width is 11 feet? A 66 ft B 60 ft C 50 ft D 45 ft

13. The cost, c, of renting a tent site can be found using the equation c = 12.50 + 2.50p, where p is the number of people you will have on the site. What would be the total cost of renting a tent site for 5 people? C $25.00 A $12.50 B $15.00 D $65.00

10. Mary Ann saved $56 when she purchased a television on clearance at an electronics store. If the sale price was 20% off the regular price, what was the regular price? F $250 G $260 H $275 J $280

If you get finished with the test before the end of the time allowed, go back and check your work.

Pre-AP Record your answers on a sheet of paper. Show your work. 14. Kevin earns a monthly salary of $1750. In addition to his salary, he receives a $250 bonus for every car that he sells. He wants to earn at least $3000 per month. a. Write an inequality to represent this situation. b. Solve the inequality that you found in part a. c. What is the minimum number of cars he must sell? d. How many cars will he have to sell if he wants to earn at least $4000 per month?

11. The cost, c, of hiring a plumber can be found using the equation c = 75 + 40h, where h is the number of hours the plumber worked. What would be the total cost of hiring a plumber for 3 hours? A $40 B $75 C $195 D $120


1

2

3

4

5

6

7

8

9

10

11

12

13

14

Go to Lesson...

8-1

6-10

1-1

3-7

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3-8

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457

Applying Algebra to Geometry Focus Analyze figures in two- and three-dimensional space. Find areas of two-dimensional figures and volumes and surface areas of three-dimensional figures.

CHAPTER 9 Real Numbers and Right Triangles Use different forms of numbers appropriate for different situations. Use indirect measurement to solve problems.

CHAPTER 10 Two-Dimensional Figures Use transformational geometry to develop spatial sense. Use geometry to model and describe the physical world.

CHAPTER 11 Three-Dimensional Figures Determine measures of three-dimensional figures. Describe how changes in dimensions affect area and volume measures. 458 Unit 4 Applying Algebra to Geometry Bill Ross/CORBIS

Algebra and Architecture Able to Leap Tall Buildings The tallest building in the United States is the Sears Tower in Chicago. It has a height of 1450 feet. Did you know that at 790 feet tall, the John Hancock Tower in Boston is the tallest building in Massachusetts? In this project, you will be exploring how geometry and algebra can help you describe unusual or large structures in the United States and from around the world. Log on to pre-alg.com to begin.

Unit 4 Applying Algebra to Geometry

459


9 •

Identify numbers in the real number system.

•

Use the Pythagorean Theorem, the Distance Formula, and the Midpoint Formula.

•

Classify angles and triangles and identify and use properties of similar figures.

•

Use trigonometric ratios to solve problems.

Key Vocabulary hypotenuse (p. 485) irrational numbers (p. 469) real numbers (p. 469) similar figures (p. 497)

Real-World Link Roller Coasters A rider at the top of the Titan roller coaster, 255 feet above the ground, can see for approximately 19.5 miles on a clear day.

Real Numbers and Right Triangles Make this Foldable to help you organize information about real numbers and right triangles. Begin with three plain sheets of 812“ × 11“ paper.

1 Fold to make a triangle. Cut off the extra paper.

3 Stack the three squares and staple along the fold.

460 Chapter 9 Real Numbers and Right Triangles Courtesy of Six Flags Over Texas

2 Repeat Step 1 twice. You now have three squares.

4 Label each section with a topic.

3IGHT S 5RIANGLE



Option 1


Replace each with , or = to make a true statement. (Prerequisite Skills, p. 742) 1. 3.2

2. 7.8

3.5

3. 5.13

5.16

4. 4.92

5. 2.62

2.6

6. 3.4

7. 0.07

0.7

8. 1.16

7.7 4.89 3.41 1.06

Example 1 Replace with , or = to make 29.19 29.2 a true statement.

29.19

Line up the decimal points.

29.2

The digits in the tenths place are not the same.

1 tenth < 2 tenths, so 29.19 < 29.2.

9. FIELD HOCKEY The following shows the winning percents of five field hockey teams. Order them from greatest to least. 0.523, 0.546, 0.601, 0.594, 0.509 (Prerequisite Skills, p. 742)

Solve each equation. (Lesson 3-4) 10. 3x = 24

11. 7y = 49

12. 120 = 2n

13. 54 = 6a

14. 90 = 10m 16. 15d = 165

Example 2 Solve 14w = 56.

14w = 56

Write the equation.

15. 144 = 12m

14w 56 _ =_


17. 182 = 14w

w=4

14

14

Simplify.

18. COOKIES A batch of cookies contains 8 tablespoons of sugar. How many batches of cookies contain 64 tablespoons of sugar?

19. (3 - 1)2 + (4 - 2)2

Example 3 Evaluate (6 - 2)2 + (9 - 7)2.

20. (5 - 2)2 + (6 - 3)2

(6 - 2)2 + (9 - 7)2 = 42 + 22

Evaluate each expression. (Lesson 4-1)

21. (4 - 7)2 + (3 - 8)2 22. (8 - 2)2 + (3 - 9)2

Simplify the expressions inside parentheses first.

23. (2 - 6)2 + [(-8) - 1]2

= 16 + 4

Evaluate 42 and 22.

24. (-7 - 2)2 + [3 - (-4)]2

= 20

Simplify.

25. BIOLOGY Suppose a virus splits into two viruses every 45 minutes. How many viruses are there after 5 hours 15 minutes? (Lesson 4-1)


461

EXPLORE

9-1

Algebra Lab

Squares and Square Roots Numbers raised to the second power are called squares. You can use a geometric model to discover the reason for the term.

ACTIVITY 1 Use algebra tiles to evaluate 62. • The expression 62 is the product 6 × 6. Products can be represented by squares with one factor as the length and the other as the width. • Arrange tiles in a 6-by-6 square. • Since 6 × 6 = 36, 62 = 36.

ANALYZE THE RESULTS Model each power. Apply what you learned to evaluate it. 1. 32

2. 52

3. 72

4. 82

5. 102

6. 122

7. Explain why n2 is often called n squared. The opposite of squaring a number is finding a square root. To find the square root of a number, find two equal factors whose product is that number. The positive square root of a number is the principal square root. The symbol for the principal square root is √.

ACTIVITY 2 Use algebra tiles to find √ 25 . • You know that a square number can be represented by the area of a square. To find the square root of 25, arrange 25 tiles into a square. • 25 tiles can be arranged in a 5-by-5 square. Therefore, 25 = 5 × 5, or 52. • The principal square root of 25 is 5.

ANALYZE THE RESULTS Model each square root. Apply what you learned to evaluate it. 8. √ 4

9. √ 16

10. √ 81

11. √49

12. √ 100

13. √ 121

14. What part of the model represents the square root of the area of the square? 462 Chapter 9 Real Numbers and Right Triangles

Suppose you try to arrange 50 tiles into a square. You discover that it’s impossible. This suggests that 50 is not a perfect square. You can estimate the square roots of numbers that are not perfect squares.

ACTIVITY 3 Use algebra tiles to estimate the principal square root of 50. • Arrange 50 tiles into the largest square possible. The largest square possible has 49 tiles, with one left over.

Ȗ{ Ç

• Add tiles until you have the next larger square. You need to add 7 tiles on top and 7 tiles on the side, and then the leftover tile from above can be placed in the upper right corner. Therefore, you added 14 new tiles to make a square that has 64 tiles.

ȖÈ{ n

• The square root of 49 is 7 and the square root of 64 is 8. Therefore the square root of 50 is between the whole numbers 7 and 8. Since 50 is closer to 49 than 50 is closer to 7 than 8. 64, you can expect that √ Ȗ{ Ȗxä Ç

ȖÈ{ n

• Verify the estimate with a calculator. 2nd [ √] 50 ENTER 7.071067812

ANALYZE THE RESULTS Model each square root. Apply what you learned to estimate the principal square root. 15. 20

16. 44

17. 58

18. 69

19. 94

20. 111

21. MAKE A CONJECTURE Describe a method that could be used to find the square root of a number by squaring numbers to estimate, rather than by taking square roots to estimate. Explore 9-1 Algebra Lab: Squares and Square Roots

463

9-1

Squares and Square Roots

Main Ideas • Find squares and square roots. • Estimate square roots.

New Vocabulary perfect square square root radical sign

Values of x2 are shown in the second column in the table. Guess and check to find the value of x that corresponds to x2. If you cannot find an exact answer, estimate with decimals to the nearest tenth to find an approximate answer.

x

x2 25 49 169 225

a. Describe the difference between the first four and the last four values of x.

8 12

b. Explain how you found an exact answer for the first four values of x.

65 110

c. How did you find an estimate for the last four values of x?

Squares and Square Roots Numbers like 25, 49, 169, and 225 are perfect squares because they are squares of integers. 5 × 5 or 52

7 × 7 or 72

13 × 13 or 132

15 × 15 or 152

25

49

169

225

A square root of a number is one of two equal factors of the number. Every positive number has a positive square root and a negative square root. A negative number like –9 has no real square root because the square of a number cannot be negative. Square Root Words

A square root of a number is one of its two equal factors.

Symbols

If x2 = y, then x is a square root of y.

Example

Since 5 · 5 or 52 = 25, 5 is a square root of 25. Since (-5) · (-5) or (-5)2 = 25, -5 is a square root of 25.

A radical sign, √, is used to indicate a positive square root. Since every positive number has both a positive and a negative square root, different notations are used to indicate one or both square roots.

EXAMPLE

Find Square Roots

Find each square root.


a. √36

√ 36 indicates the positive square root of 36.

Since 62 = 36, √ 36 = 6. b. - √ 81 - √ 81 indicates the negative square root of 81. Since 92 = 81, - √ 81 = -9.

464 Chapter 9 Real Numbers and Right Triangles

Reading Math Plus or Minus Symbol The notation ± √ 9 is read plus or minus the square root of 9.

c. ± √ 9 ± √9 indicates both square roots of 9. Since 32 = 9, √ 9 = 3 and - √9 = -3. d. √ x2 √ x2 indicates the positive square root of x2. x may be negative, but ⎪x⎥ is positive, so √ x2 = ⎪x⎥.

1A. √ 49

Reading Math Approximately Equal to Symbol The symbol ≈ is read is approximately equal to.

1C. ± 兹 144

1B. - √ 100

EXAMPLE

1D.

y2 √

Find Square Roots with a Calculator

Use a calculator to find each square root to the nearest whole number. a. √10 2nd [ √] 10 ENTER 3.16227766

Use a calculator.

10 ≈ 3.2 兹

Round to the nearest tenth.

CHECK Since (3)2 = 9, the

10

answer is reasonable. 1

2

3

4

5

6

7

8

9

10

b. - √ 27 2nd [ √] 27 ENTER 5.19615242

Use a calculator.

√ 27 ≈ -5.2


CHECK Since (-5)2 = 25 , the

⫺27

answer is reasonable. ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1

0

1

2

3

2A. √ 14

2B. - √ 12

Estimate Square Roots You can also estimate square roots mentally by using perfect squares.

EXAMPLE Appropriate Forms of Numbers Express a number as a square root if an exact answer is needed. Express a number as a decimal if an approximation is needed.

Estimate Square Roots

Estimate each square root to the nearest integer. a. √ 38 • The first perfect square less than 38 is 36. • The first perfect square greater than 38 is 49.

√ 36 = 6 √ 49 = 7

• Plot each square root on a number line. ȖÎÈ ȖÎn È

Ȗ{ Ç

The square root of 38 is between the whole numbers 6 and 7. Since 38 38 is closer to 6 than 7. is closer to 36 than 49, you can expect that √ Extra Examples at pre-alg.com

Lesson 9-1 Squares and Square Roots

465

b. - √ 175 • The first perfect square less than 175 is 169. • The first perfect square greater than 175 is 196. • Plot each square root on a number line.

√ 169 = 13 √ 196 = 14

Ȗ£È Ȗ£Çx Ȗ£È £{

£Î

The negative square root of 175 is between the integers -13 and -14. Since 175 is closer to 169 than 196, you can expect that - √ 175 is closer to -13 than -14.

3A. √ 120

3B. - √ 18

When finding square roots in real-world situations, use the positive, or principal, square root when a negative answer does not make sense.

Real-World Link To estimate how far you can see from a point above the horizon, you can use the formula D = 1.22 × 兹 A where D is the distance in miles and A is the altitude, or height, in feet.

SCIENCE Use the information at the left. The light on Cape Hatteras Lighthouse in North Carolina is 208 feet high. On a clear day, from about what distance on the ocean is the light visible? Round to the 196 = 1 ⫻ 14 = 14 nearest tenth. Estimate 1 ⫻ √ D = 1.22 × √ A Write the formula. √ = 1.22 × 208 Replace A with 208. ≈ 17.5951 Evaluate the square root first. Then multiply. On a clear day, the light will be visible from about 17.6 miles.

4. SCIENCE Spring Port Ledge Lighthouse in Maine stands roughly 55 feet high. Estimate and then find about how far a person who is standing on the observation deck can see on a clear day. Round to the nearest tenth. Personal Tutor at pre-alg.com

Example 1 (pp. 464–465)

Example 2 (p. 465)

Example 3

Find each square root. 25 1. √

2. - √ 64

3. ± √ 36

Use a calculator to find each square root to the nearest tenth. 4. √15

5. - √ 32

Estimate each square root to the nearest integer. Do not use a calculator.

(pp. 465–466)

6. √66

Example 4

8. BASEBALL A baseball diamond is actually a square with an area of 8100 square feet. The Cincinnati Reds cover their diamond with a tarp to protect it from the rain. The sides are all the same length. How long is the tarp on each side?

(p. 466)

466 Chapter 9 Real Numbers and Right Triangles Owaki-Kulla/CORBIS

7. - √ 103

HOMEWORK

HELP

For See Exercises Examples 9–14 1 15–22 2 23–28 3 29, 30 4

Find each square root. 9. √16 12. - √ 25

10. √ 49

11. - √1

13. ± √ 100

14. ± √ 196

Use a calculator to find each square root to the nearest tenth. 15. √ 30

16. √ 56

17. - √ 43

18. - √ 86

19. √180

20. √ 250

21. ± √ 0.75

22. ± √ 3.05

Estimate each square root to the nearest integer. Do not use a calculator. 79 23. √

24. √ 95

25. - √ 54

26. - √ 125

27. ± √ 200

28. ± √ 396

ROLLER COASTERS For Exercises 29 and 30, use the table shown and refer to Example 4 on page 466. 29. On a clear day, estimate how far a person can see from the top hill of the Vortex. Then calculate the distance. 30. Estimate how far a person can see on a clear day from the top hill of the Titan. Then calculate the distance.

Coaster

Maximum Height (ft)

Double Loop The Villain Mean Streak Raptor The Beast Vortex Titan Shockwave

95 120 161 137 110 148 255 116

Source: Roller Coaster Database

31. RESEARCH Use the Internet or another source to find the tallest roller coaster in the world. How far would you be able to see from the top of the roller coaster on a clear day? Complete Exercises 32–36 mentally. 54 lies between which two consecutive whole numbers? 32. The number √ 65 or 9? Explain your reasoning. 33. Which is greater, √ 120 ? Explain your reasoning. 34. Which is less, 11 or √ 35. Find a square root that lies between 17 and 18. 77 , −8, − √ 83 , 9, −10, − √ 76 , √ 65 from least to greatest. 36. Order √ GEOMETRY The area of each square is given. Estimate the length of a side of each square to the nearest tenth. Then find its approximate perimeter. 37.

38. 109 in2

39. 203 cm 2

70 m 2

40. Find the negative square root of 1000 to the nearest tenth. EXTRA PRACTICE See pages 781, 802. Self-Check Quiz at pre-alg.com

, what is the value of x to the nearest tenth? 41. If x = √5000 42. CONSTRUCTION City code requires that a reception hall must allow 4 square feet for each person on the dance floor. The reception hall wants to have a dance floor that is a square and that is large enough for 100 people at a time. What is the length of each side of the dance floor? Lesson 9-1 Squares and Square Roots

467

43. GEOMETRY Estimate the perimeter of a square that has an area of 2080 square meters. Then calculate the perimeter. Round to the nearest tenth.

H.O.T. Problems

44. OPEN ENDED Write a number for which the negative square root is not an integer. Then graph the negative square root. 45. NUMBER SENSE What are the possibilities for the ending digit of a number that has a whole number square root? Explain your reasoning. CHALLENGE For Exercises 46–48, use the following information. Squaring a number and finding the square root of a number are inverse operations. That is, one operation undoes the other operation. Use inverse operations to evaluate each expression. 64 )2 46. ( √

47. ( √ 100 )2

48. ( √ 169 )2

49. REASONING Use the pattern from Exercises 46–48 to find ( √a)2 if a ≥ 0. 50.

Writing in Math How are square roots related to factors? Give an example of a number between 100 and 200 whose square root is a whole number and an example of a number between 100 and 200 whose square root is a decimal that does not terminate.

51. Which point on the number line best ? represents √210 !

"

£Î°x £Î°Çx

A A

#

$

£{ £{°Óx £{°x £{°Çx

B B

CC

52. The area of each square is 25 square units. Find the perimeter of the figure.

£x

D D

F 60 units

H 100 units

G 75 units

J 125 units

Solve each inequality. (Lessons 8-5 and 8-6) 53. 4y > 24

a 54. _ < -7

55. 18 ≥ -2k

56. 2x + 5 < 17

57. 2t - 3 ≥ 1.4t + 6

58. 12r - 4 > 7 + 12r

0.3

59. SALES Ice cream sales increase as the temperature outside increases. Describe the slope of a line of fit that represents this situation. (Lesson 7-8) 60. Determine whether the relation (4, -1), (3, 5), (-4, 1), (4, 2) is a function. Explain. (Lesson 7-1) 61. TRAVEL Martin drives 6 hours at an average rate of 65 miles per hour. What is the distance Martin travels? Use d = rt. (Lesson 5-3)

PREREQUISITE SKILL Explain why each number is a rational number. (Lesson 5-2) 10 1 62. _ 63. 1_ 64. 0.75 65. 0.8 66. 6 67. -7 2

2


9-2

The Real Number System

Main Ideas • Identify and compare numbers in the real number system.

In this activity, you will find the length of a side of a square that has an area of 2 square units.

• Solve equations by finding square roots.

a. The small square at the right has an area of 1 square unit. Find the area of the shaded triangle.

New Vocabulary

b. Suppose four squares are arranged as shown. What shape is formed by the shaded triangles?

irrational numbers reall numbers b

c. Find the total area of the four shaded triangles. d. What number represents the length of the side of the shaded square?

Identify and Compare Real Numbers Rational numbers can be written as fractions. A few examples of rational numbers are given below. 2 -6 8_ 0.05 -2.6 5. 3 -8.12121212… 5

√ 16

Not all numbers are rational numbers. A few examples of numbers that are not rational are given below. These numbers are not repeating or terminating decimals. They are called irrational numbers. Look Back

= 3.14159…

0.101001000100001…

√2 =1.414213562…

To review rational numbers, see Lesson 5-2.

Irrational Number An irrational number is a number that cannot be expressed as _, where a b and b are integers and b does not equal 0. a

The set of rational numbers and the set of irrational numbers together make up the set of real numbers. The Venn diagram shows the relationship among the real numbers. Real Numbers

1 3

0.2

Rational Numbers 0.6 Whole Numbers

Integers ⫺4

⫺3

0 2 5

␲ 0.010010001...

⫺8 2

⫺3

0.25

Irrational Numbers

2

Natural Numbers

Lesson 9-2 The Real Number System

469

EXAMPLE

Classify Real Numbers

Name all of the sets of numbers to which each real number belongs. 3 a. 0.

This repeating decimal is a rational number because it is 1 equivalent to _ . 1 ÷ 3 = 0.33333…

b. √ 67

√ 67 = 8.185352772… It is not the square root of a perfect square so it is irrational.

28 c. -_

28 Since -_ = -7, this number is an integer and a rational number.

10 d. _

10 Since _ = 2.5, this number is a terminating decimal and thus a 4 rational number.

3

4

4

4

1A. 0.7

1B. - √ 121

9 1C. _

1D. 9

5

EXAMPLE

Compare Real Numbers on a Number Line 3 5_ a true statement.

with , or = to make √ 34

a. Replace

8

Express each number as a decimal. Then graph the numbers. 5 3 = 5.375

√ 34 = 5.830951895…

8

3

58 5.0

Look Back To review comparing fractions and decimals, see Lesson 5-1.

5.1

5.2

5.3

5.4

34 5.5

5.6

5.7

5.8

5.9

6.0

3 √ 3 Since √ 34 is to the right of 5 _ , 34 > 5 _ . 8

8

1 √ , 17 , 4.4, and √ 16 from least to greatest. b. Order 4 _ 2

Express each number as a decimal. Then compare the decimals. 1 4_ = 4.5 4. 4 = 4.444444444… 2

√ 17 = 4.123105626…

√ 16 = 4 1

16 17 4.0

4.1

4.4 4.2

4.3

4.4

42 4.5

4.6

4.7

4.8

4.9

5.0

1 , √17 , 4. From least to greatest, the order is √16 4, 4 _ . 2 3 √ 2A. Replace with , or = to make 7 _ 58 a true statement. 5 12 1 _ _ 12 , 3.3, , and 3 from greatest to least. 2B. Order √ 3

3

Solve Equations by Finding Square Roots Some equations have irrational number solutions. Just as you can solve an equation by adding the same number to each side, you can solve certain equations by taking the square root of each side. 470 Chapter 9 Real Numbers and Right Triangles

EXAMPLE

Solve Equations

Solve each equation. Round to the nearest tenth, if necessary. a. x2 = 64 x2 = 64 √x2 = √ 64

Write the equation. Take the square root of each side.

x = √ 64 or x = - √ 64 Find the positive and negative square root. x=8

or x = -8

The solutions are 8 and -8. b. 2n2 = 170 Check Reasonableness Check the results by evaluating 92 and (-9)2.

2n2 = 170 n2

= 85


√ n2 = √ 85

Take the square root of each side.

n = √ 85 or n = - √ 85 Find the positive and negative square root.

92 = 81 (-9)2 = 81 Since 81 is close to 85, the solutions are reasonable.

Write the equation.

n ≈ 9.2 or n ≈ -9.2

Use a calculator.

The solutions are 9.2 and -9.2.

3A. 363 = 3d2

3B. y2 = 30


HANG GLIDING The aspect ratio of a hang glider allows it to glide 2 through the air. The formula for the aspect ratio R is R = s, where A s is the wingspan and A is the area of the wing. What is the wingspan of a hang glider if its aspect ratio is 4.5 and the area of the wing is 50 square feet? 2

Real-World Link The record for hang gliding distance belongs to Mike Barber who hang glided 437 miles in Zapata, Texas, in June 2002.

s R=_

A 2 _ 4.5 = s 50

225 = s2 √ 225 =

√ s2

15 = s

Write the formula. Replace R with 4.5 and A with 50. Multiply each side by 50. Take the positive square root of each side. Simplify.

The wingspan of the hang glider is 15 feet.

4. SEISMIC WAVES A tsunami is caused by an earthquake on the ocean floor. s2 = 9.61, where The speed of a tsunami can be measured by the formula _ d s is the speed of the wave in meters per second and d is the depth of the ocean in meters where the earthquake occurs. What is the speed of a tsunami if an earthquake occurs at a depth of 632 meters? Round to the nearest tenth. Extra Examples at pre-alg.com G. Kalt/zefa/CORBIS

Lesson 9-2 The Real Number System

471

Example 1 (p. 470)

Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. 1. 7

Example 2 (p. 470)

3 3. -_

2. 0.5555…

with , or = to make a true statement. 6. - √ 74 -8.4

Replace each 4 5. 6_ 5

4. √ 12

4

√ 48

Order each set of numbers from least to greatest. 10 1 7. 3.7, 3 3, √ 13 , _ 8. √ 110 , 10_ , 10. 5, 10.15 5

Example 3 (p. 471)

Example 4 (p. 471)

HOMEWORK

HELP


3

5

ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 9. y2 = 25

10. 3m2 = 222

11. LANDSCAPING A sprinkler waters a circular area of the lawn as shown. The formula A = 3.14r2 measures the distance r the sprinkler shoots water within a circular area A. How far is the sprinkler shooting water if it waters an area of 572.6 square feet? Round to the nearest tenth.

R

Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. 1 13. _

2 14. _

15. 4

24 16. -_

17. 7.6

18. - √ 64

19. 0.131313…

20. 2.8

56 21. -_

22. 0. 2

23. - √ 100

12. 8

2

8

8


Replace each 1 24. 5_ 4

5

√ 26

27. - √ 18

25. √ 80

3 -4_

1 28. 1_

8

2

26. -3.3

9.2 √ 2.25

- √ 10

29. - √ 6.25

5 -_ 2

Order each set of numbers from least to greatest. 1 √ _ 2 √ √ 30. 5_ , 2.1, 4 , 6 31. 4. 23, 4 _ , 18 , 16 5

4

3

Order each set of numbers from greatest to least. 1 32. -10, -10 _ , -1.05, - √ 105 2

1 17 33. - √ 14 , -4 _ , -_ , -3.8 10

4

ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 34. a2 = 49

35. 300 = h2

36. y2 = 22

37. 0.0058 = k2

38. 5p2 = 315

39. 2d2 = 162

40. 190.5 = 1.5b2

41. 0.1x2 = 0.169


42. TRACK AND FIELD Use the information at the left. Suppose American Stacy Dragila reached a winning height of about 15 feet in the 2000 Olympics. About how fast was she running? Round to the nearest tenth. 43. PHYSICS The formula h = 16t2 measures the time t in seconds that it takes for an object to fall from a height of h feet and hit the ground. How long would it take a marble to hit the ground if it was dropped off a cliff with a height of 150 feet? Round to the nearest tenth. Determine whether each statement is sometimes, always, or never true. 44. A whole number is an integer. 45. An irrational number is a negative integer. 46. A repeating decimal is a real number. 47. An integer is a whole number. 48. FLOORING A square room has an area of 324 square feet. The homeowners plan to cover the floor with 6-inch square tiles. How many tiles will be in each row on the floor? Real-World Link To find the height h in feet that a pole vaulter can reach, coaches can v2 use the formula h = 64 where v is the velocity of the pole vaulter in feet per second. Source: American Institute of Physics

Give a counterexample for each statement. 49. All square roots are irrational numbers. 50. All rational numbers are integers. 152 ? 51. What is the value of x to the nearest tenth if x2 - 42 = √ 52. GEOMETRY Use the formula for the area of a circle A = r2, where A represents the area, r represents the radius, and is approximately equal to 3.14, to find the radius of the circle with an area of 28.26 square inches.

R

30fd , where s represents the speed of a car in miles 53. CARS The formula s = √ per hour, d represents the distance the car skidded in feet, and f is friction, can be used to determine how fast a car was traveling before it skidded to a stop. The table shows some different values of f. At an accident scene, a car made 100-foot skid marks before hitting another car. If the speed limit was 55 miles per hour, was the car speeding before applying the brakes on a dry, concrete road? Explain.

4YPE OF 3URFACE

EXTRA

PRACTICE

2OAD #ONDITIONS

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H.O.T. Problems

54. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would solve equations by finding square roots. 55. OPEN ENDED Give an example of a number that is an integer and a rational number. 56. CHALLENGE Tell whether the product of a rational number like 8 and an irrational number like 0.101001000… is rational or irrational. Explain your reasoning. Lesson 9-2 The Real Number System

Matthew Stockman/Getty Images

473

57. Which One Doesn’t Belong? Identify the number that does not belong with the other three. Explain your reasoning. 50 -_ 2

50.1

58.

√ 50

-50.1

Writing in Math Explain the relationship between the area of a square and the length of its sides. Give an example of a square whose side length is irrational and an example of a square whose side length is rational.

60. Which number can only be classified as a rational number?

59. The time t in seconds it takes an object to fall d feet can be estimated by using d = 0.5(32)t2. If a ball is dropped from the top of a 120-foot building, how long does it take to hit the ground? A 1.9 s

C 3.8 s

B 2.7 s

D 7.5 s

F -2

1 G _

H √ 2

2

1 < 61. For what value of x is _ √x true?

A -2

1 B _

1 C _ 2

4

J 2 √x

<x

D 2

Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson 9-1) 62. √ 54 63. - √ 126 64. √ 8.67 65. - √ 19.85 66. Solve 9 - 2d ≤ 23 and check your solution. Graph the solution on a number line. (Lesson 8-8) WEATHER The table shows the heat index and relative humidity for an air temperature of 75°F. (Lesson 7-8) Relative Humidity Heat Index (°F)

0% 69

5% 69

10% 70

15% 71

20% 72

25% 72

30% 73

35% 73

40% 74

45% 74

50% 75

67. Make a scatter plot and draw a line of fit. 68. Use the line of fit to predict the heat index for a relative humidity of 70%. Express each ratio as a unit rate. Round to the nearest hundredth, if necessary. (Lesson 6-1) 69. $8 for 15 cupcakes

70. 120 miles on 4.3 gallons

71. 3 feet of snow in 5 hours

72. $22 in 5 hours

PREREQUISITE SKILL Solve each equation. (Lesson 3-5) 73. 18 + 57 + x = 180

74. x + 27 + 54 = 180

75. 85 + x + 24 = 180

76. x + x + x = 180

77. 2x + 3x + 4x = 180

78. 2x + 3x + 5x = 180


Learning Geometry Vocabulary Many of the words used in geometry are commonly used in everyday language. For example, the photo shows rays of light. The everyday meanings of ray can be used to better understand its mathematical meaning. The table below shows the meanings of some geometry terms you will use throughout this chapter. Term

ray

degree

Everyday Meaning

Mathematical Meaning

any of the thin lines, or beams, of light that appear to come from a bright source • a ray of light

a part of a line that extends from a point indefinitely in one direction

extent, amount, or relative intensity • third degree burns

a common unit of measure for angles

characterized by sharpness or severity • an acute pain

an angle with a measure that is greater than 0° and less than 90°

not producing a sharp impression • an obtuse statement

an angle with a measure that is greater than 90° but less than 180°

acute

obtuse

Source: Merriam Webster’s Collegiate Dictionary

Reading to Learn 1. Write a sentence using each term listed above. Be sure to use the everyday meaning of the term. 2. RESEARCH Use the Internet or a dictionary to find the everyday meaning of each term listed below. Compare them to their mathematical meaning. Note any similarities and/or differences. a. midpoint

b. converse

c. indirect

3. RESEARCH Use the Internet or dictionary to determine which of the following words are used only in mathematics. vertex

equilateral

similar

scalene

side

isosceles

Reading Math Learning Geometry Vocabulary Digital Vision/PunchStock

475

9-3

Triangles Animation pre alg.com

Main Ideas • Find the missing angle measure of a triangle.

There is a relationship among the measures of the angles of a triangle.

• Classify triangles by properties and attributes.

New Vocabulary line segment triangle vertex acute angle right angle obtuse angle straight angle acute triangle obtuse triangle right triangle congruent scalene triangle isosceles triangle equilateral triangle

Step 1

Use a straightedge to draw a triangle on a piece of paper. Then cut out the triangle and label the vertices X, Y, and Z.

Step 2

Fold the triangle as shown so that point Z lies on side XY as shown. Label the back of ∠Z as ∠2.

Step 3

Fold again so point X meets the vertex of ∠2. Label the back of ∠X as ∠1.

Step 4

Fold so point Y meets the vertex of ∠2. Label the back of ∠Y as ∠3.

Z X

Z

Z Y

X

2

X1 2

Y

Step 2

Step 1

Z Y

X1 2 3 Y

Step 3

Step 4

MAKE A CONJECTURE What is the sum of the measures of ∠1, ∠2, and ∠3? Explain your reasoning.

Angle Measures of a Triangle A line segment is part of a line containing two endpoints and all of the points between them. A triangle is formed by three line segments that intersect only at their endpoints. Each pair of segments forms an angle of the triangle.

The vertex of each angle is a vertex of the triangle. Vertices is the plural of vertex.

Reading Math Line Segment The symbol for line . segment XY is XY

Triangles are named by the letters at their vertices. Triangle XYZ, written XYZ, is shown. vertex

X

side

angle Y

Z

The sides are XY, YZ, and XZ. The vertices are X, Y, and Z. The angles are ∠ X, ∠Y, and ∠ Z.

The activity above suggests a relationship about the angles of any triangle. Angles of a Triangle Words The sum of the measures of the angles of a triangle is 180°. Model

Symbols

x˚ y˚


z˚

x + y + z = 180

EXAMPLE

Find Angle Measures

Find the value of x in ABC.

A

m∠A + m∠B + m∠C = 180

x˚

x + 58 + 55 = 180 x + 113 = 180 x + 113 - 113 = 180 - 113

B

58˚

55˚

C

x = 67

1. Find the value of ∠E in DEF if m∠D = 62° and m∠F = 39°.

EXAMPLE

Use Ratios to Find Angle Measures

ALGEBRA The measures of the angles of a certain triangle are in the ratio 1:4:7. What are the measures of the angles? Words

The sum of the measures is 180°.

Variables

Let x represent the measure of the first angle, and 7x the measure of the third angle.

Equation

x + 4x + 7x = 180

the measure of a second angle,

x + 4x + 7x = 180 Write the equation. 12x = 180 Combine like terms. 180 12x _ =_

Check for Accuracy 15 + 60 + 105 = 180. So, the answer is correct.

12

12

x = 15

Divide each side by 12. Simplify.

Since x = 15, 4x = 4(15) or 60, and 7x = 7(15) or 105. The measures of the angles are 15°, 60°, and 105°.

2. The measures of the angles of a certain triangle are in the ratio 1:3:6. What are the measures of the angles? Personal Tutor at pre-alg.com

Classify Triangles Angles can be classified by their degree measure. Types of Angles Acute Angle

Right Angle

Obtuse Angle

Straight Angle

This symbol is used to indicate a right angle.

A

0° < m∠A < 90°

A

A

m∠ A = 90°

A

90° < m∠ A < 180°

m∠ A = 180°

Lesson 9-3 Triangles

477

EXAMPLE

Classify Angles

Classify each angle as acute, obtuse, right, or straight. a.

b.

A B

c. D

G 40˚

125˚

J

F C

E

m∠ABC > 90° So, ∠ABC is obtuse.

3A. 154°

H

m∠DEF = 90° So, ∠DEF is right.

m∠GHJ < 90° So, ∠GHJ is acute.

3B. 88°

3C. 180°

Triangles can be classified by their angles and their sides. Congruent sides have the same length.

Classify Triangles Acute Triangle

Obtuse Triangle

40˚ 80˚ 60˚

Reading Math Congruent Segments Tick marks on the sides of a triangle indicate that those sides are congruent.

Right Triangle

30˚ 110˚40

45˚

˚

45˚

all acute angles

one obtuse angle

one right angle

Scalene Triangle

Isosceles Triangle

Equilateral Triangle

at least two sides congruent

all sides congruent

no congruent sides

EXAMPLE

Classify Triangles

Classify the triangle by its angles and by its sides. R

T

45˚

Angles: RST has a right angle. 45˚

S

Sides: RST has two congruent sides.

So, RST is a right isosceles triangle.

Classify each triangle by its angles and by its sides. 4A. 4B. Ón Ón xn £Ó{ {Ó nä 478 Chapter 9 Real Numbers and Right Triangles


Example 1 (p. 477)

Find the value of x in each triangle. Then classify each triangle as acute, right, or obtuse. 1.

2.

83˚ 72˚

Example 2 (p. 477)

Example 3 (p. 478)

Example 4 (p. 478)

3. 61˚

x˚

x˚

27˚

48˚ x˚

29˚

4. ALGEBRA Triangle EFG has angles whose measures are in the ratio 1:5:9. What are the measures of the angles? Classify each angle as acute, obtuse, right, or straight. 5. 55°

6. 140°

Classify each indicated triangle by its angles and by its sides. 8.

7.

Lima

90˚ 110˚

45˚ 45˚

45˚

9.

Youngstown

Ohio

25˚

75˚

Cincinnati

75˚

30˚

HOMEWORK

HELP



11.

x˚

12. 57˚

32˚

63˚ 68˚

13.

x˚

15. 28˚

33˚

36˚

x˚

14. 45˚

x˚

x˚

x˚ 43˚ 71˚

62˚

16. ALGEBRA The measures of the angles of a triangle are in the ratio 1:3:5. What is the measure of each angle? 17. ALGEBRA Determine the measures of the angles of ABC if the measures of the angles are in the ratio 1:1:16. Classify each angle as acute, obtuse, right, or straight. 18. 40°

19. 70°

20. 65°

21. 85°

22. 95°

23. 110°

24. 155°

25. 140°

26. 38°

27. 127°

28. TIME What type of angle is formed by the hands on a clock at 6:00? Lesson 9-3 Triangles

479

Classify each indicated triangle by its angles and by its sides. 29.

30.

60˚

31. 60˚

40˚

40˚

30˚ 60˚

60˚

32.

33.

ÃÜÀÌ

34.

D[XhWiaW

40˚

V

35˚

V

70˚ 70˚

110˚

35˚

Sketch each triangle. If it is not possible to sketch the triangle, write not possible. Real-World Link Tony Hawk was the first skateboarder to perform the 900 during the X-Games. He rode off a ramp, spun 900° in mid-air, and made a perfect landing, all on a skateboard.

35. acute scalene 37. right equilateral

36. obtuse and not scalene 38. obtuse equilateral

BASEBALL If you swing the bat too early or too late, the ball will probably go foul. The best way to hit the ball is at a right angle. Classify each angle shown. 39.

J

40.

K

R

S

41.

Source: skateboardlink.com

Incoming Path Angle 90˚

D

O

Angle 90˚

T

Angle 90˚

L

T

42. SKATEBOARDING Refer to the information at the left. How many revolutions did Tony make performing the 900? EXTRA

PRACTICE


ALGEBRA Find the measures of the angles in each triangle. 43.

44.

5x ˚ 3x ˚

x˚

(x 5)˚

x˚

45. 2x ˚

85˚

(2x 15)˚ 7x ˚

H.O.T. Problems

46. OPEN ENDED Draw an obtuse isosceles triangle to represent a real-world object. Would the object still be useful if the triangle were acute? Explain. 47. CHALLENGE Numbers that can be represented by a triangular arrangement of dots are called triangular numbers. The first three triangular numbers are 1, 3, and 6. Find the next three triangular numbers. 48. SELECT A TOOL The measure of ∠1 is twice the measure of ∠2. The measure of ∠3 is 40° less than the measure of ∠2. Which of the following tools would you use to determine the measures of the three angles? Justify your selection(s). Then use the tool(s) to solve the problem. draw a model

480 Chapter 9 Real Numbers and Right Triangles Jamie McDonald/Getty Images

paper/pencil

calculator

49.

Writing in Math How do the angles of a triangle relate to each other? Include drawings of two triangles with their angles labeled.

50. The measures of the angles of a brick paver are 30°, 90°, and 60°. Which triangle most likely has these angle measures? A

51. A long piece of paper is folded so that the lower edge of the strip forms a right angle with itself. Classify ∠3.

C

B

D

F acute

H right

G obtuse

J

straight

ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. (Lesson 9-2) 52. m2 = 81

53. 196 = y2

x2 55. _ = 51

54. 168 = 2p2

2

Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson 9-1) 56. - √ 5.25

57. - √ 17.3

59. √ 140.57

58. √ 38.75

60. Twenty-six is 25% of what number? (Lesson 6-5)

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ä i « Ã V À> }i Ã } i iÃ

62. WEATHER The time a storm will hit an area can be predicted using the formula d ÷ s = t where d is the distance in miles an area is from the storm, s is the speed in miles per hour of the storm, and t is the travel time in hours of the storm. Suppose it is 11:00 A.M. and a storm is heading toward a town at a speed of 30 miles per hour. The storm is about 150 miles from the town. At what time will the storm hit? (Lesson 3-8)

Ûi ½i "ÕÌt

61. AIRPORTS The graph shows North America’s busiest cargo airports in 2003. What is the difference in cargo handling between Anchorage and New York? (Lesson 5-8)

À«ÀÌ -ÕÀVi\ À«ÀÌÃ ÕV ÌiÀ>Ì>

PREREQUISITE SKILL Find the value of each expression. (Lesson 4-1) 63. 122

64. 152

65. 182

66. 212

67. 242

68. 272

Lesson 9-3 Triangles

481

CH

APTER

9

Mid-Chapter Quiz Lessons 9-1 through 9-3 2

Find each square root. (Lesson 9-1) 1. √ 36 3. ± √ 81

2. - √ 169 2 4. √m

8t 21. CLOCKS The formula L = _ represents π2 the swing of a pendulum, where L is the length of the pendulum in feet and t is the time in seconds that it takes to swing back and forth. How long does it take a 4-foot pendulum to swing back and forth?

Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson 9-1) 5. √ 51 7. √ 17

6. √ 88 8. √ 111

(Lesson 9-2)

Classify each angle measure as acute, obtuse, right, or straight. (Lesson 9-3)

9. MULTIPLE CHOICE Which statement is NOT true? (Lesson 9-1)

22. 83°

23. 180°

24. 115°

25. 90°

Classify each triangle by its angles and by its sides. (Lesson 9-3)

A 6 < √ 39 < 7 < 10 B 9 < √89

26.

C -7 > - √ 56 > -8 D -4 < - √ 17 < -5

xÎ {Ó

{Ó

10. GARDENING Marisa wants to put a fence around her square vegetable garden that A = s, has an area of 169 square feet. If √ where s is the length of one side and A is the area, how many feet of fencing does she need to enclose the garden?

27.

È

28.

ÎÇ

29. Èäc Îäc Èäc

Èäc Èäc

(Lesson 9-1)

Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. (Lesson 9-2) 11. 0.3 − 13. 15.1

12. - √ 49 56 14. _ 8

30. MULTIPLE CHOICE Refer to the figure shown. Bob lives in Salsburg, does his grocery shopping in Richmond, and sees a doctor in Thornville. What is the measure of the angle formed when Bob travels from Salsburg to Richmond and then to Thornville? (Lesson 9-3) 2ICHMOND

ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. (Lesson 9-2) 15.

m2

= 68

16. 131 =

n2

3ALSBURG

££äc

ÎÓc

17. 600 = 1.5x2

18. 2b2 = 98

F 28°

H 48°

19. 259.2 = 5y2

20. 3.6r2 = 518.4

G 38°

J 180°


4HORNVILLE

EXPLORE

9-4

Algebra Lab

The Pythagorean Theorem ACTIVITY 1 Dot paper can be used to find the area of certain geometric figures. Consider the following examples. Find the area of each shaded region if each square square unit.

1 1 A=_ (1) or _ unit2 2

2

1 A=_ (2) or 1 unit2 2

represents one

1 A=_ (4) or 2 units2 2

The area of other figures can be found by first separating the figure into smaller regions and then finding the sum of the areas of the smaller regions.

A = 2 units2

A = 5 units2

A = 4 units2

EXERCISES Find the area of each figure. 1.

2.

3.

4.

Explore 9-4 Algebra Lab: The Pythagorean Theorem

483

ACTIVITY 2 Let’s investigate the relationship that exists among the sides of a right triangle. In each diagram shown, notice how a square is attached to each side of a right triangle. Triangle 1

Triangle 2

Triangle 3

Square C

Square C

Square C Square A

Square A

Square A Square B

Square B

Square B

Triangle 4

Triangle 5

3QUARE #

Square C Square A

3QUARE ! 3QUARE " Square B

Copy the table. Then find the area of each square that is attached to the triangle. Record the results in your table. Triangle

Area of Square A (units2)

Area of Square B (units2)

Area of Square C (units2)

1 2 3 4 5

EXERCISES 5. Refer to your table. How does the sum of the areas of square A and square B compare to the area of square C? 6. Draw a right triangle on centimeter grid paper. Count to find the measures of the legs and use the relationship you discovered to calculate the measure of the hypotenuse. Measure to verify your answer. 7. Refer to the diagram at the right. If the lengths of the sides of a right triangle are whole numbers such that a2 + b2 = c2, the numbers a, b, and c are called a Pythagorean Triple. Tell whether each set of numbers is a Pythagorean Triple. Explain. a. 3, 4, 5

b. 5, 7, 9


c. 6, 9, 12

d. 7, 24, 25

a

c

b

9-4

The Pythagorean Theorem Interactive Lab pre-alg.com

Main Ideas • Use the Pythagorean Theorem to find the length of a side of a right triangle.

In the diagram, three squares with sides 3, 4, and 5 units are used to form a right triangle.

• Use the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle.

b. What relationship exists among the areas of the squares?

New Vocabulary legs hypotenuse Pythagorean Theorem solving a right triangle converse

5 units 3 units

a. Find the area of each square.

c. Draw three squares with sides 5, 12, and 13 units so that they form a right triangle. What relationship exists among the areas of these squares?

4 units

The Pythagorean Theorem In a right triangle,

hypotenuse

the sides that are adjacent to the right angle are called the legs. The side opposite the right angle is the hypotenuse.

legs

The Pythagorean Theorem describes the relationship between the lengths of the legs and the hypotenuse for any right triangle. Pythagorean Theorem If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Words

Model c

a

c2 = a2 + b2

Example

52 = 32 + 42 25 = 9 + 16 25 = 25

b

Reading Math Hypotenuse The hypotenuse is the longest side of a right triangle.

Symbols

EXAMPLE

Find the Length of the Hypotenuse

Find the length of the hypotenuse of the right triangle. c2 = a2 + b2

Pythagorean Theorem

c2

Replace a with 12 and b with 16.

=

122

+

162

c2

= 144 + 256 Evaluate

c2

= 400

√ c2 = √ 400 c = 20

122

and

162.

Add 144 and 256.

12 ft

c ft

16 ft

Take the positive square root of each side. The length of the hypotenuse is 20 feet.

1. Find the length of the hypotenuse of a right triangle if its legs are 24 inches and 45 inches long. Lesson 9-4 The Pythagorean Theorem

485

If you know the lengths of two sides of a right triangle, you can use the Pythagorean Theorem to find the length of the third side. This is called solving a right triangle.

EXAMPLE Square Roots In Lesson 9-1, you learned that a number has both a positive and negative square root. Since we are finding the lengths of sides of triangles, we calculate the positive square root.

Solve a Right Triangle

Find the length of the leg of the right triangle. c2 = a2 + b2

14 cm

Pythagorean Theorem a cm

142 = a2 + 102

Replace c with 14 and b with 10.

196 = a2 + 100

Evaluate 142 and 102.

196 - 100 =

a2

+ 100 - 100 Subtract 100 from each side.

96 = a2 √ 96 =

10 cm

Simplify.

√ a2


2nd [ √ ] 96 ENTER 9.797958971

The length of the leg is about 9.8 centimeters. {£ V

X V

2. Find the length of the leg of the right triangle.

{ä V

GRIDDABLE A painter positions a 20-foot ladder against a house so that the base of the ladder is 4 feet from the house. About how many feet does the ladder reach on the side of the house? Round to the nearest tenth.

20 ft 4 ft

Read the Test Item The ladder, ground, and side of the house form a right triangle. You know the hypotenuse and one leg of a right triangle. You need to find the other leg. Solve the Test Item Use the Pythagorean Theorem to find how high the ladder reaches on the side of the house. c2 = a2 + b2 202 = 42 + b2 400 = 16 +

b2

Pythagorean Theorem Replace c with 20 and a with 4. Evaluate 202 and 42.

400 - 16 = 16 + b2 - 16 Subtract 16 from each side. 384 = b2 √ 384 = √ b2

19.6 ≈ b

Simplify. Take the square root of each side. Round to the nearest tenth.

The ladder reaches about 19.6 feet on the side of the house. 486 Chapter 9 Real Numbers and Right Triangles


Fill in the Answer Grid

3. GRIDDABLE A doorway is 2.7 feet wide and 8.4 feet high. What is the longest piece of drywall in feet that can be taken through this doorway? Round to the nearest tenth. Personal Tutor at pre-alg.com

Converse of the Pythagorean Theorem The Pythagorean Theorem is written in if-then form. If you reverse the statements after if and then, you have formed the converse of the Pythagorean Theorem.

Pythagorean Theorem If a triangle is a right triangle, then c2 = a2 + b2.

Since there is no place for a negative sign or a fraction on the grid, griddable answers are never negative numbers or fractions.

Write 19.6 in the answer boxes and write only one digit in each answer box. Fill in one bubble for every answer box that you have written in. Be sure not to fill in a bubble under a blank answer box.

If c2 = a2 + b2, then a triangle is a right triangle.

Converse

The converse of the Pythagorean Theorem is also true. You can use the converse to determine whether a triangle is a right triangle.

EXAMPLE

Identify a Right Triangle

The measures of three sides of a triangle are given. Determine whether each triangle is a right triangle. a. 9 m, 12 m, 15 m c2 = a2 + b2 152 92 + 122 225 81 + 144 225 = 225

Pythagorean Theorem Replace c with 15, a with 9, and b with 12. Evaluate 152, 92, and 122. Simplify.

The triangle is a right triangle. b. 6 in., 7 in., 12 in. c2 = a2 + b2 122 62 + 72 144 36 + 49 144 ≠ 85

Pythagorean Theorem Replace c with 12, a with 6, and b with 7. Evaluate 122, 62, and 72. Simplify.

The triangle is not a right triangle.

4A. 8 in., 9 in., 12 in.

4B. 15 mm, 20 mm, 25 mm Lesson 9-4 The Pythagorean Theorem

487

Example 1 (p. 485)

Find the length of the hypotenuse in each right triangle. Round to the nearest tenth, if necessary. 1.

2.

12 ft

6 ft

cm

15 m

c ft 20 m

Example 2 (p. 486)

If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 3. a = 8, b = ?, c = 17

Example 3 (pp. 486–487)

Example 4 (p. 487)

4. a = ?, b = 24, c = 25

5. GRIDDABLE Kendra is flying a kite. The length of the kite string is 55 feet, and she is positioned 40 feet away from the point directly beneath the kite. About how high is the kite in feet? The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. 6. 5 cm, 7 cm, 8 cm

HOMEWORK

HELP


7. 20 ft, 48 ft, 52 ft


9.

10.

24 in.

cm

6m

24 m 10 in.

cm

c in. 45 m

8m

11.

12.

13.

7.2 cm 2.7 cm

c ft

40 ft

12.8 m

cm

c cm 13.9 m

30 ft

If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 14. a = 30, b = ?, c = 50

15. a = ?, b = 35, c = 37

16. a = ?, b = 12, c = 19

17. a = 7, b = ?, c = 14

Find each missing measure to the nearest tenth. 18.

19. 45 ft

17 m

28 m

x ft

30 ft


xm

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. 20. a = 5, b = 8, c = 9

21. a = 16, b = 30, c = 34

22. a = 18, b = 24, c = 30

23. a = 24, b = 28, c = 32

24. a = √ 21 , b = 6, c = √ 57

25. a = 11, b = √ 55 , c = √ 177

26. GYMNASTICS The floor exercise mat measures 40 feet by 40 feet. Find the measure of the diagonal. 27. TELEVISION The size of a flat-screen television is determined by the length of the diagonal of the screen. If a 35-inch television screen is 26 inches long, what is its height to the nearest inch? Real-World Link The floor exercise mat is a square of plywood covered by a 2-inch padding and mounted on 4” springs. Source: The Gymnastics Place!

If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 28. a = 8.1, b = 3.5, c = ?

29. a = 10.4, b = 16.9, c = ?

30. a = 27, b = ?, c = 61

31. a = ?, b = 73, c = 82

123 =, c = 22 32. a = ?, b = √

33. a = √ 177 =, b = ?, c = 31

34. GEOMETRY If the vertex of an angle lies on a circle, the angle is called an inscribed angle. All angles inscribed in a semicircle are right angles. In the figure, ∠ACB is an inscribed AB is 17 and the right angle. If the measure of AC is 8, find the measure of BC. measure of

C 8

A

B

17

ANALYZE GRAPHS Find the length of the hypotenuse. Write your answer as a square root. 35.

36.

Y

37.

Y

!

Y

2 &

$ "

X

X

"

#

X

"

" 4

3

'

TRAVEL Europe’s largest town square is the Rynek Glowny located in Krakow, Poland. It covers approximately 48,400 square yards. 38. How many feet long is a side of the square? 39. To the nearest foot, approximately what is the diagonal distance across Rynek Glowny?

EXTRA

PRACTICE


B

ART For Exercises 40 and 41, use the plasterwork design shown. 40. If the sides of the square measure 6 inches, AB ? what is the length of AB 41. What is the perimeter of the design if 128 inches? measures √

A Lesson 9-4 The Pythagorean Theorem

Steven E. Sutton/DUOMO

489

H.O.T. Problems

42. OPEN ENDED State three numbers that could be the measures of the sides of a right triangle. Justify your answer. 43. FIND THE ERROR Marcus and Allyson are finding the missing measure of the right triangle shown. Who is correct? Explain your reasoning. Marcus c2 = a2 + b2 152 = 92 + b2 12 = b

Allyson c2 = a 2 + b 2 c2 = 92 + 152 c2 ≈ 125

15 ft

9 ft

c ft

44. CHALLENGE The hypotenuse of an isosceles right triangle is 8 inches. Is there enough information to find the length of the legs? If so, find the length of the legs. If not, explain why not. 45.

Writing in Math

How do the lengths of the sides of a right triangle relate to each other? Include an example of a set of numbers that represents the measures of the legs and hypotenuse of a right triangle.

46. Find the amount of edging needed to enclose the triangular flower bed. A 10 yd B 16 yd

YD

C 18 yd

47. GRIDDABLE How far is it in feet from home plate to second base? Round to the nearest tenth.

Ó` >Ãi Èä vÌ £ÃÌ >Ãi

ÎÀ` >Ãi Èä vÌ

YD

D 24 yd

i *>Ìi

Find the value of x in each triangle. Then classify each triangle as acute, right, or obtuse. (Lesson 9-3) 48.

49.

x˚ 46˚

33˚

50.

63˚ x˚

27˚

x˚ 48˚

54˚

Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. (Lesson 9-2)

51. -5

52. 0. 4

53. 63

54. 7.4

55. GRADES Tobias’ average for five quizzes is 86. If he wants to have an average of at least 88 for six quizzes, what is the lowest score he can receive on his sixth quiz? (Lesson 5-9)

PREREQUISITE SKILL Simplify each expression (Lessons 1-2 and 4-2) 56. (2 + 6)2 + (-5 + 6)2

57. (-4 + 3)2 + (0 - 2)2


58. [3 + (-1)]2 + (8 - 4)2

EXTEND

Algebra Lab

9-4

Graphing Irrational Numbers

In Lesson 2-1, you learned to graph integers on a number line. Irrational numbers can also 53 . To graph √ 53 , construct be graphed on a number line. Consider the irrational number √ √ a right triangle whose hypotenuse measures 53 units.

ACTIVITY Step 1 Find two numbers whose squares have a sum of 53. Since 53 = 49 + 4 or 72 + 22, one pair that will work is 7 and 2. These numbers will be the lengths of the legs of the right triangle. Step 2 Draw the right triangle • First, draw a number line on grid paper.

0 1 2 3 4 5 6 7 8

• Next, draw a right triangle whose legs measure 7 units and 2 units. Notice that this triangle can be drawn in two ways. Either way is correct.

7 units 2 units 01 2 3 4 5 6 7 8

01 2 3 4 5 6 7 8

53 . Step 3 Graph √ • Open your compass to the length of the hypotenuse. • With the tip of the compass at 0, draw an arc that intersects the number line at point B. 53 • The distance from 0 to B is √ 53 ≈ 7.3. units. From the graph, √

01 2 3 4 5 6 7 8

B 01 2 3 4 5 6 7 8

ANALYZE THE RESULTS Use a compass and grid paper to graph each irrational number on a number line. 1. √ 5

3. √ 45

2. √ 20

4.

√ 97

. 5. Describe two different ways to graph √34 6. Explain how the graph of √2 can be used to locate the graph of √3. Extend 9-4 Algebra Lab: Graphing Irrational Numbers

491

9-5

The Distance Formula

Main Idea • Use the Distance Formula to determine lengths on a coordinate plane.

New Vocabulary Distance Formula

The graph of points N(3, 0) and M(-4, 3) is shown. A horizontal segment is drawn from M, and a vertical segment is drawn from N. The intersection is labeled P.

y

M (⫺4, 3)

P

O

a. Name the coordinates of P.

N (3, 0) x

b. Find the distance between M and P. c. Find the distance between N and P. d. Classify MNP. e. What theorem can be used to find the distance between M and N? f. Find the distance between M and N.

The Distance Formula A line segment is a part of a line that contains two endpoints and all of the points between the endpoints. y

A line segment is named by its endpoints.

M

O

N x

The segment can be written as MN or NM.

To find the length of a segment on a coordinate plane, you can first extend horizontal and vertical segments from the vertices to form a right triangle. Then use the Pythagorean Theorem to find the length of the segment. You can also use the Distance Formula, which is based on the Pythagorean Theorem. Distance Formula Words Look Back To review the notation (x1 , y1) and (x2 , y2 ), see Lesson 7-5.

The distance d between two points with coordinates (x1, y1) and (x2, y2), is given by d = √(x - x ) 2 + (y - y ) 2 .


2

1

2

1

y

Model ( x 1, y 1)

( x 2, y 2)

d

O

x

EXAMPLE

Use the Distance Formula

Find the distance between G(-3, 1) and H(2, -4). Round to the nearest tenth, if necessary. G (⫺3, 1)

Use the Distance Formula. Substitution You can use either point as (x1, y1). The distance will be the same.

(x2 - x1)2 + (y2 - y1)2 √

Distance Formula

GH =

[2 - (-3)]2 + (-4 - 1)2 √

(x1, y1) = (-3, 1), (x2, y2) = (2, -4)

GH =

(5)2 + (-5)2 √

Simplify.

d=

y

GH = √ 25 + 25

Evaluate 52 and (-5)2.

GH = √ 50

Add 25 and 25.

GH ≈ 7.1

Take the square root.

x

O

H (2, ⫺4)

1. Find the distance between A(5, -6) and B(1, 2). Round to the nearest tenth, if necessary.

Reading Math Segment Measure The symbol GH means the measure of segment GH.

EXAMPLE

Use the Distance Formula to Solve a Problem

GEOMETRY Find the perimeter of ABC to the nearest tenth.

y

A (⫺2, 3)

First, use the Distance Formula to find the length of each side of the triangle. −− Side AB: A(-2, 3), B(2, 2) d=

B (2, 2)

O x

(x2 - x1)2 + (y2 - y1)2 √

C (0, ⫺3)

[2 - (-2)]2 + (2 - 3)2 √ AB = √ (4)2 + (-1)2 AB =

16 + 1 or √ 17 AB = √ −− Side BC : B(2, 2), C(0, -3) d=

(x2 - x1)2 + (y2 - y1)2 √

−− Side CA: C(0, -3), A(-2, 3) d=

(x2 - x1)2 + (y2 - y1)2 √

BC =

(0 - 2)2 + (-3 - 2)2 √ BC = √ (-2)2 + (-5)2

(-2 - 0)2 + [3 - (-3)]2 √ CA = √ (-2)2 + (6)2

Common Misconception

4 + 25 or √ 29 BC = √

CA = √ 4 + 36 or √ 40

To find the sum of square roots, do not add the numbers inside the square root symbols. √ 17 + √ 29 + √ 40 ≠ √ 86

Then add the lengths of the sides to find the perimeter.

CA =

√ 17 + √ 29 + √ 40 ≈ 4.123 + 5.385 + 6.325 ≈ 15.833

The perimeter is about 15.8 units.

2. Find the perimeter of XYZ with vertices X(1, 3), Y(3, -4), and Z(-4, 1) to the nearest tenth. Extra Examples at pre-alg.com

Lesson 9-5 The Distance Formula

493

TRAVEL The Yeager family is visiting Washington, D.C. A unit on the coordinate system of their map shown at the right is 0.05 mile. Find the distance between the Department of Defense at (-2, 9) and the Madison Building at (3, -3). (x2 - x1)2 + (y2 - y1)2 √

Benjamin Banneker helped to survey and lay out Washington, D.C. He also made all the astronomical and tide calculations for the almanac he published. Source: World Book

6 4

U.S. Capitol

0

[3 - (-2)]2 + (-3 - 9)2 √ d = √ (5)2 + (-12)2

Add.

d = √ 25 + 144

Simplify.

d = √ 169 or 13

Take the square root.

d=

Real-World Link

8

2

Use the distance formula. d=

Department of Defense

–2

Madison Building

–4 –4

–2

0

2

4

6

The distance between the two buildings is 13 units on the map. Since each unit equals 0.05 mile, the distance between the two buildings is 0.05 · 13 or 0.65 mile.

3. TRAVEL Find the distance between the Madison Building at (3, -3) and the U.S. Capitol at (0, 0). Personal Tutor at pre-alg.com

Example 1 (p. 493)

Find the distance between each pair of points. Round to the nearest tenth, if necessary. 1. A(-1, 3), B(8, -6)

Example 2 (p. 493)

Example 3 (p. 494)

HOMEWORK

HELP

For See Exercises Examples 5–10 1 11, 12 2 13, 14 3

3. GEOMETRY Triangle EFG has vertices E(1, 4), F(-3, 0), and G(4, -1). Find the perimeter of EFG to the nearest tenth. 4. ARCHAEOLOGY An archaeologist creates a coordinate system to record where artifacts were discovered. A unit on the grid represents 5 feet. Find the distance between two artifacts if one artifact was found at (-3, 1) and the other was found at (-6, -5) on the grid. Round to the nearest tenth.

Find the distance between each pair of points. Round to the nearest tenth, if necessary. 5. J(5, -4), K(-1, 3)

6. C(-7, 2), D(6, -4)

7. E(-1, -2), F(9, -4)

8. V(8, -5), W(-3, -5)

9. S(-9, 0), T(6, -7)

494 Chapter 9 Real Numbers and Right Triangles The Granger Collection, NY

2. M(4, -2), N(-6, -7)

10. M(0, 0), N(-7, -8)

GEOMETRY Find the perimeter of each figure. 11.

12.

y

X (⫺2, 3)

y A (4, 4)

Y (3, 0)

O

x

O

x

C (⫺2, ⫺2) B (1, ⫺4)

Z (⫺2, ⫺4)

13. LANDSCAPING Len set up a coordinate system with units of feet to locate the positions of his flowers. He planted hostas at (1, 5) and a rose bush at (-6, 3). How far apart are the two plants? Round to the nearest tenth of a foot. Real-World Link The first U.S. public zoo was established in Philadelphia in 1874. Source: philadelphiazoo.org

14. ZOO Beth is looking at a map of the zoo that is laid out on a coordinate system. Beth is at (1, -1). The gorilla house is at (-2, -4) and the reptile exhibit is at (3, 2). Is Beth closer to the gorilla house or the reptile exhibit? 15. GEOMETRY Determine whether MNP with vertices M(3, -1), N(-3, 2), and P(6, 5) is isosceles. Explain your reasoning. Find the distance between each pair of points. Round to the nearest tenth, if necessary.

(

) (

)

16. Q 51, 3 , R 2, 61 2 4 18. F(6.5, 3.2), G(-5.1, 9.3)

(

) (

)

17. A -21, 0 , B -83, -61 2

4

4

19. X(-0.4, -4.8), Y(1.8, -8.8)

20. GEOMETRY Is ABC with vertices A(8, 4), B(-2, 7), and C(0, 9) a scalene triangle? Explain.

EXTRA

PRACTICE

21. DARTS Darnell’s first dart lands 2 inches to the right and 7 inches below the bull’s-eye. What is the distance between the bull’s-eye and where his first shot hit the target? Round to the nearest tenth of an inch.


H.O.T. Problems

22. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the distance between two points. 23. OPEN ENDED Give the coordinates of the endpoints of a line segment that is neither horizontal nor vertical and has a length of 5 units. 24. CHALLENGE Find the values of x if the distance between (1, 2) and (x, 7) is 13 units. 25. SELECT A TECHNIQUE In a golf tournament, Joan’s ball landed 2 feet to the left and 3 feet short of the cup. Carolina’s ball landed 1 foot to the right and 4 feet beyond the cup. Which of the following techniques would you use to determine who is closer to the cup? Justify your selection(s). Then use the technique(s) to solve the problem. mental math

number sense

estimation

Lesson 9-5 The Distance Formula Dan Loh/AP/Wide World Photos

495

26.

Writing in Math How is the Distance Formula related to the Pythagorean Theorem? Include a comparison of the expressions (x2 - x1) and (y2 - y1) to the lengths of the legs of a right triangle.

27. Which expression shows how to find the distance between points M and N?

28. What is the distance between S and T in quadrilateral RSTU? Round to the nearest tenth.

Y

Y

4

. Ó] Î®

3 X

"

- x] Î®

X

"

2

A B C D

(2 - 5)2 + (3 - 3)2 √ [2 - (-5)]2 + [3 - (-3)]2 √ [2 - (-5)]2 + (3 - 3)2 √ (3 - 2)2 + [3 - (-5)]2 √

5

F 4.5

H 5.7

G 5.4

J 10.8

Find the length of the hypotenuse in each right triangle. Round to the nearest tenth, if necessary. (Lesson 9-4) 29.

30. 6 ft

31.

24 yd

c ft

12 km

9 yd c yd

c km

33 km

7 ft

32. ALGEBRA The measures of the angles of a triangle are in the ratio 1:4:5. Find the measure of each angle. (Lesson 9-3) 33. What number is 56% of 85? (Lesson 6-8) 34. SPACE The table shows the mass of 4 planets. About how many times bigger is Jupiter than Mercury? (Lesson 4-7)

0LANET

-ASS KG

%ARTH

*UPITER

-ERCURY

.EPTUNE -ÕÀVi\ .!3!

PREREQUISITE SKILL Solve each proportion. (Lesson 6-3) 84 _ a 4 7 12 35. _ =_ 36. _ =_ 37. _ =m x 16

15

60


52

13

2.8 4.2 38. _ =_ h

12

9-6

Similar Figures and Indirect Measurement

Main Ideas • Identify corresponding parts and find missing measures of similar figures.

Have you ever used a copy machine to make an enlargement or reduction of a drawing? In this activity, you will draw enlargements and reductions. In mathematics, these are known as dilations.

• Solve problems involving indirect measurement using similar triangles.

Step 1 On grid paper, draw a rectangle whose length is 5 inches and whose width is 2 inches. This is the original figure.

New Vocabulary similar figures indirect measurement

IN IN

Step 2 Use a scale factor of 1.5. Draw a new rectangle whose length is 1.5 × 5 inches and whose width is 1.5 × 2 inches. a. What are the dimensions of the new rectangle? b. Is the new rectangle an enlargement or a reduction? c. Repeat Steps 1 and 2 with a right triangle whose legs measure 3 inches and 4 inches. 1 . Use a scale factor of _

IN

3

d. What are the dimensions of the new triangle?

IN

e. Is the new triangle an enlargement or a reduction? f. Compare the measures of the angles of each pair of figures. Do you notice any patterns? g. How do the lengths of the sides of the figures compare? Do you notice any patterns? h. MAKE A CONJECTURE Repeat Steps 1 and 2 using different figures and different scale factors. What kinds of scale factors result in an enlargement? A reduction?

Corresponding Parts Figures that have the same shape but not Reading Math Similar The symbol ∼ is read is similar to.

necessarily the same size are called similar figures. Figure ABCD is similar to figure EFGH. This is written as figure ABCD ∼ figure EFGH, with both sets of vertices listed in the same order. & " '

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# $

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Similar figures have corresponding angles and corresponding sides. Arcs are used to show congruent angles. Lesson 9-6 Similar Figures and Indirect Measurement

497

Corresponding Parts of Similar Figures If two figures are similar, then • the corresponding angles have the same measure, and • the corresponding sides are proportional.

Words

Y

B

ABC ∼ XYZ

Model A

C

X

Z

BC AC AB ∠A ∠X, ∠B ∠Y, ∠C ∠ Z and _ = _ = _

Symbols

YZ

XY

XZ

Since corresponding sides are proportional, you can use a proportion or the direct variation equation to determine the measures of the sides of similar figures when some measures are known.


Find Measures of Similar Figures

The figures are similar. Find each missing measure. a.

J K

5 cm

b.

R

3 cm

2 Ón vÌ

M 9 cm

5

4

8 S

Scale Factor ratio of a length on a polygon to the corresponding length on a similar polygon (Lesson 6-4)

JM

x cm

{ vÌ

x = 15

Find the cross products.

y = kx

Simplify.

d = 1(49) Substitute. d=7

Mentally divide each side by 3.

1B.

)

, -

{ V (

x V

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0

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3 Ó{ V

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Multiply.

L V

Ç V

> V

Direct variation equation

7

£ä V

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:

Replace JM with 3, RT with 9, KM with 5, and ST with x.

1A. '

9

Write a proportion.

3= 5 9 x

3x = 45

7

The scale factor that relates RSTU 4 or 1. Use the scale to MNOP is 7 28 factor to relate dimensions in MNOP, x, to dimensions in RSTU, y.

KM

= RT ST

3·x=9·5

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T

The corresponding sides are proportional.

Review Vocabulary

3

{ vÌ

1

2 Extra Examples at pre-alg.com

Indirect Measurement The properties of similar triangles can be used to find measurements that are difficult to measure directly. This kind of measurement is called indirect measurement.

MAPS In the figure, ABE ∼ DCE. Find the distance across the lake.

Fox Lane

64 = 1(d) 2 Real-World Career Cartographer A cartographer gathers geographic, political, and cultural information and then uses this information to create graphic or digital maps of areas.

2(64) = d

Gazelle Road

64 yd

The scale factor that relates DCE to ABE or 1. is 48 96 2 y = kx

A E B 48 yd

C 96 yd

Direct variation equation.

d yd

Substitute. Multiply each side by 2.

D

128 = d XM

1

2. In the figure, STU ∼ VQU. Find the distance across the river.

6

M

5 3


M

M 4


MEMORIALS The lead statue of the Korean War Memorial in Washington, D.C., casts a 43.5-inch shadow at the same time a nearby tourist casts a 32-inch shadow. If the tourist is 64 inches tall, how tall is the lead statue? Explore You know the lengths of the shadows and the height of the tourist. You need to find the statue’s height. Shadow Reckoning

Plan

Write and solve a proportion.

Solve

tourist’s height 64 32 = statue’s height 43.5 h

Using similar triangles to solve problems involving shadows is called shadow reckoning.

64 · 43.5 = 32 · h 2784 = 32h 87 = h

tourist’s shadow statue’s shadow Find the cross products. Multiply. Divide each side by 32.

The height of the statue is 87 inches or 7 feet 3 inches.

Check

The tourist’s height is 2 times the length of his or her shadow. The statue should be 2 times its shadow, or 2 · 43.5, which is 87 inches.

3. MONUMENTS Suppose a bell tower casts a 27.6-foot shadow at the same time a nearby tourist casts a 1.2-foot shadow. If the tourist is 6 feet tall, how tall is the tower? Lesson 9-6 Similar Figures and Indirect Measurement Geoff Butler

499

Example 1 (p. 498)

The figures are similar. Find each missing measure. 1.

2. *

R C

+

£Ó °

Ç °

6 ft

S

° ,

1

0

ÓÇ °

T

x ft

Ý °

9 ft

D

Example 2 (p. 499)

.

E

15 ft

3. MAPS In the figure, ABC ∼ EDC. Find the distance from Austintown to North Jackson.

North Jackson

C 6 km Ellsworth

A

(p. 499)

HOMEWORK

HELP

For See Exercises Examples 5–10 1 11, 12 3 13, 14 2

B

18 km

4. SHADOWS At the same time a 10-foot flagpole casts an 8-foot shadow, a nearby tree casts a 40-foot shadow. How tall is the tree?

The figures are similar. Find each missing measure. 5.

6.

Y

H

$ Óx

5 in. 10 in.

X

4 in.

Z

£ä

Ý

#

!

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+ *

G

x in.

7. *

£Ó vÌ

Ý vÌ

/

,

&

8.

£x vÌ

'

n vÌ

J

M

È vÌ

.

Q K 6 in.

R 9 in.

9.

{

x in.

9 in.

(

£ä

C

R 2 km S K 7.5 km

D

3 ft

P

10.

4.2 ft

x km

E x ft

A 7 ft

B

U

9 km

V

For Exercises 11 and 12, write a proportion. Then determine the missing measure. 11. RIDES Suppose a roller coaster casts a shadow of 31.5 feet. At the same time, a nearby Ferris wheel casts a 19-foot shadow. A sign says the roller coaster is 126 feet tall. How tall is the Ferris wheel?


, È

J

Óä vÌ

Similar triangles and indirect measurement are used to solve problems about the height of structures. Visit pre-alg.com to continue work on your project.

Austintown x km E

D

2 km Deerfield

Example 3

/

ÎÈ °

(

12. ANIMALS At the same time a baby giraffe casts a 3.2-foot shadow, a 15-foot adult giraffe casts an 8-foot shadow. How tall is the baby giraffe? 13. PARKS How far is the pavilion from the log cabin?

14. ZOO How far are the gorillas from the cheetahs? 84 m

lake fish 51 yd

gorillas

48 yd

xm cabin cheetahs

flower garden 64 yd

35 m reptiles

x yd

birds

28 m pavilion

Determine whether each statement is sometimes, always, or never true. Explain. 15. The measures of corresponding angles in similar figures are the same. 16. Similar figures have the same shape and the same size. Real-World Link The giraffe is the tallest of all living animals. At birth, the height of a giraffe is about 6 feet tall. An adult giraffe is about 18–19 feet tall. Source:

infoplease.com

17. GEOMETRY Triangle LMN is similar to RST. What is the value of LN if RT is 9 inches, MN is 21 inches, and ST is 7 inches? 2 , draw 18. Using a scale factor of _

4 , draw 19. Using a scale factor of _

and label the new image of rectangle ABCD.

and label the new image of triangle XYZ.

3

!

3

"

8

CM

$

IN

#

CM

9

IN

:

PERIMETER AND AREA For Exercises 20–23, the figures are similar. 20. Find the perimeter of both figures. 21. Compare the scale factor of the side lengths and the scale factor of the perimeters. Explain. 22. Find the area of both figures. 23. Compare the scale factor of the side lengths and the scale factor of the areas. Explain. 3

2 }ÕÀi £

5

4

vÌ

8

7 { vÌ

}ÕÀi Ó

:

È vÌ

9

24. SCALE FACTORS Figure FGHJK ∼ figure LMNPQ. The scale factor from figure 3 . What is the perimeter of figure LMNPQ? FGHJK to figure LMNPQ is _ EXTRA

PRACTICE


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2

{ä

,

+

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Îä

*

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(

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Lesson 9-6 Similar Figures and Indirect Measurement Adam Jones/Photo Researchers

501

25. OPEN ENDED Draw two similar triangles whose scale factor is 1. Justify 3 your answer.

H.O.T. Problems

AB where 26. FIND THE ERROR Carla and Tony are finding the length of ABC ∼ DEF, BC = 16 feet, EF = 12 feet, and DE = 18 feet. Who is correct? Explain your reasoning. Carla

Tony

16 x = 12 18

16 = 12 x 18

x = 24 ft

x = 13.5 ft

27. CHALLENGE Triangle ABC has side lengths of 3 inches, 5 inches, and 6 inches. Triangle DEF has side lengths 4 inches, 6 inches, and 8 inches. Determine whether ABC ∼ DEF. Explain. 28.

Writing in Math Suppose you have two triangles. Triangle A is similar to triangle B, and the measures of the sides of triangle A are less than the measures of the sides of triangle B. The scale factor is 0.25. Which is the original triangle? Explain.

29. Quadrilateral ABCD is similar to quadrilateral EFGH. What is the length ? of FG $

&

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n

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"

A 1.5 m

C 4m

B 3.8 m

D 5.3 m

30. Two coordinates for RST are shown. Which coordinates for point T will make MNP and RST similar triangles? Y F T(3, 1)

-

G T(1, -1) H T(1, 2) J T(1, 1)

.

" 0

X

Find the distance between each pair of points. Round to the nearest tenth, if necessary. (Lesson 9-5) 31. S(2, 3), T(0, 6)

32. E(-1, 1), F(3, -2)

33. W(4, -6), V(-3, -5)

If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. (Lesson 9-4) 34. a = 10, b = 15, c = ?

35. a = 8, b = ?, c = 34

36. a = ?, b = 27, c = 82

37. GEOMETRY If each side has a length of 1 foot, find the perimeter of a figure with 9 pentagons. (Lesson 3-8)


2

3

CH

APTER


9



3IGHT S 5RIANGLE

Key Concepts Squares and Square Roots

(Lesson 9-1)

• A square root of a number is one of two equal factors of the number.


(Lesson 9-2)

• Numbers that cannot be written as terminating or repeating decimals are called irrational numbers. • The set of rational and the set of irrational numbers together make up the set of real numbers.

acute angle (p. 477)

radical sign (p. 464)

acute triangle (p. 478)

real numbers (p. 469)

congruent (p. 478)

right angle (p. 477)

converse (p. 487)

right triangle (p. 478)

equilateral triangle (p. 478)

scalene triangle (p. 478)

hypotenuse (p. 485)

similar figures (p. 497)

irrational numbers (p. 469)

solving a right triangle

isosceles triangle (p. 478)

(p. 486)

legs (p. 485)

square root (p. 464)

line segment (p. 476)

straight angle (p. 477)

obtuse angle (p. 477)

triangle (p. 476)

obtuse triangle (p. 478)

vertex (p. 476)

perfect square (p. 464)

Triangles (Lesson 9-3) • An acute angle has a measure between 0° and 90°.

Vocabulary Check Complete each sentence with the correct term. Choose from the list above.

• A right angle measures 90°. • An obtuse angle has a measure between 90° and 180°.

1. A(n) ___?_____ triangle has one angle with a measurement greater than 90°.

• A straight angle measures 180°.

2. A(n) ___?_____ has all sides congruent.

• Triangles can be classified by their angles as acute, obtuse, or right and by their sides as scalene, isosceles, or equilateral.

The Pythagorean Theorem •

c2

=

a2

+

(Lesson 9-4)

b2


3. In a right triangle, the side opposite the right angle is the ___?_____. 4. Figures that have the same shape but not necessarily the same size are called ___?_____. 5. A(n) ___?___ is a square of a whole number.

(Lesson 9-5)

2 2 • d = 兹(x 2 - x1) + (y2 - y1)

Similar Figures and Indirect Measurement (Lesson 9-6) • If two figures are similar, then the corresponding angles have the same measure, and the corresponding sides are proportional.


6. A triangle with no sides congruent is ___?_____. 7. ___?_____ polygons have the same shape and the same size. 8. Decimals that do not repeat or terminate are called ___?_____. 9. The ___?_____ of a right triangle are adjacent to the right angle.


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Squares and Square Roots

(p. 464–468)

Find each square root. 10. √ 36 11. √ 100 12. - √ 81 13. ± √ 121 14. √ 484 15. - √ 225 16. GARDENING Each tomato plant needs 3 square feet of space to grow. The gardener wants to have a garden that is a square and that is large enough for 27 tomato plants. What is the area of the garden?

49 . Example 1 Find - √ - √ 49 indicates the negative square root of 49. 49 , = -7. Since 72 = 49, - √ . Example 2 Find ± √256 ± √ 256 indicates both square roots of 256. 256 = 16 and Since 162 = 256, √ 256 = -16. - √

17. CLOCKS The period of a pendulum is the time required for it to make one complete swing back and forth. The formula of the period P of a pendulum , where is the length of is P = 2 32 the pendulum in feet. If a pendulum in a clock tower is 8 feet long, find the period. Use 3.14 for .

√

9–2


(p. 469–474)

Solve each equation. Round to the nearest tenth, if necessary. 18. n2 = 81 19. t2 = 38 20. 4y2 = 5.76 21. 37.5 = 5r 2 22. SPORTS To find the height h in meters of an object hit into the air, use the formula h = -4.9t 2 + 30t + 1.4, where t is the time in seconds. What height would a baseball reach 3 seconds after it is hit? 23. BUSINESS Kyle owns a business selling baseball cards on the internet. The function y = x2 + 50x + 1800 models the profit y that Kyle has made in month x for the first two years of his business. What is Kyle’s profit in month 15? 504 Chapter 9 Real Numbers and Right Triangles

Example 3 Solve x 2 = 72. Round to the nearest tenth. x2 = 72 √x2 = √ 72

Write the equation. Take the square root of each side.

x = √ 72 or x = - √ 72

Find the positive and negative square root.

x ≈ 8.5 or x ≈ -8.5

Simplify.



9–3

Triangles

(p. 476–481)

Example 4 Classify the triangle by its angles and by its sides.

Classify each triangle by its angles and by its sides. 25. 24. 132˚ 60˚ 24˚ 60˚

H 80˚ 24˚

60˚

J

80˚

K

26.

27. 56˚

Óx Èä

34˚

28. SIGNS Classify the yield sign by its angles and by its sides.

9–4

The Pythagorean Theorem

x

Triangle HJK has all acute angles and two congruent sides. So, HJK is an acute isosceles triangle.

YIELD

(p. 485–490)

If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary.

Example 5 Find the missing measure of the right triangle. 22

29. a = 6, b = ?, c = 15

9

30. a = ?, b = 2, c = 7 31. a = 18, b = ?, c = 24 32. BASEBALL On a baseball diamond, the bases are 90 feet apart. What is the distance from home plate to second base in a straight line? 33. TRAVEL Tiananmen Square in Beijing, China is the largest town square in the world, covering 95 acres. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot?

b

c2 = a2 + b2

Pythagorean Theorem

222 = 92 + b2

Replace c with 22 and a with 9.

484 = 81 + b2

Simplify.

403 = b2


20.1 ≈ b



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9 9–5



(p. 492–496)

Example 6 Find the distance between A(-4, 0) and B(2, 5).

Find the distance between each pair of points. Round to the nearest tenth, if necessary.

(x2 - x1)2 + (y2 - y1)2 Distance Formula d = 兹

34. J(0, 9), K(2, 7) 35. A(-5, 1), B(3, 6) 36. W(8, -4), Y(3, 3) 37. G(0, 0), H(3, 4) 38. AIRPORTS A distance of 3 units on the grid equals an actual distance of 1 mile. Suppose the locations of two airports on a map are at (121, 145) and (218, 401). Find the actual distance between these airports to the nearest mile.

9–6

Similar Figures and Indirect Measurement In Exercises 39 and 40, the figures are similar. Find each missing measure. 39.

= 兹 [2 - (-4)]2 + (5 - 0)2

(x1, y1) = (-4, 0), (x2, y2) = (2, 5)

= 兹 (6)2 + 5 2

Subtract.

= 兹 61

Simplify.

≈ 7.8

Simplify.

(p. 497–502)

Example 7 If ABC ∼ KLM, what is the value of x?

M

24 m

L

21 m 8m

L

N

40.

B

U

V

A x in. C

B x ft

BC AC _ =_

6 ft

A

LM KM x _ = _2 3 4

C J 15 ft

H

2 in.

xm

T

4 in.

x = 1.5

9 ft

K

41. WORLD RECORDS At 7 feet 8 inches, the world’s tallest woman casts a 46-inch shadow. At the same time, the world’s shortest woman casts an 18-inch shadow. How tall is the world’s shortest woman? 506 Chapter 9 Real Numbers and Right Triangles

K

3 in.

M

Write a proportion. Substitution Find cross products and simplify.

CH

A PT ER

9

Practice Test 15. HIKING Brandon hikes 7 miles south and 4 miles west. How far is he from the starting point of his hike? Round to the nearest tenth.

Find each square root, if possible. 2. - √ 121 4. √ a2

1. √256 3. ± √ 49

5. Without using a calculator, estimate - √ 42 to the nearest whole number.

16. LADDER There is a building with a 12-foot high window. You want to use a ladder to reach the window. If the bottom of the ladder is 5 feet away from the building, will a 15-foot ladder reach the window? Explain.

ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 6. x2 = 100 7. w2 = 39

FT

8. 4.5u2 = 306

FT

For Exercises 9 and 10, use the triangle shown below. Find the distance between each pair of points. Round to the nearest tenth, if necessary.

-

17. A(3, 8), B(-5, 2) 18. Q(-6, 4), R(6, -8) .

nÇc

ÎÓc

19. C(5, 9), D(-7, 3)

0

9. Find the measure of ∠M. 10. Classify MNP by its angles and by its sides.

20. MULTIPLE CHOICE In the map of the park, the triangles are similar. Find the distance to the nearest tenth from the playground to the swimming pool.

If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary.

45 ft

50 ft 36 ft

11. a = 6 yd, b = 8 yd, c = ?

Playground

12. a = 15 cm, b = ?, c = 32 cm 13.

Swimming Pool

x ft

14. V vÌ

£ä vÌ

V

Îä

A 60.5 ft B 62.5 ft

£Ó vÌ

C 63.1 ft £È


D 64.2 ft


507

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9


Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. GRIDDABLE Triangle ABC is similar to N ? LMN. What is the length, in yards, of L

9 yd

N

C

2. Mrs. Hamilton wants to have a circular swimming pool installed in her backyard, but it would intersect part of an elevated pathway. Find x, the inside length of the section of pathway that would intersect the pool. x 20 ft

1

2

3

4

F point P

J point S

B

A

0

H point R

12 yd

L

PQ R S

G point Q

M

8 yd

3. Which point on the number line shown is 7? closest to √

16 ft

4. A salesperson earned a commission of $36 on a sale worth $800. Which statement below identifies a commission that is the same rate? A a commission of $60 on a $1200 sale B a commission of $77 on a $1400 sale C a commission of $27 on a $600 sale D a commission of $35 on a $750 sale 5. Molly multiplied her age by 2, subtracted 3, divided by 9, and added 4. The result was 7. Which could be the first step in finding Molly’s age? F Add 2 and 3. G Multiply 3 by 9. H Subtract 4 from 7. J Divide 7 by 4.

A 12 ft

C 28 ft

B 24 ft

D 32 ft

6. In the equation 2s - t = s + 3t, which would NOT be an appropriate first step to solve the 1? equation for t if s = 2 A Divide each side by 2s. 1. B Replace s with 2

C Add t to each side. Pace yourself Do not spend too much time on any one question. If you’re having difficulty answering a question, mark it in your test booklet and go on to the next question. Make sure that you also skip the question on your answer sheet. At the end of the test, go back and answer the question that you skipped.


D Subtract s from each side. 7. GRIDDABLE Admission to the carnival is $6 and each ride is $0.50. If Hector wants to spend no more than $12 at the carnival, what is the maximum number of rides on which he can ride? Standardized Test Practice at pre-alg.com


4 7? 8. Which fraction is between and 3 F 4 5 G 6

12. GRIDDABLE Heather spent $172 on concert tickets. She bought 3 floor seats for $35.50 and 2 mezzanine seats. How much in dollars did the mezzanine seats cost?

8

5 8 H 9 11 J 12

Pre-AP

9. Stan has 40 baseball cards in his collection. He plans to add another 2 cards each week until he has doubled the amount in his collection. Which equation can be used to determine w, the number of weeks it will take to double the size of the baseball card collection? A B C D

13. The walls of a house usually meet to form a right angle. You can use string to determine whether two walls meet at a right angle. a. Copy the diagram shown below. Then illustrate the following situation.

40 × 2w = 80 2w + 40 = 80 w + 40 = 80 2w + 40 = 42

10. The table represents a function between x and y. What is the missing number in the table? F G H J

Record your answers on a sheet of paper. Show your work.

4 5 6 7

x

y

1

3

2

?

4

9

6

13

From a corner of the house, a 6-foot-long piece of string is extended along one side of the wall, parallel to the floor. From the same corner, an 8-foot-long piece of string is extended along the other wall, parallel to the floor. b. If the walls of the house meet at a right angle, then what is the distance between the ends of the two pieces of string? c. Draw an example of a situation where two walls of a house meet at an angle whose measure is greater than the measure of a right angle. d. Suppose the length of the walls in part c are the same length as the walls in part a, and that the 6-foot and 8-foot pieces of string are extended from the same corner. Do you think the distance between the two ends will be the same as in part b? Explain your reasoning.

11. One side of a garden is against a house as shown. If 88 feet of fencing will be used to enclose the garden, what is the width, w, of the garden? (2w 6) ft w

A 40 B 34

C 30 D 20


1

2

3

4

5

6

7

8

9

10

11

12

13

Go to Lesson...

9-6

9-4

9-1

6-1

1-1

8-1

8-6

5-2

7-1

7-2

8-2

3-5

9-4


509

10 •

Identify the relationships of parallel and intersecting lines.

•

Identify properties of congruent triangles.

• •

Identify and draw transformations.

•

Find the area of polygons and irregular figures, and find the area and circumference of circles.


Classify and find angle measures of polygons.

Key Vocabulary circle (p. 551) composite figures (p. 558) parallel lines (p. 512) transformation (p. 524)

Real-World Link Art The noses of 10 graffiti-covered Cadillacs are halfburied in a field west of Amarillo, Texas. The angles that the cars make with the ground are corresponding angles.

Compare and Contrast Polygons Make this Foldable to help you organize information about the characteristics of two-dimensional figures. Begin with four plain sheets of 11˝ × 17˝ paper, eight index cards, and glue.

1 Fold in half widthwise.

2 Fold the bottom to form a pocket. Glue the edges.

3 Repeat three times. Then glue all four pieces together to form a booklet.

510 Chapter 10 Two-Dimensional Figures Robyn Beck/Getty Images

4 Label each pocket. Place an index card in each pocket. LE S 4RAPE 4R IA N G ZOIDS E



Option 1


Solve each equation. (Lesson 3-3) 1. x + 46 = 90 2. x + 35 = 180

Example 1

Solve 7x - 2 = -72.

3. 2x - 12 = 90

4. 3x - 24 = 180

7x - 2 = -72

Write the equation.

5. 5x + 165 = 360

6. 4x + 184 = 360

7x - 2 + 2 = -72 + 2

Add 2 to each side.

7. TRAVEL Ernesto drove the same number of miles each day from Monday through Friday, and 26 miles on the weekend. If he drove a total of 176 miles during the week, how many miles does he drive each workday? (Lesson 3-3)

Find each product. Round to the nearest tenth, if necessary. (Prerequisite Skills, pp. 747–748) 8. (5.5)(8) 9. (7.5)(3.4) 10. (6.3)(11.4)

1 11. _ (8)(2.5)

12. (2)(3.14)(1.7)

13. 2(3.1)(3.14)

2

7x = - 70

Simplify.

7x _ _ = -70


7

7

x = -10

Simplify.

Example 2

Find (0.5)(3)(6.25). Round to the nearest tenth. (0.5)(3)(6.25) = [(0.5)(3)](6.25) Multiply 0.5 × 3 first.

14. FOOD Nicole poured 32.5 bowls of soup volunteering her time in a soup kitchen. If each bowl contained 16.5 ounces of soup, how much soup did Nicole pour?

= 9.375

Simplify.

≈ 9.4


(Prerequisite Skills, pp. 747–748)

Find each sum. (Lesson 5-7) 3 1 2 1 15. 5_ + 4_ 16. 2_ + 3_ 2 3 3 1 17. 1_ + 2_ 8 2 2 _ _ 19. 2 + 3 5 3 9

3 4 5 1 18. 6_ + 1_ 6 4 2 _ _ 20. 5 + 3 4 3 5

5 pounds 21. RECYCLING The class collected 12_ 6 1 _ of bottles and 8 pounds of aluminum 8

cans. How many pounds of glass and aluminum cans did the class collect? (Lesson 5-7)

Example 3 3 5 Find 1_ + 4_ .

6 4 3 5 29 7 1 _ + 4_ = _ +_ 6 6 4 4 3 29 2 7 _ =_ · +_·_ 6 2 4 3 58 21 =_ +_ 12 12 79 =_ 12 7 = 6_ 12

Write as improper fractions. Rename using the LCD, 12. Simplify. Add the numerators. Simplify.


511

10-1

Line and Angle Relationships

Main Ideas • Identify the relationships of angles formed by two parallel lines and a transversal.

A satellite dish receives signals from a satellite and directs them into a receiver. The intersecting lines from the signal lines form different angle relationships.

• Identify the relationships of vertical, adjacent, complementary, and supplementary angles.

a. What do you notice about the lines coming into the satellite dish?

New Vocabulary parallel lines transversal interior angles exterior angles alternate interior angles alternate exterior angles corresponding angles vertical angles adjacent angles complementary angles supplementary angles perpendicular lines

2

1 3 4

5 6 8 7

b. Trace the red lines onto a piece of paper. Find the measure of each numbered angle. c. What do you notice about the measures of the angles? Which angles have the same measure?

Parallel Lines and a Transversal In geometry, two lines in a plane that never intersect are parallel lines. Lines m and n are parallel. Using symbols, m n.

m

Parallel lines have no point of intersection.

n

When two parallel lines are intersected by a third line called a transversal, eight angles are formed.

Names of Special Angles The eight angles formed by parallel lines and a transversal have special names. • Interior angles lie inside the parallel lines. ∠3, ∠4, ∠5, ∠6 • Exterior angles lie outside the parallel lines. ∠1, ∠2, ∠7, ∠8 • Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. ∠3 and ∠5, ∠4 and ∠6


1

2 4

3 5

6 8

7

Arrowheads are often used in figures to indicate parallel lines.

• Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. ∠1 and ∠7, ∠2 and ∠8 • Corresponding angles are in the same position on the parallel lines in relation to the transversal. ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8

512 Chapter 10 Two-Dimensional Figures Peter Pearson/Stone/Getty Images

transversal

In Lesson 9-3, you learned that line segments are congruent if they have the same measure. Similarly, angles are congruent if they have the same measure. Parallel Lines Cut by a Transversal If two parallel lines are cut by a transversal, then • corresponding angles are congruent, • alternate interior angles are congruent, and • alternate exterior angles are congruent.

EXAMPLE

Find Measures of Angles

In the figure at the right, m n and s and t are transversals. If m∠1 = 68°, find m∠5 and m∠6.

m

Since ∠1 and ∠5 are corresponding angles, they are congruent. So, m∠5 = 68°.

n

1 3 7 4

Since ∠1 and ∠6 are alternate exterior angles, they are congruent. So, m∠6 = 68°.

9 11 15 12

5 2 8 6 13 10

s t

16 14

1. If m∠11 = 84°, find m∠10 and m∠16.

Reading Math Congruent Angles Angle 1 is congruent to angle 2. This is written ∠1 ∠2. The measure of ∠1 is equal to the measure of ∠2. This is written m∠1 = m∠2.

Intersecting Lines and Angles Other pairs of angles have special relationships. When two lines intersect, they form two pairs of opposite angles called vertical angles. Vertical angles are congruent. The symbol for is congruent to is . ∠1 and ∠2 are vertical angles. ∠1 ∠2

1

3 4

2

∠3 and ∠4 are vertical angles. ∠3 ∠4

When two angles have the same vertex, share a common side, and do not overlap, they are adjacent angles. m∠AOB = m∠1 + m∠2

∠1 and ∠2 are adjacent angles.

A 2

1

B

O

If the sum of the measures of two angles is 90°, the angles are complementary.

4 3 1 2

m∠1 = 50°, m∠2 = 40° m∠1 + m∠2 = 90° Extra Examples at pre-alg.com

m∠3 = 60°, m∠4 = 30° m∠3 + m∠4 = 90° Lesson 10-1 Line and Angle Relationships

513

If the sum of the measures of two angles is 180°, the angles are supplementary.

1

2

3

4

m∠1 = 140°, m∠2 = 40° m∠1 + m∠2 = 180°

m∠3 = 45°, m∠4 = 135° m∠3 + m∠4 = 180°

Lines that intersect to form a right angle are perpendicular lines.

right angle

TILING Jun cuts a piece of tile at a 135° angle. What is the measure of the other angle formed by the cut?

Xª

ª

The angles at the cut point are supplementary. m∠x + 135 = 180

Write the equation.

m∠x + 135 - 135 = 180 - 135 m∠x = 45°

Subtract 135 from each side. Simplify.

2. ARCHITECTURE In the semicircular window, ∠1 is complementary to ∠2. If m∠2 is 24°, find m∠1.


EXAMPLE

Find Measures of Angles

ALGEBRA Angles ABC and FGH are complementary. If m∠ABC = x + 8 and m∠FGH = x - 10, find the measure of each angle. Step 1 Find the value of x. Check your Answer To check your answer, add to see if the sum of the measures of the angles is 90. Since 54 + 36 = 90, the answer is correct.

m∠ABC + m∠FGH = 90 Complementary angles (x + 8) + (x - 10) = 90 2x - 2 = 90 2x = 92 x = 46

Substitution Combine like terms. Add 2 to each side. Divide each side by 2.

Step 2 Replace x with 46 to find the measure of each angle. m∠ABC = x + 8

m∠FGH = x - 10

= 46 + 8 or 54

= 46 - 10 or 36

So, m∠ABC = 54° and m∠FGH = 36°.

3. ALGEBRA Angles MNO and RST are supplementary. If m∠MNO = 5x and m∠RST = x - 6, find the measure of each angle. 514 Chapter 10 Two-Dimensional Figures

SAFETY A lifeguard chair is shown. If m∠1 = 105°, find m∠4 and m∠6. Since ∠1 and ∠4 are vertical angles, they are congruent. So, m∠4 = 105°.

1 6 7 4

Since ∠6 and ∠1 are supplementary, the sum of their measures is 180°.

3 8 5 2

180 - 105 = 75. So, m∠6 = 75°.

4. Find m∠2 and m∠3 in the lifeguard chair above. Explain your reasoning.


Parallel Lines

Perpendicular Lines

Vertical Angles

n 1

ab

a

m ⊥n

m

2 4

∠1 ∠3

3

∠2 ∠4

b

Adjacent Angles

Reading Math

1

A

1

2

(pp. 513, 515)

1

2

B

m∠ABC = m∠1 + m∠2

Examples 1 and 4

Supplementary Angles

C

D

Adjacent Angles Angles do not have to be adjacent to be complementary or supplementary angles.

Complementary Angles

m∠1 + m∠2 = 90°

2

m∠ + m∠2 = 180°

In the figure at the right, m and k is a transversal. k If m∠1 = 56°, find the measure of each angle. 1. ∠2

2. ∠3

ᐉ

3. ∠4

1

2 3

Example 2 (p. 514)

4.

5. 140˚

(p. 514)

4

Find the value of x in each figure. x˚

Example 3

m

152˚

6. x˚

X ÓÈ

7. ALGEBRA If m∠N = 3x and m∠M = 2x and ∠M and ∠N are supplementary, what is the measure of each angle? Lesson 10-1 Line and Angle Relationships

515

HOMEWORK

HELP

For See Exercises Examples 8–13 1, 4 14–21 2 22, 23 3

g

In the figure at the right, g h and t is a transversal. If m∠4 = 53°, find the measure of each angle. 8. ∠1

9. ∠5

10. ∠7

11. ∠8

12. ∠2

13. ∠3

4

7

8

6

t

1

3

2

5

h

Find the value of x in each figure. 14.

15. 45˚

x˚

16.

148˚ x˚

31˚ x˚

17.

x˚

18. 5˚

19. 8x ˚

4x ˚

5x ˚ 5x ˚

20. Find m∠A if m∠B = 17° and ∠A and ∠B are complementary. 21. Angles P and Q are supplementary. Find m∠P if m∠Q = 139°. 22. ALGEBRA Angles J and K are complementary. If m∠J = x - 9 and m∠K = x + 5, what is the measure of each angle? 23. ALGEBRA Find m∠E if ∠E and ∠F are supplementary, m∠E = 2x + 15, and m∠F = 5x - 38. ALGEBRA In the figure at the right, m and t is a transversal. Find the value of x for each of the following.

t

ᐉ

1 4

m

24. m∠2 = 2x + 3 and m∠4 = 4x - 7

5 8

2

3

6

7

25. m∠8 = 4x - 32 and m∠5 = 5x + 50 26. CONSTRUCTION To measure the angle between a sloped cathedral ceiling and a wall, a carpenter uses a plumb line (a string with a weight attached) as shown. If m∠YXB = 68°, what is m∠XBC? Explain your reasoning. 27. ALGEBRA The measure of the supplement of an angle is 15° less than four times the measure of the complement. Find the measure of the angle.

A

X B

Y

C

7

TIME For Exercises 28 and 29, use the clock showing 6 o’clock and 10 seconds. 28. Find m∠WXY and m∠YXZ. 29. Find the time that will show m∠WXY + m∠YXZ = 90°. 516 Chapter 10 Two-Dimensional Figures

9 8 :

30. ALGEBRA Angles R and S are complementary. The ratio of their measures is 4:5. Find the measure of each angle. ANALYZE GRAPHS For Exercises 31–33, use the graphs. y

y y 3x 2 y 2x 3 O

EXTRA

3

y 1x 1 2

See pages 783, 803.

H.O.T. Problems

x

O y 1x 1

PRACTICE


x

31. How are each pair of graphs related? 32. What seems to be true about the slopes of the graphs? 33. MAKE A CONJECTURE about the slopes of the graphs of perpendicular lines. 34. OPEN ENDED Draw a pair of adjacent, supplementary angles. 35. CHALLENGE Suppose two parallel lines are cut by a transversal. How are the interior angles on the same side of the transversal related? 36.

Writing in Math

How are parallel lines and angles related? Illustrate with a drawing of parallel lines intersected by a transversal and a list of the congruent and supplementary angles.

37. The hedge shears have two different sets of angles. Find x.

38. Which angles are NOT supplementary?

A 32

#

B 58

$

" x°

C 102 D 121

! 59°

:

%

F ∠EZC, ∠CZA

H ∠AZB, ∠BZE

G ∠BZC, ∠CZD

J

∠DZE, ∠AZD

39. FARMING At the same time a 40-foot silo casts a 22-foot shadow, a fence casts a 3.3-foot shadow. Find the height of the fence. (Lesson 9-6) 40. ARCHAEOLOGY Two artifacts are found at a dig. If a coordinate plane is set up, one artifact was found at (1, 5) and the other artifact was found at (3, 1). How far apart were the two artifacts? Round to the nearest tenth. (Lesson 9-5) ALGEBRA Solve each inequality. (Lesson 8-5) a 42. _ >3

41. 5m < 5

43. -4x ≥ -16

-2

PREREQUISITE SKILL Use a protractor to draw an angle having each measurement. (pp. 757–758) 44. 20°

45. 45°

46. 65°

47. 145°

48. 170°

Lesson 10-1 Line and Angle Relationships

517

10-2 Main Idea • Identify congruent triangles and corresponding parts of congruent triangles.

Congruent Triangles

Ivy is a type of climbing plant. Most ivy leaves have five major veins. In the photo shown, the outlines form two triangles.

New Vocabulary congruent corresponding parts

a. Trace the triangles shown at the right onto a sheet of paper. Then label the triangles.

C

E

B

F

b. Measure and then compare the lengths of the sides of the triangles. c. Measure the angles of each triangle. How do the angles compare?

D

A

d. Make a conjecture about the triangles.

Congruent Triangles Figures that have the same size and shape are congruent. The parts of congruent triangles that “match” are corresponding parts.

Vocabulary Link Corresponding Everyday Use having the same relationship Math Use having the same position

Corresponding Parts of Congruent Triangles Words

If two triangles are congruent, their corresponding sides are congruent and their corresponding angles are congruent.

Model

1

9

Tick marks are used to indicate which sides are congruent.

Arcs are used to indicate which angles are congruent.

8

:

0

2

Symbols Congruent Angles: ∠X ∠P, ∠Y ∠Q, ∠Z ∠R −− −− −− −− −− −− Congruent Sides: XY PQ, YZ QR, XZ PR

518 Chapter 10 Two-Dimensional Figures SuperStock

When writing a congruence statement, the letters must be written so that corresponding vertices appear in the same order. For example, for the diagram below, write FGH JKM. G

K

FGH J K M

Congruence Statements You can also write a congruence statement

F

H

J

Vertex F corresponds to vertex J. Vertex G corresponds to vertex K. Vertex H corresponds to vertex M.

M

as GHF KMJ, HFG MJK, FHG JMK, GFH KJM, and HGF MKJ.

EXAMPLE

Name Corresponding Parts

Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement.

A

Z

Corresponding Angles B

∠A ∠Z, ∠B ∠Y, ∠C ∠X

C

X

Y

ABC ?

Corresponding Sides −− −− −− −− −− −− AB ZY, BC YX, CA XZ One congruence statement is ABC ZYX. &

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1. Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement DEF ? . $

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Identify Congruent Triangles

Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement.

R 20 m

16 m

Explore The drawing shows which S angles are congruent and the lengths of all sides.

U

24 m

20 m

T

W

24 m

16 m

V

Plan

Note which segments have the same length and which angles are congruent. Write corresponding vertices in the same order.

Solve

Angles: The arcs indicate that ∠S ∠W, ∠R ∠V, and ∠T ∠U. −− −−− WV , RT Sides: The side measures indicate that SR VU, and −−− TS UW. Since all pairs of corresponding angles and sides are congruent, the two triangles are congruent. One congruence statement is SRT WVU. (continued on the next page)


Lesson 10-2 Congruent Triangles

519

Check Draw SRT and WVU so that they are oriented in the same way. Then compare the angles and sides.

R 16 m

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V 16 m

20 m

24 m

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You can use corresponding parts to find the measures of angles and sides in a figure that is congruent to a figure with known measures.

LANDSCAPING A brace is used to support a tree and to help it grow straight. In the figure, TRS ERS. a. At what angle is the brace placed against the ground?

∠E and ∠T are corresponding angles. So, they are congruent. Since m∠T = 65°, m∠E = 65°.

R brace

The brace is placed at a 65° angle with the ground.

8 ft

b. What is the length of the brace? −− −− and RE corresponds to RT. So, RE RT are congruent. Since RT = 8 feet, RE = 8 feet.

65˚

T

E S 3 ft

The length of the brace is 8 feet.

3. QUILTING A quilt design is shown. In the figure, ABC ADE. What is the measure of ∠BCA? What is the perimeter of the design?

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Example 1 (p. 519)

For each pair of congruent triangles, name the corresponding parts. Then complete the congruence statement. 1.

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5. TOWERS A tower that supports highvoltage power lines is shown at the right. In the tower, ADC BFC. FC if AC = 10 feet What is the length of and DC = 15 feet?

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B C

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Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. 10.

11.

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M

M M

M

M 2

1

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Q V

12.

R 4 cm S F

6 cm 21˚

6 cm

G 9 cm

4 cm 32˚ H

Real-World Link Trusses were used in the construction of the Eiffel Tower in Paris, France. The tower contains more than 15,000 pieces of steel and 2.5 million rivets. Source: paris.org

13. (

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104˚

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R ARCHITECTURE For Exercises 14 and center top left chord 15, use the diagram of the roof truss web metal plate at the right. In the figure, TRU connector SRU. 30˚ T 14. Find the distance from the right U metal plate connector to the center bottom chord web. 32 ft 15. What is the measure of the angle formed by the top left chord and the bottom chord?

S

Find the value of x for each pair of congruent triangles. 16. A

12

D

17.

C 16

x

3

2 *

X

20

B

E

4 +

-

Determine whether each statement is true or false. If false, give a counterexample. 18. If two triangles are congruent, then the perimeters are equal. 19. If the perimeters of two triangles are equal, then the triangles are congruent. 20. ALGEBRA If ABC XYZ, what is the value of x?

A

Y 3x ⫹ 12

18

Z B EXTRA

PRACTICE


21. BUTTERFLIES Butterfly wings are triangular in shape. Using the photograph of the butterfly as a model, draw two sets of congruent triangles, label the vertices, and write a congruence statement for each.

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24

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30

H.O.T. Problems

22. FIND THE ERROR Jade and Fernando are writing a congruence statement for the congruent triangles at the right. Who is correct? Justify your reasoning.

3

4 8

2 :

9

Fernando −− −− ST YZ

Jade ∠YXZ ∠STR

D

23. CHALLENGE In the figure at the right, there are two pairs of congruent triangles. Write a congruence statement for each pair. 24.

Explain where congruent triangles are present in nature. Include a definition of congruent triangles and an example of an object in nature that contains congruent triangles.

10

Y U

A

I

H

F

G

F 12 ft

2

G 24 ft H 48 ft

24

Z

A 10

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26. Guy wires create two congruent triangles PQR and SQR. Find the −− length of QS.

W 26

E J

B

Writing in Math

−− 25. Find the measure of UY if XYZ UYW.

X

C

J 65 ft B 20

C 24

D 26

1

0

3

FT

27. Angles P and Q are supplementary. Find m∠P if m∠Q = 129°. (Lesson 10-1) −− 28. Find AC if ABC ∼ DEF, AB = 15, DE = 10, and DF = 5. (Lesson 9-6) 29. GEOMETRY The table shows how the perimeters of an equilateral triangle and a square change as side lengths increase. Compare the rates of change. (Lesson 8-5) 30. FOOD A recipe for butter cookies requires 12 tablespoons 12 as a of sugar for every 16 tablespoons of flour. Write _ 16 fraction in simplest form. (Lesson 4-4)

Perimeter y

Side Length x

Triangle

Square

0

0

0

2

6

8

4

12

16

PREREQUISITE SKILL Graph each point on a coordinate plane. (Lesson 2-6) 31. A(2, 4) 32. J(-1, 3) 33. H(0, 5) 34. D(2, 0) 35. W(-2, -4)


523

10-3

Transformations on the Coordinate Plane

Main Idea • Draw translations, reflections, and dilations on a coordinate plane.

The physical motions used in recreational activities such as skateboarding or riding a scooter are related to mathematics.

New Vocabulary transformation image translation reflection line of symmetry dilation center

a. Describe the motion involved in making a 180° turn on a skateboard. b. What type of motion does a scooter display when moving forward?

Transformations A mapping of a geometric figure that may change its shape or position is a transformation. Every corresponding point on the figure after a transformation is called its image. Three types of transformations are shown below. Translation • In a translation, you slide a figure from one position to another without turning it. Translations are also called slides.

y

x

O

Reflection

• In a reflection, you flip a figure over a line. This line is called a line of symmetry. The figures are mirror images of each other. Reflections are also called flips.

y

line e of symmetry y e

O

x

$ILATION

Look Back To review scale factors, see Lesson 6-4.

• In a dilation, you enlarge or reduce a figure by a scale factor with respect to a fixed point called the center. The resulting image is similar to the original figure. The scale factor in the graph at the right is 3.

Y

CENTER "


X

Reading Math Notation The notation M’ is read M prime. It corresponds to point M.

When translating a figure, every point of the original figure is moved the same distance and in the same direction. The image is congruent to the original figure and the orientation is the same as the original figure. Translation 5 units down

Translation 4 units left y

y

A

y

R'

B

X

X'

Translation 6 units right, 3 units up

R Y'

D

Y

A'

x

O

x

O C

O

x

T'

B'

Z

Z'

S' T

D'

C'

S

To translate a point in the coordinate plane, describe the translation using an ordered pair. Then add the coordinates of the ordered pair to the coordinates of the original point.

y 5 4 3 N 2 1

Triangle MNP is shown on the coordinate plane. Find the coordinates of the vertices of the image of MNP translated 5 units left and 3 units up. A B C D

M'(-1, 1), N'(0, 5), P'(-5, 5) M'(4, 1), N'(0, 5), P'(5, 5) M'(-1, 1), N'(-5, 5), P'(0, 5) M'(-1, -2), N'(-5, 2), P'(0, 2)

⫺4⫺3⫺2⫺1O

P 1 2 3 4x

⫺2 ⫺3

M

Read the Test Item Common Misconception In a translation, the order in which a figure is moved does not matter. For example, moving 3 units down and then 2 units right is the same as moving 2 units right and then 3 units down.

This translation can be written as (-5, 3). To find the coordinates of the translated image, add -5 to each x-coordinate and add 3 to each y-coordinate. Solve the Test Item vertex M(4, -2) N(0, 2) P(5, 2)

5 left, 3 up

+ + +

(-5, 3) (-5, 3) (-5, 3)

translation

→ → →

M’(-1, 1) N’(-5, 5) P’(0, 5)

The answer is C.

1. Triangle ABC is translated so that B is mapped to B’. Which coordinate pair best represents C’? F (-4, 1)

H (-1, 1)

G (0, 3)

J (1, 3)

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Personal Tutor at pre-alg.com Lesson 10-3 Transformations on the Coordinate Plane

525

When reflecting a figure, every point of the original figure has a corresponding point on the other side of the line of symmetry. The image is congruent to the original figure, but the orientation is different from the original figure. To reflect a point over the x-axis, use the same x-coordinate and multiply the y-coordinate by -1. To reflect a point over the y-axis, use the same y-coordinate and multiply the x-coordinate by -1.

Reflection over the x-axis y

Reflection over the y-axis y

E

x

O

I'

E'

EXAMPLE

H

H' F F' x

G O G'

I

J' J

Reflection in a Coordinate Plane

The vertices of a figure are A(-2, 3), B(0, 5), C(3, 1), and D(3, 3). Graph the figure and the image of the figure after a reflection over the x-axis. To find the coordinates of the vertices of the image after a reflection over the x-axis, use the same x-coordinate and multiply the y-coordinate by -1.

B

opposite same

A(-2, 3) B(0, 5) C(3, 1) D(3, 3)

→ → → →

(-2, -1 · 3) (0, -1 · 5) (3, -1 · 1) (3, -1 · 3)

D

A C C' x

O

→ → → →

A'(-2, -3) B'(0, -5) C'(3, -1) D'(3, -3)

A' D'

B'

2. The vertices of polygon DEFG are D(4, -2), E(5, -5), F(2, -4), and G(1, -1). Graph the polygon and the image of the figure after a reflection over the y-axis. The diagrams below show how dilations result in similar figures that are larger and smaller than the original. The center of each dilation is the origin. Scale Factor ⴝ 2 (enlargement) Vocabulary Link Dilate Everyday Use to expand or widen Math Use to enlarge or reduce by a scale factor

3CALE FACTOR

y

Y

B'

REDUCTION

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Suppose k is the scale factor. • If k > 1, the dilation is an enlargement. • If 0 < k < 1, the dilation is a reduction. • If k = 1, the dilation is congruent to the original figure. When the center of a dilation is the origin, you can find the coordinates of the image by multiplying the coordinates of a polygon by the scale factor. 526 Chapter 10 Two-Dimensional Figures


EXAMPLE

Dilation in a Coordinate Plane

A figure has vertices J(2, 4), K(2, 6), M(8, 6), and N(8, 2). Graph the figure and the image of the figure after a dilation centered at the 1 origin with a scale factor of _ .

Y

To dilate the polygon, multiply the coordinates

+g

1 . of each vertex by _

*g

+

2

* -g

2

Corresponding Parts Dilated figures have congruent angles and sides that are proportional.

-

J(2, 4) → J’(1, 2)

K(2, 6) → K’(1, 3)

M(8, 6) → M’(4, 3)

N(8, 2) → N’(4, 1)

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3. A figure has vertices R(-1, 2), S(1, 4), and T(1, 1). Graph the figure and the image of the figure after a dilation centered at the origin with a scale factor of 3. Transformations Translations and Reflections produce images that are the same shape and the same size. The figures are congruent to the images. Dilations produce images that are similar (same shape, but not the same size). The figures are not congruent to the images, except when the scale factor k = 1.

Example 1 (p. 525)

1. MULTIPLE CHOICE Rectangle RSTU has been translated. Which describes the translation?

Y

2

3

A 4 units left, 2 units up B 4 units right, 2 units down

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C 2 units right, 4 units down

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D 2 units left, 4 units up Example 2 (p. 526)

2. Suppose the figure graphed is reflected over the y-axis. Find the coordinates of the vertices after the reflection. 3. Hold your hands in front of you with your palms down. What kind of transformation exists from your left hand to your right hand?

Example 3 (p. 527)

4. Triangle ABC is shown. Graph the image of ABC after a dilation centered at the origin with a scale factor of 2.

X y W

x

O

Y Z

Y

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Lesson 10-3 Transformations on the Coordinate Plane

527

HOMEWORK

HELP


Find the vertices of each figure after the given translation. Then graph the translation image. 5. (2, 3)

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2

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y

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A D H

B x

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x

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x

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8. The vertices of a figure are D(1, 2), E(1, 4), F(-4, 4), and G(-1, 2). Graph the image of the figure after a translation 4 units down. Find the vertices of each figure after a reflection over the given axis. Then graph the reflection image. 9. x-axis

10. x-axis

11. y-axis

y

R

y

y

N

S

B

A M O C x

O

P x

OQ

x

O

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T

12. The vertices of a figure are W(-3, -3), X(0, -4), Y(4, -2), and Z(2, -1). Graph the image of its reflection over the y-axis. Find the vertices of each figure after a dilation with the given scale factor centered at the origin. Then graph the dilation image. 13. scale factor: 4

14. scale factor: 1.5

1 15. scale factor: _

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For Exercises 16–18, use the graph at the right. 16. Graph the image of the figure after a dilation with a scale factor of 2 with the center at the origin. 17. Graph the image of the original figure after a dilation 1 with the center at the origin. with a scale factor of _

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18. Find the vertices of the original figure after a dilation with a scale factor of 1.5 with the center at the origin. 528 Chapter 10 Two-Dimensional Figures

9

19. BIOLOGY A microscope dilates the image of objects by a scale factor of 12. How large will a 0.0016-millimeter paramecium appear? 20. GAMES What type of transformation is used when moving a knight in a game of chess? Explain. 21. MIRRORS Which transformation exists when you look into a mirror? Explain.

Real-World Link In chess, each player has 16 game pieces, or chessmen. There are two rooks, two knights, two bishops, a queen, a king, and eight pawns.

22. Give a counterexample for the following statement. The image of a figure’s reflection is never the same as the image of its translation. 23. PRESENTATIONS Felicia wants to project a 2-inch by 2-inch slide onto a wall to create an image 128 inches by 128 inches. If the slide projector makes the image twice as large for each yard that it is moved away from the wall, how far away should Felicia place the projector? ANALYZE GRAPHS Identify each transformation as a translation, a reflection, or a dilation. 25. 26. 24. Y Y Y

EXTRA

PRACTICE

See pages 784, 803.

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X

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X

X

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H.O.T. Problems

27. OPEN ENDED Draw a triangle on grid paper. Then draw the image of the 1 . triangle after it is moved 5 units right and then dilated by a scale factor of _ 3

Y

28. CHALLENGE Suppose figure ABCD is dilated by a scale factor of 2 and then reflected over the x-axis and the y-axis. Describe the resulting figure and explain how it relates to a reflection. Then graph the image A’B’C’D’ on a coordinate plane.

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29. Which One Doesn’t Belong? Without graphing, identify the pair of points that does not represent a reflection over the y-axis. Justify your reasoning. E(0, 1) E’(0, -1)

F(- 2 , 5) F’(2, 5)

G(-3, -4) G’(-3, 4)

H(5, 0) H’(-5, 0)

CHALLENGE Discuss the results of the following transformations. Suppose the figure is in Quadrant I. 30. Reflect a figure over the x-axis. Then reflect the image over the x-axis. 31. Reflect a figure over the x-axis. Then reflect the image over the y-axis. 32.

Writing in Math

How are transformations involved in recreational activities? Include an example of a recreational activity that represents each type of transformation. Lesson 10-3 Transformations on the Coordinate Plane

Pete Saloutos/CORBIS

529

33. Figure DEFG is dilated by a scale factor of _1 with the center at the origin. Which graph 3 shows this transformation?

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Complete each congruence statement if ABC DEF. (Lesson 10-2) −− −− 34. ∠D ? 35. AC ? 36. DE ?

37. ∠C ?

38. ALGEBRA Angles A and B are complementary. If m∠A = (x + 3)° and m∠B is twice m∠A, write an equation that can be used to find the value of x. (Lesson 10-1) to estimate how many 39. SKYSCRAPERS Use the formula D = 1.22 × √A miles you can see from a point above the horizon. Suppose you are standing in the observation area of the Sears Tower in Chicago. About how far can you see on a clear day if the deck is 1353 feet above the ground? (Lesson 9-1) 40. Evaluate ⎪-4⎥ - ⎪3⎥. (Lesson 2-1)

PREREQUISITE SKILL Solve each equation. (Lesson 3-5) 41. 2x + 134 = 360

43. 5x + 125 = 360

42. 3x + 54 = 360

44. 4x + 92 = 360


EXTEND

10-3

Geometry Lab

Rotations

Another type of transformation is a rotation. A rotation turns a figure with respect to a point called the center of rotation.

ACTIVITY 1 Step 1 Use grid paper to draw a triangle on a coordinate plane. Label the triangle ABC. Label the origin as O. Tape the coordinate plane to the desktop.

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Step 2 Trace the triangle onto a sheet of tracing paper. This will become the rotated image A’B’C’. Position the traced triangle over ABC on the grid paper. Then draw segment OA’ on the tracing paper.

Step 3 Place the point of your pencil through the hole of a protractor and −−− position it at the origin. Place the 0 line of the protractor on OA’. Step 4 Holding the protractor still, turn the tracing paper clockwise −−− until OA’ is at the 90° line.

ANALYZE THE RESULTS 1. How was ABC transformed to the image A’B’C’? 2. Copy and complete the tables using the coordinates of your triangles. 3. What do you observe about the coordinates of points A and A'?

x

y

x

A

A’

B

B’

C

C’

y

4. Repeat the Activity, rotating the figure 90° counterclockwise. Describe what you observe about the coordinates of points A and A'. 5. Repeat the Activity, rotating the figure 180° about the origin. What do you observe about the coordinates of points A and A’? 6. MAKE A CONJECTURE Write a rule to describe what happens to coordinates (x, y) of a figure after a clockwise rotation of 90°, a counterclockwise rotation of 90°, and a rotation of 180° about the origin. For Exercises 7–9, use the graph at the right.

y

7. Graph the image of the figure after a rotation of 90° counterclockwise. 8. Find the coordinates of the vertices of the figure after a 180° rotation. 9. Graph the image of the figure after a rotation of 90° clockwise.

B x

O A

C E

D

Extend 10-3 Geometry Lab: Rotations

531

10-4

Quadrilaterals

Main Ideas • Find the missing angle measures of a quadrilateral. • Classify quadrilaterals.

New Vocabulary quadrilateral

Geometric figures are often used to create various designs. Notice how the brick walkway at the right is formed using different-shaped bricks to create circles. a. Describe the bricks used to create the smallest circles. b. Describe how the shape of the bricks change as the circles get larger.

Quadrilaterals Squares, rectangles, and trapezoids are examples of quadrilaterals. A quadrilateral is a closed figure with four sides and four vertices. The segments of a quadrilateral intersect only at their endpoints. Quadrilaterals

Not Quadrilaterals

As with triangles, a quadrilateral can be named by its vertices. Two names of the quadrilateral below are quadrilateral ABCD and quadrilateral CBAD. Naming Quadrilaterals

The vertices are A, B, C, and D.

When you name a quadrilateral, you can begin at any vertex. However, it is important to name vertices in order.

The angles are A, B, C, and D.

A

B The sides are AB, BC, CD, and DA. C

D

A quadrilateral can be separated into two triangles. Since the sum of the measures of the angles of a triangle is 180°, the sum of the measures of the angles of a quadrilateral is 2(180°) or 360°. 532 Chapter 10 Two-Dimensional Figures Richard Hamilton Smith/CORBIS

F G E

H

Angles of a Quadrilateral Th e su m o f th e m e asu re s o f th e an gle s o f a q u ad rilate ral is 3 6 0 °.

EXAMPLE

Find Angle Measures

ALGEBRA Find the value of x. Then find each missing angle measure.

B 62˚ 2x ˚

The sum of the measures of the angles is 360°. Let m∠A, m∠B, m∠C, and m∠D represent the measures of the angles. m∠A + m∠B + m∠C + m∠D = 360

88 + 62 + 2x +

= 360

x

3x + 150 = 360 Check Your Work To check the answer, find the sum of the measures of the angles. Since 88° + 62° + 140° + 70° = 360°, the answer is correct.

A

88˚

C

x˚

D

Angles of a quadrilateral Substitution Combine like terms.

3x + 150 - 150 = 360 - 150 Subtract 150 from each side. 3x = 210 x = 70


So, m∠D = 70° and m∠C = 2(70) or 140°.

1. ALGEBRA In quadrilateral EFGH, m∠E = 3x°, m∠F = 70°, m∠G = x°, and m∠H = 82°. Find the value of x. Then find each missing angle measure. Personal Tutor at pre-alg.com

Classify Quadrilaterals The diagram below shows how quadrilaterals are related. It goes from the most general quadrilateral to the most specific. The best description of a quadrilateral is the one that is the most specific. Animation pre-alg.com

+Õ>`À>ÌiÀ>

*>À>i}À> /À>«iâ` QUADRILATERAL WITH EXACTLY ONE PAIR OF PARALLEL SIDES

QUADRILATERAL WITH BOTH PAIRS OF OPPOSITE SIDES PARALLEL AND CONGRUENT

,LÕÃ PARALLELOGRAM WITH CONGRUENT SIDES

,iVÌ>}i PARALLELOGRAM WITH RIGHT ANGLES


-µÕ>Ài PARALLELOGRAM WITH CONGRUENT SIDES AND RIGHT ANGLES

Lesson 10-4 Quadrilaterals

533

BASKETBALL The photograph shows the free-throw lane used during international competitions. Classify the quadrilateral using the name that best describes it. The quadrilateral has exactly one pair of opposite sides that are parallel. It is a trapezoid.

Classify each quadrilateral using the name that best describes it. 2A. 2B.

ALGEBRA Find the value of x. Then find the missing angle measures. 1.

x˚

68˚

125˚

2x ˚

118˚

x˚

Classify each quadrilateral using the name that best describes it. 3.

4.

HOMEWORK

HELP

For See Exercises Examples 6–11 1 12–17 2

8

10

ALGEBRA Find the value of x. Then find the missing angle measures. 6.

65˚

7. 109˚

128˚ 3x ˚ x˚

115˚

9.

8.

x˚ 96˚ 110˚

x˚

x˚

x˚

2x ˚

534 Chapter 10 Two-Dimensional Figures John Kolesidis/Reuters/CORBIS

8

5. SPORTS Classify the quadrilaterals that are found on the scoring region of a shuffleboard court.

10

(p. 534)

100˚

10

Example 2

2.

64˚

7

(p. 533)

7

Example 1

120˚

120˚

52˚

10.

2x ˚

x˚

11.

90˚ (2x 20)˚

135˚

(x 5)˚

x˚

(x 10)˚

Classify each quadrilateral using the name that best describes it. 12.

13.

14.

15.

16.

17.

18. ART Classify the quadrilaterals that are outlined in the painting at the right.

Real-World Career Artist An artist uses math to create paintings, sculptures, or illustrations to communicate ideas. For more information, go to pre-alg.com.

EXTRA

PRACTICE


H.O.T. Problems

19. GAMES Identify a game that is played on a board that is shaped like a square. Describe the characteristics that make the board a square. 20. COOKING Name an item found in a kitchen that is rectangular in shape. Explain why the item is a rectangle. Determine whether each statement is sometimes, always, or never true. 21. A square is a rhombus. 22. A parallelogram is a rectangle. 23. A rectangle is a square. 24. A parallelogram is a quadrilateral.

Irene Rice Perlera. Untitled, 1951

Make a drawing of each quadrilateral. Then classify each quadrilateral using the name that best describes it. 25. In quadrilateral JKLM, m∠J = 90°, m∠K = 50°, m∠L = 90°, and m∠M = 130°. −−− −− −− −− 26. In quadrilateral CDEF, CD and EF are parallel, and CF and DE are parallel. Angle C is not congruent to ∠D. 27. ART The abstract painting at the right is an example of how shape and color are used in art. Write a few sentences describing the geometric shapes used by the artist. CHALLENGE For Exercises 28 and 29, use the following information. An equilateral figure is one in which all sides have the same measure. An equiangular figure is one in which all angles have the same measure. 28. Is it possible for a quadrilateral to be equilateral Elizabeth Murray Painter’s Progress, 1981 without being equiangular? If so, explain with a drawing. 29. Is it possible for a quadrilateral to be equiangular without being equilateral? If so, explain with a drawing. Lesson 10-4 Quadrilaterals

(tl)Pat LaCroix; (cr)“Untitled”, 1951. Irene Rica Pereira. Oil on board, 101.5 X 61 cm. Solomon R. Guggenheim Museum, New York, NY. Gift of Jerome B. Lurie, 1981.; (bl)“Painter’s Progress”, 1981. Elizabeth Murray. Museum of Modern Art, New York, NY. Aquired through the Bernhill Fund and gift of Agnes Gund/Art Resource, NY

535

30. OPEN ENDED Choose four cities on a map of the United States that when connected form a rectangle. How do you know it is a rectangle? 31.

Writing in Math How are quadrilaterals used in design? Include an example of a real-world design that contains quadrilaterals and an explanation of the figures used in the design.

32. Which figure is best described as a square? A C

B

33. GRIDDABLE Mrs. Smith used the parallelogram below to design a pattern for a paving stone. She will use the paving stone for a sidewalk. Find x.

D

Ýc

£Îäc

Ýc

£Îäc

34. A figure has vertices D(1, 2), E(1, 4), F(-4, 4), and G(-2, 2). Graph the figure and its image after a translation 4 units down. (Lesson 10-3) Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. (Lesson 10-2) 35.

% £È vÌ

&

Èäc £Ó vÌ

xxc

Èxc £{ vÌ

36. ,

(

'

n vÌ

*

Èäc Èxc

0

È vÌ xxc

Ç vÌ

-

.

/

)

37. GARDENING Suppose you plant a square garden with an area of 300 square feet. How much fencing will you need to buy to enclose the garden if the fencing only comes in whole-foot sections? (Lesson 9-2) 38. ALGEBRA Solve x - 3.4 ≥ 6.2. Graph the solution on a number line. (Lesson 8-4) 39. GEOGRAPHY Forty-six percent of the world’s water is in the Pacific Ocean. What fraction is this? (Lesson 6-5) 40. TRACK AND FIELD Heather needs to average 11.4 seconds in the 100-meter dash in six races to qualify for the championship race. The mean of her first five races was 11.2 seconds. What is the greatest time that she can run and still qualify for the race? (Lesson 5-9)

PREREQUISITE SKILL Simplify each expression. (Lesson 1-2) 41. (5 - 2)180 42. (7 - 2)180 43. (10 - 2)180 536 Chapter 10 Two-Dimensional Figures

44. (9 - 2)180

CH

APTER

10


1. FENCING A diagonal brace strengthens the wire fence and prevents it from sagging. The brace makes a 60° angle with the post as shown. Find y. (Lesson 10-1)

10. MULTIPLE CHOICE Triangle JKL has vertices J(3, 5), K(5, 7), and L(6, 3) and is dilated by a scale factor of 1 with the origin as 3 the center of dilation. What are the coordinates of L’? (Lesson 10-3)

Èäc

Þc

A (2, 1)

B 1, 1 3 C (18, 9)

2. If m∠Y = 23° and ∠Y and ∠Z are complementary, what is m∠Z? (Lesson 10-1)

D 3, 3

3. Angles G and H are supplementary. If m∠G = x + 11 and m∠H = x - 13, what is the measure of each angle? (Lesson 10-1) In the figure m and t is a transversal. Find the value of x for each of the following. (Lesson 10 -1)

11. After a translation of 4 units left and 2 units up, the coordinates of the vertices of the image ABC are A(3, 2), B(0, 4), and C(-3, 5). What were the coordinates of the vertices before the translation? (Lesson 10 -3)

t

m n

x Ç

£ Ó { Î È

Find the value of x. Then find the missing angle measures. (Lesson 10 - 4)

4. m∠3 = 3x + 9 and m∠5 = 6x + 12 5. m∠6 = 8x + 7 and m∠8 = 9x - 10 6. UMBRELLAS An umbrella has eight ) congruent triangular sections with spokes of equal length. " Name one pair of congruent triangles. Then find the # corresponding parts. (Lesson 10-2)

2

12.

(

25˚

'

175˚

x˚

13.

95˚

!

2x ˚

x˚

&

PARKING SPACES Classify each quadrilateral using the name that best describes it. (Lesson 10 - 4) %

14.

15.

$

7. Suppose ABC DEF. Which angle is congruent to ∠D? (Lesson 10-2) 8. Triangle QRS has vertices Q(3, 3), R(5, 6), and S(7, 3). Find the coordinates of the vertices after the triangle is reflected over the x-axis.

16. MULTIPLE CHOICE Which figure is best described as a parallelogram? (Lesson 10-4) F

H

G

J

(Lesson 10-3)

9. GAMES What type of transformation is used when a checker piece is moved on a checkerboard? (Lesson 10-3)

Chapter 10 Mid-Chapter Quiz

537

Learning Mathematics Prefixes Quadruplets are four children born at the same tim to the same mother. The prefix quad- also appears in the term quadrilateral—a polygon with four sides. The table shows some of the prefixes that are used in mathematics. These prefixes are also used in everyday language. In order to use each prefix correctly, you need to understand its meaning.

Prefix

Meaning

Everyday Words

four

quadruple quadruplet quadriceps

a sum four times as great as another one of four offspring born at one birth a muscle with four points of origin

Pentagon pentathlon pentad

headquarters of the Department of Defense a five-event athletic contest a group of five

quad-

Meaning

pent-

five

hex-

six

hexapod hexagonal hexastich

having six feet having six sides a poem of six lines

hept-

seven

hepted heptagonal heptarchy

a group of seven having seven sides a government by seven rulers

oct-

eight

octopus octet octennial

a type of mollusk having eight arms a musical composition for eight instruments lasting eight years

dec-

ten

decade decameter decathlon

a period of ten years ten meters a ten-event athletic contest

Reading to Learn 1. Refer to the table above. For each prefix listed, choose one of the everyday words listed and write a sentence that contains the word. 2. RESEARCH Use the Internet, a dictionary, or another reference source to find a mathematical term that contains each of the prefixes listed. Write the definition of each term. 3. RESEARCH Use the Internet, a dictionary, or another reference source to find a different word that contains each prefix. Then define the term. 538 Chapter 10 Two-Dimensional Figures Ace Stock Limited/Alamy Images

10-5

Polygons

Main Ideas • Classify polygons. • Determine the sum of the measures of the interior and exterior angles of a polygon.

New Vocabulary polygon diagonal interior angles regular polygon

The tiled patterns below are called regular tessellations. Notice how the figures repeat to form patterns that contain no gaps or overlaps.

Square Tessellation

Triangle Tessellation Hexagon Tessellation

a. Which figure is used to create each tessellation?

vertex

b. Refer to the diagram at the right. What is the sum of the measures of the angles that surround the vertex? c. Does the sum in part b hold true for the square tessellation? Explain. d. Make a conjecture about the sum of the measures of the angles that surround a vertex in the hexagon tessellation.

Classify Polygons A polygon consists of a sequence of consecutive line segments placed end to end to form a simple closed figure. The figures below are examples of polygons. The line segments meet only at their endpoints.

The points of intersection are called vertices.

The line segments are called sides.

The following figures are not polygons.

This is not a polygon because it has a curved side. n-gon A polygon with n sides is called an n-gon. For example, an octagon can also be called an 8-gon.

This is not a polygon because it is an open fugure.

This is not a polygon because the sides overlap.

Polygons can be classified by the number of sides they have. Name of Polygon

pentagon

hexagon

heptagon

octagon

nonagon

decagon

Number of Sides

5

6

7

8

9

10

Lesson 10-5 Polygons

539

EXAMPLE

Classify Polygons

Classify each Polygon. a.

b.

The polygon has 8 sides. It is an octagon.

1A.

The polygon has 6 sides. It is a hexagon.

1B.

Measures of the Angles of a Polygon A diagonal is a line segment in a polygon that joins two nonconsecutive vertices. In the diagram below, all possible diagonals from one vertex are shown. quadrilateral

pentagon

hexagon

heptagon

octagon

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Sides

4

5

6

7

8

Diagonals

1

2

3

4

5

Triangles

2

3

4

5

6

Notice that the number of triangles is 2 less than the number of sides.

You can use the property of the sum of the measures of the angles of a triangle to find the sum of the measures of the interior angles of any polygon. An interior angle is an angle inside a polygon. Interior Angles of a Polygon If a polygon has n sides, then n - 2 triangles are formed. The sum of the degree measures of the interior angles of the polygon is (n - 2)180.

EXAMPLE

Measures of Interior Angles

Find the sum of the measures of the interior angles of a heptagon. A heptagon has 7 sides. Therefore, n = 7. (n - 2)180 = (7 - 2)180 Replace n with 7. = 5(180) or 900° Simplify.

2. Find the sum of the measures of the interior angles of a 13-gon. 540 Chapter 10 Two-Dimensional Figures


A regular polygon is a polygon that is equilateral (all sides are congruent) and equiangular (all angles are congruent). Since the angles of a regular polygon are congruent, their measures are equal.

SNOW Snowflakes are some of the most beautiful objects in nature. Notice how they are regular and hexagonal in shape. What is the measure of one interior angle in a snowflake? Step 1 Find the sum of the measures of the angles. A hexagon has 6 sides. Therefore, n = 6. Real-World Link Snowflakes are also called snow crystals. It is said that no two snowflakes are alike. They differ from each other in size, lacy structure, and surface markings.

(n - 2)180 = (6 - 2)180 Replace n with 6. = 4(180) or 720 Simplify. The sum of the measures of the interior angles is 720°. Step 2 Divide the sum by 6 to find the measure of one angle. 720 ÷ 6 = 120 So, the measure of one interior angle in a snowflake is 120°.


3. KALEIDOSCOPE This kaleidoscope is regular and nonagonal in shape. What is the measure of one interior angle of the nonagon?


Example 1 (p. 540)

Classify each polygon. Then determine whether it appears to be regular or not regular. 1.

2.

3. TESSELLATIONS Identify the polygons that are used to create the tessellation shown at the right. Example 2 (p. 540)

Example 3 (p. 541)

4. Find the sum of the measures of the interior angles of a nonagon. 5. What is the measure of each interior angle of a regular heptagon? Round to the nearest tenth. Lesson 10-5 Polygons

(l)Steve Austin/Papilio/CORBIS; (r)Becky Hayes/age fotostock

541

HOMEWORK

HELP



7.

8.

9.

10.

11.

Find the sum of the measures of the interior angles of each polygon. 12. pentagon 15. hexagon

13. octagon 16. 18-gon

14. decagon 17. 23-gon

Find the measure of an interior angle of each polygon. 18. regular nonagon 21. regular decagon

19. regular pentagon 22. regular 12-gon

20. regular octagon 23. regular 25-gon

ART For Exercises 24 and 25, use the painting below.

Roy Lichtenstein. Modern Painting with Clef. 1967

24. List five polygons used in the painting. 25. RESEARCH The title of the painting mentions the music symbol, clef. Use the Internet or another source to find a drawing of a clef. Is a clef a polygon? Explain. TESSELLATIONS For Exercises 26 and 27, identify the polygons used to create each tessellation. 27. 26.

EXTRA

PRACTICE


28. BASEBALL The star in the Houston Astros’ logo is regular and pentagonal in shape. What is the measure of one interion angle in the pentagon? 29. ART Refer to Exercise 3 on page 541. The tessellation design contains regular polygons. Find the perimeter of the design if the measure of the sides of the 12-gon is 5 centimeters.

542 Chapter 10 Two-Dimensional Figures (t)“Modern Painting with Clef”, 1967. Roy Lichtenstein. Oil on synthetic polymer and pencil on canvas, 252 x 458 cm. Hirschhorn Museum and Sculpture Garden, Smithsonian Institution, Washington D.C.; (b)Courtesy of Houston Astros & Major League Baseball

H.O.T. Problems

30. SELECT A TOOL Study the dot pattern shown at the right. Which of the following tools would you use to find four line segments connecting all of the points that were drawn without lifting a pencil from the paper? Justify your selection. Then use the tool to solve the problem. draw a model

paper/pencil

technology

31. OPEN ENDED Draw a polygon that is both equiangular and equilateral. CHALLENGE When a side of a polygon is extended, an exterior angle is formed. In any polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°.

72˚

35.

33. regular triangle

exterior angle

72˚

Find the measure of each exterior angle of each regular polygon. 32. regular octagon

72˚

72˚ 72˚

34. regular decagon

Writing in Math

How are polygons used in tessellations? Draw an example of a tessellation in which the pattern is formed using only one type of polygon and an example of a tessellation in which the pattern is formed using more than one polygon.

36. A landscape architect is looking for a brick paver shape that will tessellate. Which shape will allow her to tessellate a patio area? A

C

B

D

37. Which term identifies the shaded part of the design shown?

F heptagon

H octagon

G hexagon

J pentagon

Classify each quadrilateral using the name that best describes it. (Lesson 10-4) 38.

39.

40.

41. Triangle MNP has vertices M(-1, 1), N(5, 4), and P(4, 1). Graph the triangle after a translation 3 units left and 4 units down. (Lesson 10-3) 42. ALGEBRA Simplify 4.6x + 2.5x + 9.3x. (Lesson 3-2)

PREREQUISITE SKILL Find each product. (pages 747–748) 43. (3)(4.8) 44. (5.4)(6) 45. (9.2)(3.1)

46. (10.5)(5.7)

Lesson 10-5 Polygons

543

Geometry Lab

EXTEND

10-5

Tessellations

A tessellation is a pattern of repeating figures that fit together with no overlapping or empty spaces. Tessellations can be formed using transformations.

ACTIVITY 1 Create a tessellation using a translation. Step 1 Draw a square. Then draw a triangle inside the top of the square as shown.

Step 3 Repeat this pattern unit to create a tessellation. It is sometimes helpful to complete one pattern, cut it out, and trace it for the other pattern units.

Step 2 Translate or slide the triangle from the top to the bottom of the square. Step 1

Step 2

ACTIVITY 2 Create a tessellation using a rotation. Step 1 Draw an equilateral triangle. Then draw another triangle inside the left side of the triangle as shown below.

Step 3 Repeat this pattern unit to create a tessellation.

Step 2 Rotate the triangle so you can trace the change on the side as indicated. Step 1

Step 2

EXERCISES Use a translation to create a tessellation for each pattern unit shown. 1.

2.

3.

Use a rotation to create a tessellation for each pattern unit shown. 4.

5.

6.

7. Make a tessellation that involves a translation, a rotation, or a combination of the two. 544 Chapter 10 Two-Dimensional Figures

10-6

Area: Parallelograms, Triangles, and Trapezoids

Main Ideas • Find areas of parallelograms. • Find the areas of triangles and trapezoids.

The area of a rectangle can be found by multiplying the length and width. The rectangle shown below has an area of 3 6 or 18 square units. Suppose a triangle is cut from one side of the rectangle and moved to the other side. The new figure is a parallelogram.

New Vocabulary base altitude

UNITS UNITS

a. Compare the area of the rectangle to the area of the parallelogram. b. What parts of a rectangle and parallelogram determine their area?

Areas of Parallelograms The area of a parallelogram can be found by multiplying the measures of the base and the height.

TThe e base can a be e an side any e of the e p a eo parallelogram.

The e height he g is the length eng of an n altitude, a line segment eg p rp perpendicular c a to the e base as with w e po endpoints on o the e base a e and n the side de opposite pp the base.

altitude base

Area of a Parallelogram Words

If a parallelogram has a base of b units and a height of h units, then the area A is bh square units.

Model

b h

Symbols A = bh

EXAMPLE Generating Formulas For a lesson on generating formulas involving area, see page 807.

Find Areas of Parallelograms

Find the area of each parallelogram. a.

The base is 14 feet. The height is 12 feet. 12 ft

Estimate A = 15 · 10 or 150 14 ft

A = bh

Area of a parallelogram

= 14 • 12 Replace b with 14 and h with 12. = 168

Multiply.

The area is 168 square feet. The answer is close to the estimate so the answer is reasonable. Lesson 10-6 Area: Parallelograms, Triangles, and Trapezoids

545

b.

The base is 5.9 centimeters. The height is 7.5 centimeters. Estimate A = 6 8 or 48

Altitudes

7.5 cm

An altitude can be outside the parallelogram.

A = bh


= (5.9)(7.5) Replace b with 5.9 and h with 7.5. = 44.25

5.9 cm

Multiply.

The area is 44.25 square centimeters. Is the answer reasonable? Find the area of each parallelogram. 1A.

1B. Î

°Ç

È £ä°Î

Area of Triangles and Trapezoids A diagonal of a parallelogram separates the parallelogram into two congruent triangles. The area of each triangle is one-half the area of the parallelogram. diagonal

The area of parallelogram ABCD is 7 ⭈ 4 or 28 square units.

B

C 4

A

The area of triangle ABD is 21 ⭈ 28 or 14 square units.

D

7

Using the formula for the area of a parallelogram, we can find the formula for the area of a triangle. Area of a Triangle If a triangle has a base of b units and a height of h units, then the 1 area A is _ bh square units.

Words

1 Symbols A = _ bh

Model h

2

b

2

EXAMPLE Alternative Method Multiplication is commutative and associative. So you can also find 12 of 6 first and then multiply by 5.

Find Areas of Triangles

Find the area of each triangle. a.

The base is 5 inches. The height is 6 inches.

b.

The base is 7 meters. The height is 4.2 meters.

4.2 m

6 in. 7m

1 A=_ bh

5 in.

1 A=_ bh

Area of a triangle 2 1 =_ (5)(6) Replace b with 5 and h with 6. 2 1 =_ (30) Multiply. 5 × 6 = 30 2

= 15

in2


Simplify.

2 1 _ = (7)(4.2) 2 1 =_ (29.4) 2

= 14.7 m2

Area of a triangle Replace b with 7 and h with 4.2. Multiply 7 × 4.2 = 29.4 Simplify.

2A.

2B.

Î vÌ

Ó V £ Î

x vÌ

£ä V

A trapezoid has two bases. The height of a trapezoid is the distance between the bases. A trapezoid can be separated into two triangles. F

G

a

K

h

E

&

h

%

A

+

'

H

H

H

b

J

&

(

B

(

area of trapezoid EFGH = area of EFH + area of FGH

_1 bh

=

1 =_ h(a + b) 2

_1 ah

+

2

2


Area of a Trapezoid Words

If a trapezoid has bases of a units and b units and a height of h units, then the area A of 1 the trapezoid is _ h(a + b) 2 square units.

a

Model h

b

1 Symbols A = _ h(a + b) 2

EXAMPLE Look Back To review multiplying fractions, see Lesson 5-3.

Find Area of a Trapezoid

Find the area of the trapezoid. 1 The height is 4 inches. The bases are 6 _ inches 2 1 _ and 3 inches. 4 Estimate 12(4)(7 + 3) or 20 1 A=_ h(a + b) 2 1 =_ · 4 61 + 31 2 4 2 3 1 =_ · 4 · 9_ 4 2 39 1 _ 4 _ _ = · · 2 1 4 39 = or 19 1 in2 2 2

( _ _)

Area of a trapezoid

6 1 in. 2

4 in.

3 1 in. 4

Replace h with 4, a with 6 12 and b with 314. 6 12 + 314 = 9 34 Divide out the common factors.

1 3. Find the area of a trapezoid with a height of 8_ yards and bases that 4 1 _ are 5 yards and 3 yards long. 3


Lesson 10-6 Area: Parallelograms, Triangles, and Trapezoids

547

FLAGS The signal flag shown represents the number five. Find the area of the blue region. Estimate The blue region is about

_1 of the whole flag.

IN

1 1 So, A = _bh = _(14)(14) or about 50. 4

4

IN

IN

4

To find the area of the blue region, subtract the areas of the triangles from the area of the square. Area of the square

IN

Area of each triangle 1 A=_ bh

A = bh = 14 · 14 = 196

2

1 = _ · 12 · 6 2 = 36

The total area of the triangles is 4(36) or 144 square inches. So, the area of the blue region is 196 - 144 or 52 square inches. The answer is close to the estimate so the answer is reasonable.

4. FLAGS The flag shown below is the international signal for the number three. Find the area of the red region. 18 in.

9 in.

3 in.

7 in. 12 in.


Examples 1–3 (pp. 545–547)

Find the area of each figure shown or described. 1.

2. 4 ft

3. 5.4 cm

2 ft

15 m 6m 8m

3 cm

7.6 m 5m

4. parallelogram: base, 5.6 m; height, 9.4 m 5. triangle: base, 12 km; height 13 km 6. trapezoid: height: 16 in.; bases, 3.1 in. and 7.6 in. Example 4 (p. 548)

7. Find the approximate area of the state of Nevada. Round to the nearest square mile.

MI MI .EVADA


MI

HOMEWORK

HELP

For See Exercises Examples 8–19 1–3 20, 21 4

Find the area of each figure. 8.

9.

Ç°{ °

10.

15 cm

Î°x °

11.

12. 10 in.

12 cm

5.5 m

2m

13.

6.7 cm

5.3 cm

7.2 in.

Ó{ vÌ £ ££ vÌ x

£Ó vÌ

9.9 cm

£{ vÌ

6 in.

Find the area of each figure described. 14. triangle: base, 8 in.; height, 7 in. 15. trapezoid: height, 2 cm; bases, 3 cm, 6 cm 16. parallelogram: base, 3.8 yd; height, 6 yd 17. triangle: base, 9 ft; height, 3.2 ft 18. trapezoid: height, 3.5 m; bases, 10 m and 11 m 19. parallelogram: base, 5.6 km; height, 4.5 km GEOGRAPHY For Exercises 20 and 21, use the approximate measurements to estimate the area of each state. 20.

287 mi

Real-World Link The largest U.S. state is Alaska. It has an area of 615,230 square miles. Rhode Island is the smallest state. It has an area of 1231 square miles. Source: The World Almanac

21.

332 mi

270 mi

ARKANSAS

OREGON

235 mi

165 mi

22. Find the base of a parallelogram with a height of 9.2 meters and an area of 36.8 square meters. 23. Suppose a triangle has an area of 20 square inches and a base of 2 1 inches. 2 What is the measure of the height? 24. A trapezoid has an area of 54 square feet. What is the measure of the height if the bases measure 16 feet and 8 feet? Find the area of each figure with the vertices shown. 25. rectangle: A(-3, 4), B(5, 4), C(5, -1), D(-3, -1) 26. triangle: E(-2.5, 2), F(3, -2.5), G(3, 2) Find the area of each figure. 27.

EXTRA

28.

12 km

PRACTICE

2 km

3m

4m

8 km 15 km

8 ft

9m

8m


29. 6 ft

6m 8 ft 11 ft

Lesson 10-6 Area: Parallelograms, Triangles, and Trapezoids Lewis Kemper/SuperStock

549

30. LAWNCARE Mrs. Malone plans to fertilize her lawn. The fertilizer she will be using indicates that one bag fertilizes 2000 square feet. How many bags of fertilizer should she buy?

H.O.T. Problems

15 ft

125 ft Lawn 15 ft

Flower bed

84 ft

Patio 12 ft

31. OPEN ENDED Draw and label a parallelogram that has an area of 24 square inches.

100 ft

32. CHALLENGE Explain how the formula for the area of a trapezoid can be used to find the formulas for the areas of parallelograms and triangles. 33.

Writing in Math How is the area of a parallelogram related to the area of a rectangle? Describe the similarities and differences between a rectangle and a parallelogram, and draw a diagram that shows how the area of a parallelogram is related to the area of a rectangle.

34. The developer of a park wants to 320 ft change the base 160 ft measures but keep 560 ft the park the same size. Which base measures are NOT possible if the height stays the same? A 220 ft, 660 ft

C 231 ft, 649 ft

B 353 ft, 527 ft

D 370 ft, 610 ft

35. Which figure does NOT have an area of 48 square meters? F

H

8m

3m 1.6 m

6m

G

J

8m

12.8 m 5m

12 m 6.4 m

Find the measure of an interior angle of each polygon. (Lesson 10-5) 36. regular hexagon

37. regular decagon 38. regular octagon

39. regular 15-gon

Find the value of x. Then find the missing angle measures. (Lesson 10-4) 40.

60˚

x˚ 60˚

120˚

41. 110˚ 4x ˚

130˚

42. TAXI FARES The table shows taxi ride fares for two companies. Compare the rates of change. (Lesson 7-5)

x˚

Miles (min) x 0 3 9

Fare y Taxi 1 Taxi 2 $0 $0 $2.40 $2.70 $7.20 $8.10

PREREQUISITE SKILL Use a calculator to find each product. Round to the nearest tenth. (pages 747–748) 43. 3.14 · 4.3 44. 2 · 3.14 · 5.4 45. 3.14 · 42 46. 3.14(2.4)2 550 Chapter 10 Two-Dimensional Figures

10-7

Circles: Circumference and Area

Main Ideas • Find circumference of circles. • Find area of circles.

New Vocabulary circle diameter center circumference radius (pi)

Coins, paper plates, cookies, and CDs are all examples of objects that are circular in shape. d

Object

a. Collect three different-sized circular objects. Then copy the table shown.

C

C d

1 2 3

b. Using a tape measure, measure each distance below to the nearest millimeter. Record your results. • the distance across the circular object through its center (d) • the distance around each circular object (C) C c. For each object, find the ratio _ . Record the results in the table. d

d. Write an equation that relates the circumference C of a circle to its diameter d.

Circumference of Circles A circle is the set of all points in a plane that are the same distance from a given point.

Pi Although π is an irrational number, 3.14 22 and _ are two 7 generally accepted approximations for π.

The distance across the circle through its center is its diameter.

The given point is called the center.

The distance around the circle is called the circumference.

The distance from the center to any point on the circle is its radius.

The relationship you discovered above is true for all circles. The ratio of the circumference of a circle to its diameter is always equal to 3.1415926… The Greek letter π (pi) stands for this number. Using this ratio, you can derive a formula for the circumference of a circle. C _ =π

d C _ ·d =π·d d

C = πd

The ratio of the circumference to the diameter equals pi. Multiply each side by d. Simplify.

Circumference of a Circle Words

The circumference of a circle is equal to its diameter times π, or 2 times its radius times π.

Model C d

r

Symbols C = πd or C = 2πr Lesson 10-7 Circles: Circumference and Area

551


If an exact answer is required, leave the answer in terms of . A decimal 22 can be used for estimating answers. approximation of , 3.14, or _ 7

EXAMPLE

Find the Circumference of a Circle

Find the circumference of each circle to the nearest tenth. C = d

a. 5 cm

Circumference of a circle

=·5

Replace d with 5.

= 5

Simplify. This is the exact circumference.

To approximate the circumference, first use ≈ 3 to get an estimate. Then use a calculator. Estimate: 5 × ≈ 5 × 3 or about 15 Calculating with π Unless otherwise specified, use a calculator to evaluate expressions involving and then follow any instructions regarding rounding.

5 × 2nd [] ENTER 15.70796327 The circumference is about 15.7 centimeters. b.

C = 2r

3.2 ft


= 2 · · 3.2

Replace r with 3.2.

= 20.1

Simplify. Use a calculator.

3 1A. diameter = 3_ ft

1B. radius = 7 mm

4


TREES A tree in Madison’s yard was damaged in a storm. Her parents want to replace the tree with another whose trunk is the same size as the original tree. Suppose the circumference of the original tree was 14 inches. What should be the diameter of the replacement tree? Explore You know the circumference of the original tree. You need to find the diameter of the new tree. Plan Solve

Use the formula for the circumference of a circle to find the diameter. C = d


14 = · d Replace C with 14. 14 _ =d

Divide each side by .

4.5 ≈ d

Simplify. Use a calculator.

The diameter of the tree should be about 4.5 inches. Check

Is the solution reasonable? Check by replacing d with 4.5 in C = d. C = d


= · 4.5 Replace d with 4.5. ≈ 14.1

Simplify. Use a calculator. The solution is reasonable.

2. MUSIC A CD has a diameter of 120 millimeters. A Universal Media Disc (UMD) has a diameter of 60 millimeters. Compare the circumferences of both disc sizes. 552 Chapter 10 Two-Dimensional Figures


Areas of Circles A circle can be separated into parts as shown below. The parts can then be arranged to form a figure that resembles a parallelogram. #IRCUMFERENCE 2ADIUS #IRCUMFERENCE

Review Vocabulary

Since the circle has an area that is relatively close to the area of the figure, you can use the formula for the area of a parallelogram to find the area of a circle.

Exponents in a power, the number of times the base is used as a factor; Example: 53; 3 is the exponent (Lesson 4-1)

A = bh


1 A= _ × C r

The base of the parallelogram is one-half the circumference, and the radius is the height.

A = 1 × 2r r 2

Replace C with 2r.

2

A=×r×r

Simplify.

A = r2

Replace r × r with r2.

Area of a Circle Words

The area of a circle is equal to times the square of its radius.

Model r

Symbols A = r2

EXAMPLE

Find Areas of Circles

Estimation

Find the area of each circle. Round to the nearest tenth.

To estimate the area of a circle, square the radius and then multiply by 3.

a.

Estimate 3 · 36 or 108 6 in.

b.

A = r2 =·

Area of a circle

62

Replace r with 6.

= · 36

Evaluate 62.

≈ 113.1 in2

Use a calculator. The answer is reasonable.

Estimate 3 · 256 or 768

A = r2

31 m

Area of a circle

= · 31 2

3A.

2

Since d is 31, r is _. 31 2

= · (15.5)2

31 _ = 15.5

= · 240.25

Evaluate (15.5)2.

≈ 754.8 m2

Use a calculator. The answer is reasonable.

2

3B. £

£Î Ó ° ££

Lesson 10-7 Circles: Circumference and Area

553

Examples 1, 3 (pp. 552, 553)

Find the circumference and area of each circle. Round to the nearest tenth. 1.

2.

3.

5 mi

4 in. 8m

4. The radius is 1.3 kilometers. Example 2 (p. 552)

HOMEWORK

HELP

For See Exercises Examples 7–16 1, 3 17, 18 2

5. The diameter is 6.1 centimeters.

6. MUSIC During a football game, the marching band can be heard within a radius of 1.7 miles. What is the area of the neighborhood that can hear the band?


8.

9. 13 in.

6 cm

10 m

10.

11.

12.

21 km

13. The radius is 4.5 meters. 4 feet. 15. The diameter is 7 _ 5

1

9 2 ft

12.7 m

14. The diameter is 7.3 centimeters. 3 16. The radius is 15 _ inches. 8

17. LAWN CARE Sunki has a sprinkler positioned in her lawn that directs a 12-foot spray in a circular pattern. About how much of the lawn does the sprinkler water? 18. SCIENCE The circumference of the Moon is about 6790 miles. What is the distance to the center of the Moon?

Real-World Link The world’s largest fountain is the Suntec City Fountain of Wealth in Singapore. Made of cast bronze, it covers a total area of 18,117 square feet. Source: guinnessworldrecords.com

Match each circle described in the column on the left with its corresponding measurement in the column on the right. 19. radius: 4 units a. circumference: 37.7 units 20. diameter: 7 units b. area: 7.1 units2 21. diameter: 3 units c. area: 50.3 units2 22. radius: 6 units d. circumference: 22.0 units 23. What is the diameter of a circle if its circumference is 25.8 inches? Round to the nearest tenth. 24. Find the radius of a circle if its area is 254.5 square inches. 25. CAROUSEL The Carousel in Spring Green, Wisconsin is the world’s largest carousel. If it has a diameter of 80 feet, what is the distance a seat on it travels in 10 revolutions? Round to the nearest foot. 26. FOUNTAINS A circular fountain at a park has a radius of 4 feet. The mayor wants to build a fountain that is quadruple the size of the current fountain. Find the length of the radius of the new fountain.

554 Chapter 10 Two-Dimensional Figures eye35.com/Alamy Images

MI

Ü iÀV>Ã >Ì > ««i ANALYZE GRAPHS For Exercises 27–29, the circle graph at the right has a radius of 1 inch. Suppose the circle graph is {Ç¯ redrawn onto a poster board so Ìi Ì Ì that the diameter is tripled. Î¯ ½Ì Ü 27. How much space on the poster board will the circle graph ££¯ *ii Ì Î¯ cover?

ÕÌ Ì Ì 28. How much of the total space ÃViÃ will each section of the graph cover? -ÕÀVi\ "« ,iÃi>ÀV vÀ -«i`> 29. How many times greater is the area of the new graph than the area of the original graph at the right? Is there, if any, a relationship with the increase in diameter and increase in area from one graph to the other? Explain.

Find the distance around and area of each figure. Round to the nearest tenth. Look Back

30. semicircle

31. semicircle

To review slope, see Lesson 7-5.

32. quarter circle

£ä vÌ x °

n

EXTRA

PRACTICE


H.O.T. Problems

33. FUNCTIONS Graph the circumference of a circle as a function of the diameter. Use values of d like 1, 2, 3, 4, and so on. What is the slope of this graph? How is the slope related to the formula for finding circumference? 34. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the circumference or area of a circle. 35. OPEN ENDED Draw and label a circle that has an area between 5 and 8 square units. 36. FIND THE ERROR Dario and Peter are finding the area of a circle with a diameter of 7. Who is correct? Explain your reasoning. Dario A = πr2 = π(7)2 ≈ 153.9 units2

Peter A = πr2 = π(3.5)2 ≈ 38.5 units2

37. CHALLENGE The radius of circle B is 2.5 times the radius of circle A. If the area of circle A is 8 square yards, what is the area of circle B?

!

"

Lesson 10-7 Circles: Circumference and Area

555

38. NUMBER SENSE The numerical value of the area of a circle is twice the numerical value of the circumference. What is the radius of the circle? (Hint: Use a table of values for radius, circumference, and area.) 39.

Writing in Math How are circumference and diameter related? Give the ratio of the circumference to the diameter and describe what happens to the circumference as the diameter increases or decreases.

40. The Blackwells have a circular pool with a radius of 10 feet. They plan on installing a 3-foot-wide walkway around the pool. What will be the area of the walkway?

41. A sprinkler is set to cover the area shown. Find the area of the grass being watered if the sprinkler reaches a distance of 20 feet.

3 ft 20 ft 10 ft

A 216.8 ft2

C 314.2 ft2

B 285.9 ft2

D 442.2 ft2

F 78.5 ft2

H 942.5 ft2

G 314.2 ft2

J 1,256.6 ft2

Find the area of each figure described. (Lesson 10-6) 42. trapezoid: height, 2 m; bases, 20 m and 18 m 43. parallelogram: base, 6 km; height, 8 km Find the sum of the measures of the interior angles of each polygon. (Lesson 10-5) 44. pentagon

45. quadrilateral

46. octagon

47. ALGEBRA Solve 2x - 7 > 5x + 14. (Lesson 8-6) 48. INTERNET SHOPPING For every order submitted, an online bookstore charges a $5 shipping fee plus a charge on the weight of the items being shipped of $2 per pound. The total shipping charges y can be represented by y = 2x + 5, where x represents the weight of the order in pounds. Graph the equation. (Lesson 7-6) 49. TRAVEL Jessica’s flight to Chicago leaves Rome, Italy at 4:30 P.M. on Tuesday. The flight time is 8.5 hours. If Rome is 7 hours ahead of Chicago, use Chicago time to determine when she is scheduled to arrive. (Lesson 1-1)

PREREQUISITE SKILL Find each sum. (page 745) 50. 200 + 43.9 51. 23.6 + 126.9 556 Chapter 10 Two-Dimensional Figures

52. 345.14 + 23.8

53. 720.16 + 54.7

EXTEND

10-7

Spreadsheet Lab

Circle Graphs and Spreadsheets

In the following example , you will learn how to use a computer spreadsheet program to graph the results of a probability experiment in a circle graph.

EXAMPLE A spinner like the one shown at the right was spun 20 times each for two trials. The data are shown below. Use a spreadsheet to make a circle graph of the result. Step 1 Enter the data in a spreadsheet as shown.

Circle Graphs.xls B C D E A Blue Red Yellow 1 10 17 13 2 Total Trials 4 8 8 3 Trial 1 6 9 5 4 Trial 2 5 Total Trials 6 7 8 9 10 Blue 11 Red 12 Yellow 13 14 15 Sheet 1

Sheet 2

The spreadsheet evaluates the formula ⫽SUM(D3:D4) to find the total.

Sheet 3

Step 2 Select the data to be included in your graph. Then use the graph tool to create the graph. The spreadsheet will allow you to add titles, change colors, and so on.

EXERCISES 1. Describe the results you would theoretically expect for one trial of 20 spins. Explain your reasoning. 2. Make a spinner like the one shown above. Collect data for five trials of 20 spins each. Use a spreadsheet program to create a circle graph of the data. 3. A central angle is an angle whose vertex is the center of a circle and whose sides intersect the circle. After 100 spins, what kind of central angle would you theoretically expect for each section of the circle graph? Explain. 4. Predict how many trials of the experiment are required to match the theoretical results. Test your prediction. 5. When the theoretical results match the experimental results, what is true about the circle graph and the spinner? Extend 10-7 Spreadsheet Lab: Circle Graphs and Spreadsheets

557

10-8

Area: Composite Figures

Main Idea • Find area of composite figures.

New Vocabulary composite figures

California is the most populous state in the United States. It ranks third among the U.S. states in area. In the diagram, the area of California is separated into polygons.

210 mi 213.3 mi 546.7 mi

280 mi

133.3 mi

a. Identify the polygons. 160 mi

b. Explain how polygons can be used to estimate the total land area.

40 mi 160 mi

c. What is the area of each region? What is the total area?

Area of Composite Figures So far in this chapter, we have discussed the following area formulas. Area Formulas Triangle

1 A = _ bh 2

Composite Figures

Trapezoid

Parallelogram

1 A = _ h(a + b)

Circle A = r2

A = bh

2

These formulas can be used to help you find the area of composite figures. A composite figure is a figure that cannot be classified as a single polygon or circle. Some composite figures are shown.

There can be more than one way to separate a composite figure. For example, another way to separate the first figure at the right is shown below.

IDAHO

To find the area of a composite figure, separate the composite figure into figures whose areas you know how to find. half of a circle or semicircle

triangle

parallelogram

IDAHO

trapezoid


rectangle

rectangle

EXAMPLE

Find Areas of Composite Figures

Find the area of the figure to the nearest tenth.

10 cm 12 cm

25 cm

Explore You know the dimensions of the figure. You need to find its area. Plan

First, separate the figure into a parallelogram and a semicircle. Then find the sum of the areas of the figures. Estimate: The area of the entire figure should be a little greater than the area of the rectangle. One estimate is 10 × 25 or 250.

Solve Semicircle

Area of Parallelogram

Area of Semicircle

A = bh

1 2 A= _ ␲r

The area of a

= 25 · 12

semicircle is _ the 1 2

b = 25 and h = 12

r=5

= 300 Simplify. ≈ 39.3 Simplify. The area of the figure is 300 + 39.3 or about 339.3 square centimeters.

area of a circle with the same radius. A = _ ␲r2 1 2

2 1 =_ · ␲ · 52 2

Check

Check the reasonableness of the solution by solving the problem another way. Separate the figure into two rectangles and a semicircle. 15 cm 10 cm 12 cm 25 cm

The area of one rectangle is 10 · 12 or 120 square centimeters, the area of the other rectangle is 12 · 15 or 180 square centimeters, and the area of the semicircle remains 39.3 square centimeters. 120 + 180 + 39.3 = 339.3 So, the answer is correct.

Find the area of each figure to the nearest tenth, if necessary. £Ç vÌ 1A. 1B. £È

vÌ n vÌ

È £ä vÌ

Personal Tutor at pre-alg.com Lesson 10-8 Area: Composite Figures

559

LANDSCAPE DESIGN Suppose one bag of mulch covers an area of about 9 square feet. How many bags of mulch will be needed to cover the flower garden?

28 ft

Step 1 Find the area of the flower garden. 38 ft

Area of rectangle A = bh

Area of a rectangle 8 ft

= 24 · 38 Substitute. = 912

Simplify. 24 ft

Area of parallelogram A = bh


= 28 · 8

Replace b with 28 and h with 8.

= 224

Simplify.

The area of the garden is 912 + 224 or 1136 square feet. Step 2

Find the number of bags of mulch needed. 1136 ÷ 9 ≈ 126.2 So, 127 bags of mulch will be needed. Why is the answer rounded up? FT

2. CARPETING Mr. Reyes wants to carpet his family room. If carpet costs $2.25 per square foot, how much would it cost to carpet Mr. Reyes’ family room if there is no leftover carpet?

FT FT FT

FT

Example 1 (p. 559)

Find the area of each figure. Round to the nearest tenth. 1.

3 yd 4 yd

2.

5 in.

5 in.

9 yd

Example 2 (p. 560)

HOME IMPROVEMENT For Exercises 3 and 4, use the diagram and the following information. The Slavens are painting their porch. One gallon of paint costs $19.95 and covers about 200 square feet.

9 ft 7 ft 12 ft 20 ft

10 ft

3. Suppose the Slavens need to apply only one coat of paint. How many gallons of paint will they need to buy?

18 ft

4. Find the total cost of the paint, not including tax. 560 Chapter 10 Two-Dimensional Figures


HOMEWORK

HELP


Find the area of each figure. Round to the nearest tenth, if necessary. 5.

6.

4 ft

3m

7.

5m

4 cm

3m

6 ft

9.5 cm 4 ft

4.2 m

12 ft

3.8 cm

8.

9.

3.2 in.

KM

3.2 in. KM

KM

10.

11.

17 m 3m 12 m

6 ft 1.5 ft

8 ft

8m

3 ft

4m

12. What is the area of a figure that is formed using a square with sides 8 meters and a semicircle with a diameter of 5.6 meters? 13. Find the area of a figure formed using a rectangle with base 3.5 yards and height 2.8 yards and a semicircle with radius 7 yards. 14. SPORTS In the diagram shown, a track surrounds a football field. How many bags of fertilizer would be needed to fertilize the grass region inside the track if a bag of fertilizer covers approximately 555 square yards?

100 yd 50

50 yd 25 yd

15. WALKWAYS A sidewalk forms a 3-foot wide border with the grass as shown. How many bags of concrete are needed to make the sidewalk if a bag of concrete covers approximately 0.75 square feet?

The design of various buildings use areas of composite figures. Visit pre-alg.com.

50

12 ft

GEOGRAPHY For Exercises 16–18, use the diagram below. 16. Tell how you would separate the 170 mi 290 mi composite figure into polygons to find its area. 35 mi Oklahoma 17. Use your method to find the total 130 mi Oklahoma City land area of Oklahoma. 18. RESEARCH Use the Internet or another source to find the actual total land area of Oklahoma. How does it compare to your answer in Exercise 17? Why do you think it is less or more?

12 ft

225 mi

305 mi

Lesson 10-8 Area: Composite Figures

561

Find the area of each blue region. Round to the nearest tenth if necessary. (Hint: Find the total area and subtract the tan area.) 19.

20.

15

21.

10

7

5

7

7 15

EXTRA

12

PRACTICE

See pages 785, 803.

3

8

8

11

3

4

14 Self-Check Quiz at pre-alg.com

H.O.T. Problems

22. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the area of composite figures. 23. OPEN ENDED Draw two examples of composite figures. Describe how you would find the area of each figure. 24. CHALLENGE To find people lost in Antarctica snowstorms without getting lost themselves, rescue teams plant a stake in the ground and tie a rope to it. They make knots in the rope at 20-meter intervals. They then put six people between the stake and the first knot and walk a circle. If the lost person is not found, the six people move between the first and second knots and walk another circle. They continue walking circles until the lost person is found. About how much area will a rescue team cover in three sweeps of the circle? 25.

Writing in Math How can polygons help you find the area of a composite figure? Illustrate with an example of a composite figure and an explanation as to how the figure can be separated to find its area.

26. GRIDDABLE In the diagram, a 4-foot wide patio surrounds a swimming pool. What is the area of the patio in square feet? Round to the nearest tenth.

27. The Lin family is buying a cover for their swimming pool shown below. The cover costs $3.19 per square foot. How much will the cover cost? FT

FT

FT

FT

A $219.27

C $699.47

B $258.54

D $824.74

Find the circumference and area of each circle. Round to the nearest tenth. (Lesson 10-7) 28. diameter: 8.5 ft

29. radius: 7 cm

30. diameter: 19 in.

Find the area of each figure described. (Lesson 10-6) 31. triangle: base, 9 in.; height, 6 in. 32. trapezoid: height, 3 cm; bases, 4 cm, 8 cm 33. CROSS COUNTRY Jeremy can run 31 miles in 25 minutes. How many minutes would it take 3 him to run 5 miles at this same rate? (Lesson 6-3) 562 Chapter 10 Two-Dimensional Figures

EXTEND

10-8

Spreadsheet Lab

Dilations and Perimeter and Area

Dilations change the dimensions of a figure proportionally. You can use a spreadsheet to investigate how the perimeter and area are affected when dimensions of a figure are changed proportionally.

ACTIVITY Step 1 In Column A, enter the labels as shown. Columns B, C, D, E, and F will be used for five similar rectangles. Step 2 Enter the formula =2*(B1+B2) for the perimeter of the rectangle in cell B3. Copy the formula into the other cells in row 3. Step 3 Write a formula to find the area of the rectangle. Copy the formula in the cells in row 4. Step 4 Enter the formula =B1*C5 in cell C1 and enter =B2*C5 in cell C2. These formulas find the dimensions of rectangle C based on the dimensions of rectangle B and the scale factor you enter. Enter similar formulas in the cells for columns D, E, and F. Step 5 Type the formula =C3/B3 in cell C6, type =D3/B3 in cell D6, and so on. This formula will find the ratio of the perimeter of each of the other rectangles to the perimeter of rectangle B. Step 6 Write a formula for the ratio of the area of rectangle C to the area of rectangle B. Enter the formula in cell C7. Enter similar formulas in the cells in row 7.

%XCEL SAMPLEXLS "

!

,ENGTH 7IDTH 0ERIMETER !REA 3CALE &ACTOR 2ATIO OF 0ERIMETERS 2ATIO OF !REAS 3HEET

3HEET

#

3HEET

Step 7 Use Columns D, E, and F to find the perimeters, areas, and ratios for rectangles with scale factors of 3, 4, and 5.

ANALYZE THE RESULTS 1. Compare the ratios in rows 5, 6, and 7 of columns C, D, and E. What do you observe? 2. What happens to the perimeter of a rectangle if the dimensions are doubled? multiplied by 4? multiplied by n? Change the original dimensions of the rectangle and the scale factors in the spreadsheet to verify your conclusion. 3. Describe the effect on the area of a rectangle if its dimensions are doubled. multiplied by 4. multiplied by n. Change the original dimensions of the rectangle and the scale factors in the spreadsheet to verify your conclusion. 4. Change the scale factors in cells C5, D5, E5, and F5 to 0.1, 0.2, 0.3 and 0.5. Describe the ratios of the perimeters and areas for these reductions. 5. Use the perimeter formula P = 2( + w) and the area formula A = w to explain the effects of changing the dimensions of a rectangle proportionally. Extend 10-8 Spreadsheet Lab: Dilations and Perimeter and Area

563

CH

APTER

10



Key Vocabulary adjacent angles (p. 513) alternate exterior angles


LE S 4RAPE 4R IA N G ZOIDS E

alternate interior angles (p. 512)

Key Concepts Angle Relationships

(p. 512)

(Lesson 10-1)

• When two parallel lines are cut by a transversal, the corresponding angles, the alternate interior angles, and the alternate exterior angles are congruent. • Two angles are complementary if the sum of their measures is 90°. • Two angles are supplementary if the sum of their measures is 180°.

Congruent Triangles and Transformations (Lessons 10-2 and 10-3)

altitude (p. 545) base (p. 545) circle (p. 551) circumference (p. 551) complementary (p. 513) composite figures (p. 558) congruent (p. 518) corresponding angles (p. 512)

diameter (p. 551)

dilation (p. 524) parallel lines (p. 512) parallelogram (p. 533) perpendicular lines (p. 514) quadrilateral (p. 532) radius (p. 551) reflection (p. 524) regular polygon (p. 539) rhombus (p. 533) supplementary (p. 514) transformation (p. 524) translation (p. 524) trapezoid (p. 533) vertical angles (p. 513)

• Figures that have the same size and shape are congruent. • Three types of transformations are translations, reflections, and dilations.

Quadrilaterals and Polygons (Lessons 10-4 and 10-5) • A trapezoid, parallelogram, rhombus, square, and rectangle are examples of quadrilaterals. • Polygons are classified by the number of sides they have. • If a polygon has n sides, then the sum of the degree measures of the interior angles of the polygon is (n - 2)180.

Area and Circumference (Lessons 10-6 to 10-8) 1 • Area of a trapezoid: A = _ h(a + b) 2

• Area of a parallelogram: A = bh 1 • Area of a triangle: A = _ bh 2

• Circumference of a circle: C = 2␲r • Area of a circle: A = ␲r2


Vocabulary Check Choose the correct term to complete each sentence. 1. When two angles have the same vertex, share a common side, and do not overlap, they are (adjacent, vertical) angles. 2. A (rhombus, trapezoid) has four congruent sides. 3. In congruent triangles, the (corresponding angles, adjacent angles) are congruent. 4. In a (dilation, translation), a figure is enlarged or reduced around a fixed point. 5. Two lines in a plane that never intersect are called (parallel, perpendicular) lines. 6. A (regular polygon, composite figure) has congruent sides and congruent angles. 7. The (altitude, base) can be outside the parallelogram.


Lesson-by-Lesson Review 10-1


(pp. 512–517)

Example 1 In the figure below m n and t is a transversal. If m∠6 = 77°, find m∠1 and m∠2.

In the figure below m and t is a transversal. If m∠1 = 109°, find the measure of each angle. t 8. ∠5 9. ∠3

1 4

10. ∠2 5

11. ∠6

8

ᐉ

2 3

m

6 7

N

M

Since ∠6 and ∠1 are alternate exterior angles, they are congruent. So, m∠1 = 77°.

£ Î Ç {

x Ó n È

Since ∠6 and ∠2 are supplementary, m∠6 + m∠2 = 180.

12. ∠4 13. ∠8 14. DOORS A door can swing open 180°. The door is open at an angle of 99°. What is the measure of the angle between the door and the door jam?

T

m∠6 + m∠2 = 180 Supplementary angles 77 + m∠2 = 180 Replace m∠6 with 77. m∠2 = 103 Subtract 77 from each side. So, m∠2 = 103°.

10-2

Congruent Triangles

(pp. 518–523)

15. Name the corresponding parts in the congruent triangles shown. Then write a congruence statement. '

A

-

(

&

,

.

16. BASEBALL In a softball diamond, the triangle formed by home plate, first base, and second base is congruent to the triangle formed by home plate, third base, and second base. If it is 65 feet from home plate to first base, how far is it from third base to home plate? 3ECOND "ASE

4HIRD "ASE

Example 2 Name the corresponding parts in the congruent triangles shown. Then write a congruence statement.

&IRST "ASE

X

B

C

Y

Z

䉭ABC 䉭XYZ

A corresponds to X, so ∠A ∠X. B corresponds to Y, so ∠B ∠Y. C corresponds to Z so ∠C ∠Z. −− −− −− −− YZ corresponds to BC, so YZ BC. −− −− −− −− ZX corresponds to CA , so ZX CA. −− −− −− −− XY corresponds to AB, so XY AB. CAB ZXY.

FT


565

CH

A PT ER

10 10-3


Transformations on the Coordinate Plane

(pp. 524–530)

Graph each figure and its image. 17. The vertices of a rectangle are C(0, 2), D(2, 0), F(-1, -3), and G(-3, -1). The rectangle is translated 4 units right and two units down. 18. The vertices of a triangle are H(-1, 4), I(-4, -2), and J(-2, -1). The triangle is reflected over the y-axis. 19. A triangle has vertices N(6, 3), P(3, 9), and Q(9, 6). The triangle is dilated by 1 . a scale factor of _ 3

20. ESCALATORS What type of transformation is used when moving up an escalator? 21. PHOTOGRAPHS Percy wants to increase the size of his 5-inch by 7-inch photograph by a scale factor of 1.5 on his computer. What is the new size of the photograph?

Example 3 The vertices of JKL are J(1, 2), K(3, 2), and L(1, -1). Find the vertices of the image after a translation 3 units left and 2 units up, after a reflection over the x-axis, and after a dilation by a scale factor of 2 centered at the origin. Translation: This translation can be written as (-3, 2). vertex 3 left, 2 up → J(1, 2) + (-3, 2) K(3, 2) + (-3, 2) → L(1, -1) + (-3, 2) →

translation J(-2, 4) K(0, 4) L(-2, 1)

Reflection: Use the same x-coordinate and multiply the y-coordinate by -1. vertex reflection J(1, 2) → (1, 2 · -1) → J(1, -2) K(3, 2) → (3, 2 · -1) → K(3, -2) L(1, -1) → (1, -1 · -1) → L(1, 1) Dilation: Multiply the x- and y-coordinates by the scale factor. vertex dilation J(1, 2) → (1 · 2, 2 · 2) → J(2, 4) K(3, 2) → (3 · 2, 2 · 2) → K(6, 4) L(1, - 1) → (1 · 2, -1 · 2) → L(2, -2)

10-4

Quadrilaterals

(pp. 532–536)

Example 4 Find the value of x. Then find the missing angle measures.

Find the value of x. Then find the missing angle measures. 23. 22. 110˚ 110˚ 2x˚ 78˚

x˚

x˚

24. STREET SIGNS Name two classifications of this quadrilateral. (Hint: Think about different viewing angles.) 566 Chapter 10 Two-Dimensional Figures Robert Brenner/PhotoEdit

x˚

x + 2x + 96 + 87 3x + 183 3x x

= = = =

360 360 177 59

x˚

2x˚ 96˚ 87˚

Write the equation. Combine like terms. Subtract. Divide each side by 3.

The value of x is 59. So, the missing angle measures are 59° and 2(59) or 118°.



10-5

Polygons

(pp. 539–543)

Classify each polygon. Then find the sum of the measures of the interior angles. 26. 25.

27.

(n - 2)180 = (5 - 2)180

28.

Replace n with 5.

= 3(180) or 540

29. STREET SIGNS What is the measure of each interior angle of a stop sign?

10-6

Example 5 Find the measure of one interior angle in a regular pentagon. Step 1 Find the sum of the measures of the angles. A pentagon has 5 sides. Therefore, n = 5.

Area: Parallelograms, Triangles, and Trapezoids

Simplify.

The sum of the measures of the interior angles is 540°. Step 2 Divide the sum by 5 to find the measure of one angle. So, the measure of one interior angle in a regular pentagon is 540 ÷ 5 or 108°.

(pp. 545–550)

Example 6 Find the area of the trapezoid.

Find the area of each figure. 31. 30.

2 cm 1

13 in.

5 2 yd

1.8 cm

32.

5 cm

4 yd

9 in.

33.

8.7 m

{ {

6.2 m

1 A=_ h(a + b)

2 _ = 1 (1.8)(2 + 5) 2

= 6.3 5.0 m

34. CARPET Mrs. Jackson wants to carpet her bedroom. The carpet that she wants costs $2.99 a square foot. How much will it cost Mrs. Jackson to carpet her bedroom? £ä vÌ £Ó vÌ

Area of a trapezoid Substitution Simplify.

The area of the trapezoid is 6.3 square centimeters. Example 7 Find the area of the triangle. 1 A=_ bh

Î°

Area of a triangle

2 1 =_ (9)(3.9) Replace b with 9 and h with 3.9. 2 = 17.55 Simplify.

The area of the triangle is 17.55 square meters.


567

CH

A PT ER

10 10-7


Circles: Circumference and Area

(pp. 551–556)

Find the circumference and area of each circle. Round to the nearest tenth. 36. 35.

Example 8 Find the circumference of the circle. Round to the nearest tenth.

2.1 m

5 cm

C = 2␲r = 2 · ␲ · 7.5 37.

38.

≈ 47.1

18 ft

Ç °

Area: Composite Figures

7 in.

34 cm

6 in.

A = ␲r2 =␲·

15 cm

=␲·9

Evaluate 32.

≈ 28.3

The area is about 28.3 m2.

Example 10 Find the area of the figure. Area of a parallelogram A = bh

5 cm

Area of a square A=

42. MUSEUM The floor plan of a new museum is shown. What is the area of the museum?

FT FT FT

FT

Replace r with 3.

= 3(5) or 15

16 in.

FT

Area of a circle

32

(pp. 558–562)

Find the area of each figure. Round to the nearest tenth, if necessary. 41. 40. 7 in.

Circumference of a circle 15 Replace r with _ or 7.5. 2 The circumference is about 47.1 m.

Example 9 Find the area of the circle. Round to the nearest tenth. Î

39. FERRIS WHEEL The wheel section of the first Ferris wheel had a diameter of 250 feet. What is the distance passengers would travel if they stayed on the ride for 15 rotations?

10-8

15 m

FT


3 cm

s2

= 32 or 9

3 cm

The area of the figure is 15 + 9 or 24 square centimeters.

CH

A PT ER

10

Practice Test

In the figure below, a b, and c is a transversal. If m∠5 = 58°, find the measure of each angle.

a

1. ∠6 2. ∠7

2 5 6 8

3. ∠4

ALGEBRA Find the value of x. Then find the missing angle measure(s). 9.

b 3 1 4 7

c

121⬚

10.

4. ∠3 5. For the pair of congruent triangles, name the corresponding parts. Then write a congruence statement. "

x⬚

63⬚

( 2x 1)⬚

(x 11)⬚

70⬚ x⬚

40⬚

Find the area of each figure described.

$

11. triangle: base, 21 ft; height, 16 ft 12. parallelogram: base, 7 m; height, 2.5 m

!

%

#

6. MULTIPLE CHOICE On the graph below, point F is translated to F. Y &

'

" ( X &

Which coordinate pair represents G ? A (-3, -8)

C (2, -6)

B (2, -7)

D (2, -2)

Find the circumference and area of each circle. Round to the nearest tenth. 13. The radius is 3 miles. 14. The diameter is 10 inches. 15. MULTIPLE CHOICE What is the diameter of a bicycle tire if its circumference is 54.8 inches? Round to the nearest tenth. F 8.7 in.

H 17.4 in.

G 15.6 in.

J 34.9 in.

16. In the diagram, 1 square unit equals 5 square feet. What is the area of the figure?

Classify each polygon. Then find the sum of the measures of the interior angles. 7.

8.



569

CH

A PT ER

10


Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The figure shows a square inside of a circle.

Which procedure should be used to find the area of the shaded region? A Find the area of the square and then subtract the area of the circle. B Find the area of the circle and then subtract the area of the square. C Find the perimeter of the square and then subtract the circumference of the circle. D Find the circumference of the circle and then subtract the perimeter of the square. 2. A golf cart wheel travels about 38 inches in 1 full rotation. What is the diameter of the wheel to the nearest inch? F 12 in. H 15 in. G 14 in. J 18 in.

4. The area of a quilt block is 150 square inches. Which best represents the length of a side of the square? A 10.9 in. B 11.5 in. C 12.2 in. D 15.0 in. 5. If the equation for a line is y = 3x + 2, which table of ordered pairs contains only points on this line? F

G

x

y

-2

-1

-1

1

0 1

H

x

y

-1

-3

0

0

3

1

3

5

2

6

x

y

J

x

y

-2

-4

-1

-2

-1

-1

0

5

0

2

1

12

1

5

2

19

6. GRIDDABLE What is the eighth term of the sequence given by the expression below? 5(n + 2)

Questions 1 and 2 Most standardized tests include any necessary formulas in the test booklet. It helps to be familiar with formulas such as the area of a square and the circumference of a circle, but use any formulas that are given to you.

3. GRIDDABLE A ship left the harbor and traveled 20 kilometers due north. Then the ship turned and traveled 8 kilometers due east. What is the shortest distance, in kilometers, from the ship’s current location to the harbor? Round your answer to the nearest tenth.

570 Chapter 10 Two-Dimesional Figures

7. Mrs. Collins plans to buy 4 hamburgers for each family attending an open house. If 20 families are coming to the open house and if hamburgers cost $8.75 per package, what other information is needed to find the cost of the hamburgers? A The cost of the buns B The number of people who eat hamburgers C The number of hamburgers in each package D The number of times the hamburgers will be served



8. The distance from Neptune to the Sun is 28,000,000,000 miles. Which expression represents this number in scientific notation? F 2.8 × 10-10 G 28 × 109 H 2.8 × 1010 J 0.28 × 1011

11. A circle with a radius of 4 units has its center at (-3, 1) on a coordinate grid. Ç È x { Î Ó £ Ç Èx{ ÎÓ £" £ Ó Î { x È Ç

9. The two rectangles below are similar. !

"

1

IN

$

#

2 IN

! IN

4

3

Pre-AP Record your answers on a sheet of paper. Show your work. Y 12. a. Copy JKM onto a * sheet of grid paper. Label its vertices. Graph the image + " of JKM after a translation 2 units left and 3 units down. Label the translated image J’K’M’. b. Graph the image of J’K’M’ after a reflection over the y-axis. Label the reflected image J’’K’’M’’. c. On another sheet of grid paper, graph JKM. Then graph JKM after a reflection over the y-axis.

10. The turnstile at a subway station counted 3424 passengers during a four-hour period. If the number of people passing through the turnstile remains the same, which proportion can be used to find x, the number of passengers through the turnstile during a 10-hour period? x 4 =_ F _ 10

10 x G_ =_ 4

3424

10 4 H_ =_ x

£ Ó Î { x È ÇX

If the circle is translated 2 units down and 3 units right, what will be the coordinates of the new center? A (0, -1) C (1, -1) B (-5, 4) D (-2, -3)

If the area of the larger rectangle is 392 square inches, find the area of the smaller rectangle. A 64 in2 C 106 in2 B 82 in2 D 128 in2

3424

Y

3424

10 4 J _ =_ 3424 x

X


1

2

3

4

5

6

7

8

9

10

11

12

Go to Lesson...

10-8

10-7

9-4

9-1

8-2

3-7

1-1

4-7

9-6

6-3

10-3

10-3


571

11 •

Use geometry to model and describe the physical world.

•

Find volumes and surface areas of three-dimensional figures.

•

Describe how changes in dimensions affect linear, area, and volume measures.


Key Vocabulary lateral area (p. 597) polyhedron (p. 575) similar solids (p. 608) surface area (p. 597) volume (p. 583)

Real-World Link Buildings The Louvre museum in Paris has a square glass pyramid at the entrance with a slant height of about 92 feet. Its square base is 116 feet on each side.

Compare and Contrast Three-Dimensional Figures Make this Foldable to help you organize information about surface area and volume of three-dimensional figures. Begin with a plain piece of 11’’ × 17’’ paper.

1 Fold the paper in thirds

2 Fold a 2’’ tab along the short side. Then fold the rest in fourths.

3 Draw lines along the fold and label as shown.

3URFACE !REA

6OLUME #H 0RI SM S #Y LIND ERS

lengthwise.

572 Chapter 11 Three-Dimensional Figures Walter Bibikow/Index Stock Imagery

AM 0YR

IDS

S

NE #O



Option 1


Determine whether each figure is a polygon. If it is, classify the polygon. (Lesson 10-5)

1.

2.

Example 1 Determine whether the figure is a polygon. If it is, classify the polygon.

The polygon has 4 sides. It is a quadrilateral. 3. SIGNS Classify the shape of the sign shown. (Lesson 10-5)

Find each product. (Prerequisite Skills, pp. 747–748) 4. 8.5 · 2 5. 3.2(3.2)10 1 (50)(9.3) 6. _

1 _ 1 7. _ ·3·8 3(2

3

)

Example 2 1 Find _ (24)(5.8). 3

_1 (24)(5.8) = ⎡_1 (24)⎤(5.8) ⎣3

3

8. MUSIC Suppose you practice the cello 2 hour every day. How many hours for _ 3 do you practice every week? (Lesson 5-3)

Determine whether each pair of ratios forms a proportion. (Lesson 6-2) 3 _ 9. _ , 9 8 24 1.2 _ , 6 11. _ 5 25

7 _ 10. _ , 14 2 6 1.6 _ 12. _ , 3.6 2 6

13. WATER Determine whether the set of numbers in the table are proportional. Explain your reasoning. (Lesson 6-2) Bottles

1

2

3

4

Cost ($)

1.25

2.50

3.75

5.00

⎦

= (8)(5.8) = 46.4

Associative Property Simplify. Simplify.

Example 3 12 2 Determine whether _ and _ form a 7 42 proportion. 12 _2 = _

Write a proportion.

42

7

2(42) = 7(12) Find the cross products. 84 = 84

Simplify.

2 12 _ 12 Since _ =_ , 2 and _ form a proportion. 7

42 7

42


573

EXPLORE

11-1

Geometry Lab

Building Three-Dimensional Figures Different views of a stack of cubes are shown. A point of view is called a perspective. You can build or draw a three-dimensional figure using different perspectives.

top

side

front

ACTIVITY Build the figure corresponding to the views shown. Then use isometric dot paper to draw the model. top Step 1 Use the top view to build the base of the figure. The top view shows that the base is a 2-by-3 rectangle. Step 2 Use the side view to complete the figure. The side view shows that the height of the first row is 1 unit, and the height of the second and third rows is 2 units. Step 3 Use the front view to check the figure. The front view is a 2-by-2 square. This shows that the overall height side front and width of the figure is 2 units. So, the figure is correct.

EXERCISES The top view, a side view, and the front view of three-dimensional figures are shown. Use cubes to build each figure. Draw your model on isometric dot paper. 1. top

side

front

2. top

side

front

3.

top

side

front

4. top

side

front

5.

top

side

front

6. top

side

front

Draw and label the top view, a side view, and the front view for each figure. 7.

574 Chapter 11 Three-Dimensional Figures

8.

9.

11-1


Main Ideas • Identify threedimensional figures. • Draw various views of three-dimensional figures.

New Vocabulary plane solid polyhedron edge vertex face prism base pyramid cylinder cone

USX Tower, Pittsburgh

Great Pyramid, Egypt

a. If you observed the Great Pyramid or the USX Tower from directly above, what geometric figure(s) might you see? b. If you stood directly in front of each structure, what geometric figure would you see? c. Explain how you can see different polygons when looking at a 3-dimensional figure.

Identify Three-Dimensional Figures and Parts A plane is a twodimensional flat surface that extends in all directions. There are different ways that planes may be related in space. Intersect in a Line Dimensions A two-dimensional figure has two dimensions, length and width. A threedimensional figure has three dimensions, length, width, and depth (or height).

P

ᐉ

Intersect in a Point

Q

No Intersection

Q

P A

These are called parallel planes.

Intersecting planes can also form three-dimensional figures or solids. A polyhedron is a solid with flat surfaces that are polygons. An edge is where two planes intersect in a line. A face is a flat surface. A vertex is where three or more planes intersect in a point.

Lesson 11-1 Three-Dimensional Figures (l)Yann Arthus-Bertrand/CORBIS, (r)Andre Jenny/Alamy

575

A prism is a polyhedron with two parallel, congruent faces called bases that are polygons. A pyramid is a polyhedron with one base that is any polygon. Its other faces are triangles. Prisms and pyramids are named by the shape of their bases. pyramid

prism

base

bases

Polyhedrons Common Misconception

Polyhedron

In a rectangular prism, the bases do not have to be on the top and bottom. Any two parallel rectangles are bases. In a triangular pyramid, any face is a base.

Number of Bases Polygon Base

triangular prism

rectangular prism

triangular pyramid

rectangular pyramid

2

2

1

1

triangle

rectangle

triangle

rectangle

Figure

There are solids that are not polyhedrons. A cylinder is a solid with congruent, parallel bases that are circles connected with a curved side. A cone is a solid with one circular base and a vertex, connected by a curved side. cone

cylinder

h

bases

h

base

r r

Vocabulary Link Cone Everyday Use a crispy wafer used to hold ice cream Math Use a solid with one circular base connected to a vertex by a curved side

You can use the labels on the vertices to name a base or a face of a solid.

EXAMPLE

Identify Solids

Identify each solid. Name the bases, faces, edges, and vertices. a. A

C B F

D E

This figure has two parallel congruent bases that are triangles, ABC and DEF, so it is a triangular prism. faces: ABC, ADEB, BEFC, CFDA, DEF −− −− −− −−− −− −− −− −− −− edges: AB, BC, CA, AD, BE, CF, DE, EF, FD vertices: A, B, C, D, E, F 576 Chapter 11 Three-Dimensional Figures

b.

J

M

N

K

L

This figure has one rectangular base, KLMN, so it is a rectangular pyramid. faces: JKL, JLM, JMN, JNK, KLMN −− −− −− −− −−− −− −−− −−− edges: JK, JL, JM, JN, NK, KL, LM, MN vertices: J, K, L, M, N c. 1

0

The solid has two parallel, circular bases. So it is a cylinder. bases: circles P and Q

I

1A.

J

E

1B.

1C.

4

F R

L H

K G

5

3 2

Draw Views of Three–Dimensional Figures Three-dimensional objects drawn on isometric dot paper can be shown using different two-dimensional views.

ARCHITECTURE An architect’s sketch shows the plans for a new office building. Each unit on the drawing represents 40 feet.

a. Draw a top view and find the area of the ground floor. The drawing is 6 × 5, so the actual dimensions are 6(40) × 5(40) or 240 feet by 200 feet. A=·w = 240 · 200 or 48,000 Extra Examples at pre-alg.com

Formula for area

ft2

top view Lesson 11-1 Three-Dimensional Figures

577

b. Draw a top-count view of the building. Using the top view from part a from the previous page, write the number of levels for each unit of the building.

Top-Count View A top-count view shows the height or number of levels in a top view of a threedimensional figure.

Ó Ó Ó Ó Ó

Ó Î Î Î Ó

Ó Î { Î Ó

Ó Î { Î Ó

Ó Î Î Î Ó

Ó Ó Ó Ó Ó

c. How many floors are in the office building if each floor is 20 feet high? You can see from the side view and top-count view that the height of the building is 4 units.

side view

total height: 4 units × 40 feet per unit = 160 feet number of floors: 160 feet ÷ 20 feet per floor = 8 floors

2. PHOTOGRAPHY The sketch shows a stand photographers use for group photographs. Each unit in the drawing represents 1.5 feet. Draw a top-count view of the podium and find the height of the stand. Personal Tutor at pre-alg.com

Example 1 (pp. 576–577)

Identify each solid. Name the bases, faces, edges, and vertices. 1.

2.

Q

D

C

A F

R

T

E

3.

9

B G H

S

Example 2 (pp. 577–578)

BUILDING For Exercises 4–6, use the information below. The sketch below shows the plans for porch steps.

4. Draw and label the top, front, and side views. 5. Draw a top-count view. 6. If each unit on the drawing represents 4 inches, what is the height of the steps in feet? 578 Chapter 11 Three-Dimensional Figures

HOMEWORK

HELP


Identify each solid. Name the bases, faces, edges, and vertices. 7. L

8.

M N

P

-

.

I

Q

R

J

F

S

T

10.

9.

K H

G A

11.

W

12. +

B

E

Z

X

D

C

Y

ARCHITECTURE For Exercises 13–15, use the information below. The sketch at the right shows the plans for a high-rise building. Each unit on the drawing represents 75 feet. 13. Draw and label the top, front, and side views and a top-count view. 14. Find the area of the ground floor. 15. How many floors are in the office building if each floor is 15 feet high? COMICS For Exercises 16 and 17, use the comic below. SHOE

16. Which view of the Washington Monument is shown? 17. RESEARCH Use the Internet or another source to find a photograph of the Washington Monument. Draw and label the top, side, and front views. 18. BUILDINGS Use the top-count view at the right to sketch the three-dimensional model of the building.

{

{

{

{

{

Î

Î

Î

Î

{

Ó

Ó

Ó

Î

{

19. STATUES Use the top, front, and side views of the statue below to draw a top-count view. (Hint: Use isometric grid paper to help you draw the statue.) /«

-ì

ÀÌ

Lesson 11-1 Three-Dimensional Figures SHOE 8/31/94 ©Tribune Media Services. All rights reserved. Reprinted by permission.

579

ART For Exercises 20 and 21, refer to Picasso’s painting The Factory, Horta de Ebro shown at the right. 20. Describe the polyhedrons shown in the painting. 21. Explain how the artist portrayed three-dimensional objects on a flat canvas. 22. RESEARCH Find other examples of art in which polyhedrons are shown. Describe the polyhedrons. Real-World Link Pablo Picasso (1881–1973) was one of the developers of a movement in art called Cubism. Cubist paintings are characterized by their angular shapes and sharp edges.

Determine whether each statement is sometimes, always, or never true. Explain. 23. The bases of cylinders have different radii. 24. Two planes intersect in a single point. 25. Three planes do not intersect in a point.

Source: World Wide Arts Resources

EXTRA

PRACTICE

26. PATTERNS The top-count views of a block pattern are shown. How many blocks does it take to build Stage 10?

See pages 786, 804.

Î Ó


£

-Ì>}i £

H.O.T. Problems

£

£

-Ì>}i Ó

Ó

£

Ó £

-Ì>}i Î

27. OPEN ENDED Choose a solid object from your home and draw the top, front, and side views. 28. REASONING Are cylinders polyhedrons? Explain. B

CHALLENGE Skew lines are lines that are neither intersecting nor parallel. They lie in different planes. In the rectangular −− prism at the right, EC is a diagonal of the prism because it −− joins two vertices that have no faces in common. So EC is −−− skew to AD. For Exercises 29–31, use the rectangular prism below. 29. Identify a diagonal. −−− 30. Name four segments skew to QR. −− −−− 31. State whether WR and XY are parallel, skew, or intersecting.

C

A D F

G

E H

X W

Y Z

Q

32.

Writing in Math

How are two-dimensional figures related to three-dimensional figures? Include a description of how two-dimensional figures can form a three-dimensional figure.

580 Chapter 11 Three-Dimensional Figures (tl)Roberto Otero/Black Star Publishing/PictureQuest; (tr)“The Factory in Horta de Ebbo”, 1909. Pablo Picasso. Oil on canvas. Hermitage Museum, St. Petersburg, Russia/SuperStock

P

R S

33. Which drawing shows a threedimensional view of the solid figure represented below?

/« 6iÜ

34. Which figure does NOT have the same dimensions as the other figures? F

H

G

J

-ì 6iÜ

A

C

B

D

35. Find the area of ABCDEF at the right if each unit represents 1 square centimeter. (Lesson 10-8) 36. Find the circumference and the area of a circle whose radius is 6 meters. Round to the nearest tenth. (Lesson 10-7)

A

B

F

C

E

D

37. Triangle ABC is similar to triangle MNP. Write a proportion to find the missing measure. Then find the value of x. (Lesson 9-7) P

A

6 cm

N 2 cm

6 cm

B

M x cm

C

38. ALGEBRA Triangle DEF has angles whose measures are in the ratio 1 : 4 : 7. What is the measure of the largest angle? (Lesson 9-3) Solve each inequality. Check your solution. (Lesson 7-4) 39. c + 4 < 12

40. 7 ≥ t - 2

42. k + (-4) ≥ 3.8

1 1 43. y - _ < 1_ 4

41. -26 < n + (-15) 3 1 44. 3_ >a-_

2

33 41 preferred vanilla ice cream, and _ 45. ICE CREAM In a survey of teenagers, _ 50 60

5

10

preferred chocolate ice cream. Of those surveyed, do a greater fraction prefer vanilla or chocolate? Explain. (Lesson 5-1)

46. Simplify (4x)(-6y). (Lesson 3-2)

PREREQUISITE SKILL Find the area of each triangle described. (Lesson 10-5) 47. base, 4 in.; height, 7 in.

48. base, 10 ft; height, 9 ft

49. base, 6.5 cm; height, 2 cm

50. base, 0.4 m; height, 1.3 m Lesson 11-1 Three-Dimensional Figures

581

EXPLORE

11-2

Geometry Lab

Volume In this activity, you will investigate volume by making containers of different shapes and comparing how much each container holds.

ACTIVITY Step 1 Use three 5-inch by 8-inch index cards to make three different containers, as shown below.

square base with 2-inch sides

circular base with 8-inch circumference

triangular base with sides 2 inches, 3 inches, and 3 inches

Step 2 Tape one end of each container to another card as a bottom, but leave the top open, as shown at the right. Step 3 Estimate which container would hold the most (have the greatest volume) and which would hold the least (have the least volume), or whether all the containers would hold the same amount. Step 4 Use rice to fill the container that you believe holds the least amount. Then pour the rice from this container into another container. Does the rice fill the second container? Continue the process until you find out which container, if any, has the least volume and which has the greatest.

ANALYZE THE RESULTS 1. Which container holds the greatest amount of rice? Which holds the least amount? 2. How do the heights of the three containers compare? What is each height? 3. Compare the perimeters of the bases of each container. What is each base perimeter? 4. Trace the base of each container onto grid paper. Estimate the area of each base. 5. Which container has the greatest base area? 6. Does there appear to be a relationship between the area of the bases and the volume of the containers when the heights remain unchanged? Explain. 582 Chapter 11 Three-Dimensional Figures

11-2

Volume: Prisms and Cylinders

Main Ideas • Find volumes of prisms. • Find volumes of circular cylinders.

The rectangular prism is built from 24 cubes. a. Build three more rectangular prisms using 24 cubes. Enter the dimensions and base areas in a table.

New Vocabulary volume

Prism

Length (units)

Width (units)

Height (units)

Area of Base (units2)

1

6

1

4

6

2 3 4

b. Volume equals the number of cubes that fill a prism. How is the volume of each prism related to the product of the length, width, and height? c. Make a conjecture about how the area of the base B and the height h are related to the volume V of a prism.

Measures of Volume A cubic centimeter (cm3) is a cube whose edges measure 1 centimeter. 1 cm

Volumes of Prisms The prism above has a volume of 24 cubic centimeters. Volume is the measure of space occupied by a solid region. To find the volume of a prism, use the area of the base and the height. Volume of a Prism Words

The volume V of a prism is the area of the base B times the height h.

Models

Symbols V = Bh

h

B

1 cm

h

B

1 cm

EXAMPLE

Volume of a Rectangular Prism

Find the volume of the prism. 4 in. 2 in. 7.5 in.

V = Bh Formula for volume of a prism = ( · w)h The base is a rectangle, so B = · w. = (7.5 · 2)4 = 7.5, w = 2, h = 4 = 60 in3 Simplify.


1. Find the volume of a rectangular prism with a length of 10 meters, a width of 13 meters, and a height of 21 meters.

Lesson 11-2 Volume: Prisms and Cylinders

583

EXAMPLE

Volume of a Triangular Prism

Find the volume of the triangular prism. V = Bh

Formula for volume of a prism

1 = _ · 4 · 3 h B = area of base or _12 · 4 · 3 3 cm

6 cm

(2 ) 1 = (_ · 4 · 3)6 2 = 36 cm3

The height of the prism is 6 cm. Simplify.

4 cm

1 2. Find volume of a triangular prism with the base of the triangle 6_ inches, 2 altitude of the triangle 8 inches, and height of the prism 15 inches.

Estimation Estimate before solving the problem. h ≈ 200 ÷ 10 ÷ 5 ≈ 4 ft

AQUARIUMS A wall is being constructed to enclose three sides of an aquarium that is a rectangular prism 8 feet long and 5 feet wide. If the aquarium is to contain 220 cubic feet of water, what is its height? V = Bh

Formula for volume of a prism

V=·w·h

Replace B with · w.

220 = 8 · 5 · h

Replace V with 220, with 8, and w with 5.

220 = 40h

Simplify.

5.5 = h


The height of the aquarium is 5.5 feet. How does this compare to the estimate?

3. POOLS A children’s rectangular pool holds 480 cubic feet of water. What is the depth of the pool if its length is 30 feet and its height is 16 feet?

Find the volume of the model of the house at the right.

3 ft 8 ft

A 180 ft3 B 960 ft3 C 1140 ft3

10 ft 12 ft

D 1320 ft3 Read the Test Item The solid is made up of a rectangular prism and a triangular prism. The volume of the solid is the sum of both volumes. 584 Chapter 11 Three-Dimensional Figures

Solve the Test Item Estimate You can eliminate A and B as answers because the volume of the rectangular prism is 960 ft3, so the volume of the whole solid must be greater.

V(solid) = V(rectangular prism) + V(triangular prism) V(solid) = · w · h + Bh

Volume formulas

1 = 12 · 10 · 8 + (_ · 10 · 3) · 12

Substitute.

= 960 + 180 or 1140 ft3

Simplify.

2

£ V

The answer is C.

Ó V

4. Find the volume of the package at the right. F 120

cm3

H 195

G 127.5 cm3

£x V

Ó°x V

cm3

{ V

J 214.5 cm3


Reading Math

Volumes of Cylinders Like prisms, the volume of a cylinder is the product of

Cylinders In this text, cylinder refers to a cylinder with a circular base.

the base area and the height. Volume of a Cylinder The volume V of a cylinder with radius r is the area of the base B times the height h.

Words

Symbols

Model

r

V = Bh, where B = r2 or V = r2h

EXAMPLE

h

Volume of a Cylinder

Find the volume of each cylinder. Round to the nearest tenth. a.

5 ft

Estimate

3 · 52 · 15 = 1125

V = Bh 15 ft

V=

r 2h

Formula for volume Replace B with r 2.

= · 52 · 15 Replace r with 5 and h with 15. ≈ 1178.1 ft3 Compare to the estimate. b. diameter of base 16.4 mm, height 20 mm Since the diameter is 16.4 mm, the radius is 8.2 mm. V = r 2h

Formula for volume of a cylinder

= · 8.22 · 20 ≈ 4224.8

5A.

mm3

Replace r with 8.2 and h with 20. Simplify.

5B. diameter of base 11.4 m, height 5 m

È°£ vÌ

£Ó vÌ


Lesson 11-2 Volume: Prisms and Cylinders

585

Examples 1, 2, 5 (pp. 583–585)

Find the volume of each solid. Round to the nearest tenth, if necessary. 1.

2. 9 cm

3.

15 in.

8 ft

8 in. 8 ft

5.1 cm

7 in.

4 cm

4. rectangular prism: length 6 in., width 6 in., height 9 in. 5. triangular prism: base of triangle 7 cm, altitude of triangle 20 cm, height of prism 10 cm 6. cylinder: radius 3 m, height 10.3 m Example 3 (p. 584)

Example 4 (pp. 584 –585)

7. ENGINEERING A cylindrical storage tank is being manufactured to hold at least 1,000,000 cubic feet of natural gas and have a diameter of no more than 80 feet. What height should the tank be to the nearest tenth of a foot? 8. MULTIPLE CHOICE Find the volume of the stand at the right.

2 in. 1

A 6 in3

C 13 in3

12 in.

B 10 in3

D 16 in3

1 in.

2 in. 5 in.

HOMEWORK

HELP

For See Exercises Examples 9–18 1, 2, 5 19, 20 3 36–39 4


10. 8 in.

11. 4.5 m 2.6 m

11 cm

7m

16 in.

8 cm

4 in.

12.

13. 7.6 m

10 m

17 cm

2 ft

7 ft

14.

2.7 m

30 m

15 m

15. rectangular prism: length 3 mm, width 5 mm, height 15 mm 16. triangular prism: base of triangle 8 in., altitude of triangle 15 in., height of 1 in. prism 6_ 2 17. cylinder: d = 2.6 m, h = 3.5 m 18. triangular prism: base of triangle 6.2 yd, altitude of triangle 20 yd, height of prism 14 yd 19. Find the height of a rectangular prism with a length of 4.2 meters, width of 3.2 meters, and volume of 83.3 cubic meters. 20. Find the height of a cylinder with a radius of 2 feet and a volume of 28.3 cubic feet. 586 Chapter 11 Three-Dimensional Figures


22.

Ó

Î vÌ

È n vÌ È £È

23.

vÌ

È vÌ

24. n V

ÓÎ V

n V

È Þ` È Þ` È Þ`

25. Find the volume of an octagonal prism with a base area of 25 square meters and a height of 1.5 meters.

Real-World Link Gold bars can be denominated in different units of weight. Grams is the International standard. Englishspeaking countries like the U.S., U.K., and Australia prefer ounces. Source: www.gold.org

CONVERTING UNIT OF MEASURE For Exercises 26–28, use the cubes at the right. The volume of the left cube is 1 yd3. In 1 yd the right cube, only the units have been changed. So, 1 yd3 = 3(3)(3) or 27 ft3. Use a similar process to convert each measurement. 26. 1 ft3 = in3

27. 1 cm3 = mm3

PRACTICE


H.O.T. Problems

1 yd

3 ft 3 ft

3 ft

28. 1 m3 = cm3

29. METALS The density of gold is 19.29 grams per cubic centimeter. Estimate the mass in grams of a gold bar that is 2 centimeters by 3 centimeters by 2 centimeters. 30. BATTERIES The current of an alkaline battery corresponds to its volume. Find the volume of each cylinder-shaped battery shown in the table. Write each volume in cm3. (Hint: 1 cm3 = 1000 mm3)

EXTRA

1 yd

Battery Size

Diameter (mm)

Height (mm)

D

33.3

61.1

C

25.5

50.0

AA

14.5

50.5

AAA

10.5

44.5

31. MICROWAVES The inside of a microwave oven has a volume of 1.2 cubic feet and measures 18 inches wide and 10 inches long. To the nearest tenth, how deep is the inside of the microwave? (Hint: Convert 1.2 cubic feet to cubic inches.)

32. OPEN ENDED Describe a problem from an everyday situation in which you need to find the volume of a cylinder or a rectangular prism. Explain how to solve the problem. 33. FIND THE ERROR Ian and Marissa are describing what happens to the volume of a cube when the length is doubled. Ian says that the volume doubles. Marissa thinks that the volume is 8 times greater. Who is correct? Explain your reasoning. Lesson 11-2 Volume: Prisms and Cylinders

PhotoLink/Getty Images

587

1 -inch by 11-inch piece of paper is rolled to form a 34. CHALLENGE An 8 _ 2 cylinder. Will the volume be greater if the height is 81 inches or 11 inches, 2 or will the volumes be the same? Explain your reasoning.

35.

Writing in Math

Explain how measurements for length and width (one dimension), area (two dimensions), and volume (three dimensions) are related in a prism. Include in your answer a description of the formula for volume that involves area and one other dimension.

36. Find the maximum amount of water that can fill the trough.

38. Which is the best estimate for the volume of an ice cube whose sides measure 18.79 millimeters?

2.5 ft

10 ft

A 20.5 ft3

C 48 ft3

B 24.5 ft3

D 49 ft3

F 80 mm3

H 8,000 mm3

G 800 mm3

J 80,000 mm3

39. Mr. Toshio is filling a 20-foot by 35-foot garden framed by two levels of bricks with topsoil. If the topsoil costs $9 per cubic foot, what other information is needed to find s, the cost of the soil?

37. GRIDDABLE Find the height of a box with a length of 3 meters, width of 1.5 meters, and a volume of 60.3 cubic meters. Round to the nearest tenth.

A The area of the garden. B The perimeter of the garden. C The price per cubic yard of soil. D The height of the bricks.

41. Find the area of the figure below. Round to the nearest tenth. (Lesson 10-8)

40. Draw and label the top, front, and side views of the table. (Lesson 11-1)

Ó°n Þ` Î°x Þ`

= m2. (Lesson 9-2) 42. Find the value of m if √256 Solve each inequality. Check your solution. (Lesson 8-4) 43. x + 5 > -3

44. k + (-9) ≥ 1.8

45. y - 10.2 ≤ -13.4

PREREQUISITE SKILL Find each product. (Lesson 5-3) 1 46. _ · 5 · 15 3

1 47. _ ·4·9 3


1 48. _ ·2·2·3 3

1 49. _ · 22 · 21 3

11-3

Volume: Pyramids, Cones, and Spheres Animation pre-alg.com

Main Ideas • Find volumes of pyramids. • Find volumes of cones and spheres.

New Vocabulary sphere

You can see that the volume of the pyramid shown at the right is less than the volume of the prism in which it sits. If the pyramid were made of sand, it would take three pyramids to fill a prism having the same base dimensions and height. a. Compare the base areas and compare the heights of the prism and the pyramid. b. How many times greater is the volume of the prism than the volume of one pyramid? c. What fraction of the prism volume does one pyramid fill? d. Write an equation that relates the volume y of one pyramid to the volume x of the prism.

Volumes of Pyramids A pyramid has one-third the volume of a prism with the same base area and height. Volume of a Pyramid

Reading Math Height of Pyramid The height of a pyramid is the distance from the vertex, perpendicular to the base.

The volume V of a pyramid is one-third the area of the base B times the height h.

Words

Model h

1 Symbols V = _ Bh

B

3

EXAMPLE

Volume of a Pyramid

Find the volume of the pyramid. Round to the nearest tenth, if necessary. 1 V=_ Bh 3 _ =1 3 1 =_ 3

(_12 · 8 · 6)h (_12 · 8 · 6)20

= 160 ft3

Formula for volume of a pyramid The base is a triangle, so B = _ · 8 · 6. 1 2

20 ft 8 ft

6 ft

The height of the pyramid is 20 feet. Simplify.

10 ft

1. Find the volume of a pyramid with a base area of 125 square meters and a height of 6.5 meters. Personal Tutor at pre-alg.com Lesson 11-3 Volume: Pyramids, Cones, and Spheres

589

Reading Math Cones In this text, cone refers to a circular cone.

Volumes of Cones and Spheres A cone is a three-

vertex

dimensional figure with one circular base. A curved surface connects the base and the vertex.

height

The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related.

base

1 The volume of a cone is _ the volume of a cylinder with the same base 3 area and height.

Volume of a Cone Words

The volume V of a cone with radius r is one-third the area of the base B times the height h.

Model h

Symbols V = _ Bh or V = _ r2h, where B = r2 1 3

EXAMPLE Checking Your Solution 1 Since _ · is about 1, 3

you can use estimation to determine whether your solution is reasonable.

1 3

r

Volume of a Cone

Find the volume of the cone. Round to the nearest tenth. 1 2 V=_ r h

3 1 =_ · · 52 · 12 3

≈ 314.2

cm3

Formula for volume of a cone

5 cm

Replace r with 5 and h with 12. 12 cm

Simplify.

V ≈ r2 · h ≈ 52 · 12 ≈ 300

2A. radius 6 ft, height 20 ft

3 2B. radius 3 _ yd, height 9 yd 5

2C. Consider the diagram at the top of the page. Write an equation that relates the volume y of one cone to the volume x of the cylinder. A sphere is a set of points in space that are a given distance r from the center. Suppose a sphere with radius r is placed inside a cylinder with the same radius r and height 2r. The 2 of the volume of the cylinder. The volume of the sphere is _ 3 volume of the cylinder is shown below. V = r2h

À ÓÀ

Volume of a cylinder

= r2(2r) Replace h with 2r. = 2r3

Simplify. 2 Since the sphere is _ the size of the cylinder, you can find the volume of 3

the sphere.

2 V= _ 2r3

(3)

4 3 =_ r 3

2 The sphere is _ the size of the cylinder. 3

Simplify.


Volume of a Sphere The volume V of a sphere is fourthirds times pi times the radius cubed.

Words

Model À

4 3 Symbols V = _ r 3

EXAMPLE

Volume of a Sphere

Find the volume of the sphere. Round to the nearest tenth. 4 3 V=_ r

3 4 =_ · · 83 3

≈ 2144.7 in3

Formula for the volume of a sphere Replace r with 8. n °

Simplify.

3. Find the volume of a sphere with a radius of 4 meters. Round to the nearest tenth.

Finding the volumes of threedimensional figures will help you analyze structures. Visit pre-alg.com to continue work on your project.

AUTO MAINTENANCE The funnel shown is used to fill the oil reservoir of a car.

{ °

a. Find the volume of the funnel. Round to the nearest tenth. 1 Estimate _ · 3 · 22 · 9 = 36

°

3

1 2 V=_ r h

3 _ = 1 · · 22 · 9 3

≈ 37.7 in3

Formula for volume of a cone Since d = 4, replace r with 2. Replace h with 9. Simplify.

b. Suppose a container with 57.75 cubic inches of oil can be emptied into the funnel in 4 seconds. Will the funnel overflow if it is able to drain oil at a rate of 6 cubic inches per second? Explain. 3

57.75 in or about The container of oil is poured into the funnel at a rate of _ 4s

14.4 cubic inches per second. So, find the amount of oil in the funnel after every second. oil poured into funnel - oil drained from funnel = 14.4 - 6 or 8.4 in3 After 4 seconds, the amount of oil remaining in the funnel is 8.4 · 4 or 33.6 cubic inches. Since the capacity of the funnel is greater (37.7 cubic inches), the funnel will not overflow.

4. ICE CREAM A spherical scoop of ice cream with a diameter of 6.3 centimeters is placed in a cup. Find the volume of the ice cream. If the ice cream melts at a rate of 2.1 cubic centimeters every minute, how long would it take for the scoop of ice cream to completely melt?


Lesson 11-3 Volume: Pyramids, Cones, and Spheres

591

Examples 1–3 (pp. 589-591)


2.

3.

4 cm

4.

5 cm

Ç Þ`

10 in.

15 m 3 cm 5m

A ⫽ 48 in2

4m

5. rectangular pyramid: length 9 ft, width 7 ft, height 18 ft 6. cone: radius 4 mm, height 6.5 mm 7. sphere: radius 5 in. Example 4 (p. 591)

HOMEWORK

HELP

For See Exercises Examples 9–12, 18, 19 1 13, 14, 20, 21 2 15–17, 22, 23 3 24–27 4

8. HISTORY The Great Pyramid of Khufu in Egypt was originally 481 feet high and had a square base 756 feet on a side. What was its volume? Use an estimate to check your answer.


10.

11. 12 cm

6 ft

9.2 mm

4 ft 4 ft

A ⫽ 40.6 mm2

10 cm 10.3 cm

12.

15 in.

13.

6 in.

14.

12 m 5m

10 in.

12 in.

5 in.

13 in.

15.

16. {

18. 19. 20. 21. 22. 23.

17.

Óä vÌ

x°Ó

square pyramid: length 5 in., height 6 in. hexagonal pyramid: base area 125 cm2, height 6.5 cm cone: radius 3 yd, height 14 yd cone: diameter 12 m, height 15 m sphere: radius 7.2 km sphere: diameter 1.8 mm

WEATHER A conical icicle 2.5 feet long with a diameter of 1.5 feet has formed at the bottom of a roof. 24. Find the amount of ice in the icicle to the nearest tenth of a cubic foot. 25. The icicle melts at a rate of 0.1 cubic foot every 5 min on a sunny day. How long would it take for the icicle to melt? 592 Chapter 11 Three-Dimensional Figures

GEOLOGY A stalactite in a cavern is cone-shaped. It is 20 feet high and has a diameter at its base of 7 feet. 26. Estimate the volume of the stalactite. Then find the volume to the nearest tenth. 27. The stalactite is made of calcium carbonate, which weighs 131 pounds per cubic foot. What is the weight of the stalactite? Find the volume of each solid. Round to the nearest tenth, if necessary. 28.

15 ft

29.

1.5 m

30.

8 cm

5 cm

31.

£È °

3m 20 ft

20 cm 2m

20 ft 20 ft

Real-World Link A stalactite is a type of cave formation that hangs from the ceiling or wall of limestone caves. It takes approximately 100 years for a formation to grow 1 cubic inch.

32. SCIENCE In science, a standard funnel is shaped like a cone, and a buchner funnel is shaped like a cone with a cylinder attached to the base. Which funnel has the greater volume? 48 mm 34 mm 20 mm

40 mm

Source: nbcham.org and naturalbridgecaverns.com

18 mm

standard

buchner

33. BASKETBALL The volume of a mini-basketball is about 230 cubic inches. What is its radius? Round to the nearest inch. 34. PACKAGING A gift set of three golf balls is packaged in a clear rectangular box 13.1 centimeters long, 4.5 centimeters wide, and 4.5 centimeters tall. If each ball is 4.3 centimeters in diameter, find the volume of the empty space in the box. EXTRA

PRACTICE

CM


H.O.T. Problems

CM CM

35. OPEN ENDED Draw and label a cone whose volume is between 100 cm3 and 1000 cm3. 36. SELECT A TECHNIQUE Mackenzie is scooping ice cream into a container that is in the shape of a square pyramid. Each scoop contains about 4.8 cubic centimeters of ice cream. The base of the container has a length of 4.2 centimeters and the height is 8.4 centimeters. Which technique might Mackenzie use to find about how many scoops it will take to fill the cone? Justify your selection(s). Then use the technique(s) to solve the problem. Mental math

Number sense

Estimation

Lesson 11-3 Volume: Pyramids, Cones, and Spheres Jonathan A. Meyers/Photo Researchers, Inc.

593

CHALLENGE Suppose a cone has a height h and radius r and a sphere has radius r. 37. If you double the height of a cone, how does the volume change? 38. If you double the radius of the base of a cone, how does the volume change? Explain. 39. What happens to the volume of a sphere if you double the radius? Explain your reasoning. 40.

Writing in Math How is the volume of a pyramid related to the volume of a prism? Include in your answer a discussion of the similarities between the dimensions and base area of the pyramid and prism and a description of how the formulas for the volume of a pyramid and the volume of a prism are similar.

41. The solids shown have the same base area and height. If the cone is filled with water and poured into the cylinder, how many cones filled with water would it take to fill the cylinder? 2 A 2_ 3

1 C 3_ 3

B 3

2 D 3_

42. Choose the best estimate for the volume of a rectangular pyramid that is 4.9 centimeters long, 3.3 centimeters wide, and 6.8 centimeters high. F 7 cm3 G 35 cm3 H 70 cm3 J 105 cm3

3

Find the volume of each prism or cylinder. Round to the nearest tenth, if necessary. (Lesson 11-2) 43. rectangular prism: length 4 cm, width 8 cm, height 2 cm 44. cylinder: diameter 1.6 in., height 5 in.

D

45. Identify the solid at the right. Name the bases, faces, edges, and vertices. (Lesson 11-1)

C F

46. TRAVEL Loretta drives due north for 22 miles and then east for 11 miles. How far is Loretta from her starting point? Round to the nearest tenth of a mile. (Lesson 9-5) 47. ALGEBRA Copy and complete the table. Use the results to write four solutions of y = x + 5. Write the solutions as ordered pairs. (Lesson 7-2) Express each percent as a fraction in simplest form. (Lesson 6-5) 48. 25%

1 49. 87_ % 2

50. 150%

52. 2 · 1.7 · 9


x

x⫹5

3

3 5

1 0 1

PREREQUISITE SKILL Estimate each product. (page 746) 51. 4.9 · 5.1 · 3

G

53. 2 · · 6.8

54. 2 · 6.2 · 7

y

CH

APTER

11


Identify each solid. Name the bases, faces, edges, and vertices. (Lesson 11-1) K 1. J 2. 4

L

P N

3.

B 606.1 in3 4.

2

B

D

6. RETAIL The sketch shows a display containing tissue boxes. How tall is the display if each unit in the drawing 3 inches? represents 3_

H 2400 ft3

3 _ G 2 2 ft3 5

J 2880 ft3

(Lesson 11-3)

13. What is the volume of the salt-sand mixture to the nearest cubic foot? 14. How many square feet of roadway can be salted if 500 square feet can be covered by 1 cubic foot of salt?

4

Find the volume of each solid. Round to the nearest tenth, if necessary. (Lessons 11-2 and 11-3) 7. 8. n°x vÌ È V Ó vÌ

2 3 F 1_ ft

HIGHWAY MAINTENANCE Salt and sand mixtures are often used on icy roads. When the mixture is dumped from a truck into the staging area, it forms a cone-shaped mound with a diameter of 10 feet and a height of 6 feet.

-ì 6iÜ

£Î V £ä V

£Ó vÌ

12. MULTIPLE CHOICE How much water can fit into an aquarium with length 15 inches, width 12 inches, and height 16 inches? (Lesson 11-2)

C

5. ARCHITECTURE The drawings show the top view and the side view of a solid figure built with cubes. Draw a three-dimensional view of the solid. (Lesson 11-1) /« 6iÜ

D 2424.5 in3

A

E

(Lesson 11-1)

2

1 dissolves in 3_ cans of water, how much 2 water must be added? (Lesson 11-2) A 173.2 in3 C 692.7 in3

7

M

11. MULTIPLE CHOICE A can of lemonade concentrate has a diameter of 7 inches and 1 a height of 4_ inches. If the concentrate

15. MULTIPLE CHOICE A spherical balloon is inflated with helium. How much helium is in the balloon if the balloon has a radius of 9 centimeters? (Lesson 11-3) A 339.3 cm3 B 381.7 cm3 C 1357.2 cm3

9.

10.

£È Þ`

D 3053.6 cm3

Ó

Þ` £x Þ`

16. FOOD A cake-decorating bag is in the shape of a cone. How much frosting can fit in the bag if its diameter is 4 inches and its height is 9 inches? Round to the nearest tenth. (Lesson 11-3) Chapter 11 Mid-Chapter Quiz

595

EXPLORE

11-4

Geometry Lab

Exploring Lateral Area and Surface Area The lateral area of a three-dimensional object is the sum of the areas of its lateral faces. The surface area is the sum of the lateral area and the areas of the bases.

ACTIVITY 1 Find the lateral area and the surface area of a rectangular prism. Step 1 Outline the faces of the prism on blank paper and cut them out. Mark the lateral faces with an X. Step 2 Measure each face. Record the Face Length Width Area measures in a table. Then find and record the area of each face. Top Step 3 The lateral area is the sum of the Bottom areas of the faces marked with Left an X. Find the lateral area. Right Step 4 Find the surface area. Step 5 Tape the faces together so that Front they can be re-folded to make Back the original prism. This figure is called a net.

ANALYZE THE RESULTS 1. What do you observe about the areas of opposite faces? 2. MAKE A CONJECTURE Write a formula for the surface area S of a rectangular prism with length , width w, and height h.

ACTIVITY 2 Find the lateral area and surface area of a cylinder. Step 1 Outline the bases on blank paper and cut them out. Find the area of the bases. Step 2 Wrap paper around the curved surface of the cylinder. Tape in place. Then cut the paper to the height of the can. Step 3 Cut the paper along the height of the can. Unroll the paper and measure the rectangle. The lateral area is the area of the curved surface. Find the lateral area. Step 4 Find the surface area. Step 5 Tape the faces together so they can be re-folded to make the original cylinder. This is the net.

ANALYZE THE RESULTS 3. What do you observe about the length of the lateral surface? 4. MAKE A CONJECTURE Write a formula for the surface area S of a cylinder with radius r and height h. 596 Chapter 11 Three-Dimensional Figures

11-4

Surface Area: Prisms and Cylinders

Main Ideas The sizes and prices of shipping boxes are shown in the table. a. For each box, find the area of each face Box Size (in.) Price ($) and the sum of the areas.

• Find lateral area and surface areas of prisms. • Find lateral area and surface areas of cylinders.

b. Find the volume of each box. Are these values the same as the values you found in part a? Explain.

A

8×8×8

$1.50

B

15 × 10 × 12

$2.25

C

20 × 14 × 10

$3.00

New Vocabulary net lateral face lateral area surface area

Prisms If you open up a box or prism and lay it flat, the result is a net. A net is a two-dimensional pattern for a three-dimensional figure. A net allows you to see all the surfaces. The surfaces of prisms have two characteristics • The prism has two bases, which are parallel. • Faces that are not bases are called lateral faces. The lateral area is the sum of the areas of the lateral faces. The surface area is the sum of the lateral area plus the area of the bases. Lateral Area h

h

Area of Bases

top (bottom)

side

side

side

base (bottom)

w

w

Area of Base (B )

w

w

w

side

h

top (base)

P Lateral Area

side Area of Base (B )

front

side

h bottom (base)

Lateral Area and Surface Area of Prisms Words

The lateral area L of a prism is the perimeter of the base P times the height h.

Symbols

L = Ph

Words

The surface area S of a prism is the lateral area L plus the area of the two bases 2B.

Symbols

S = L + 2B or S = Ph + 2B


Model

0

!REA OF "ASE " !REA OF "ASE "

!REA OF "ASE "

H W

0

!REA OF "ASE "

Lesson 11-4 Surface Area: Prisms and Cylinders

H

597

EXAMPLE

Surface Area of Prisms

Find the lateral area and surface area of each prism. a. Find the lateral area. L = Ph = (2 + 2w)(h) = (2 · 20 + 2 · 14)(10) = 680 in2

Find the surface area. S = L + 2B = L + 2w = 680 + 2(20)(14) = 1240 in2

10 in. 14 in.

20 in.

b. The lateral area is made up of bases that are not parallel. Nets Another way to find the surface area of any prism is to draw its net on grid paper and find the area of each face.

5 cm

L = Ph Write the formula. = (3 + 4 + 5)(6) P is the perimeter of the triangular base. h is the height of the prism.

= 72

cm2

4 cm

Simplify.

6 cm

Find the surface area. S = L + 2B

3 cm

Write the formula.

1 =L+2 _ bh

(2 ) 1 = 72 + 2(_ · 3 · 4) 2

B = _ bh (area of triangle)

= 84

Simplify.

cm2

1 2

Substitution

Find the lateral area and surface area of each prism. 1A. 1B. £Ó vÌ £Ó

£{

£x vÌ

£Î

x vÌ £Î vÌ


Cylinders You can also find surface areas of cylinders by finding the sum of the lateral area and the area of the bases. If you unroll a cylinder, its net is a rectangle (lateral face) and two circles (bases). LATERAL AREA

R

TOP BASE

R

Circles and Rectangles

# PR

To see why = 2␲r, find the circumference of a soup can by using the formula C = 2␲r. Then peel off its label and measure the length (minus the overlapping parts).


H

H

H

# PR

R

#

BOTTOM BASE

Surface Area of a Cylinder lateral area

Model

Area

rectangle

w or 2␲r(h)

area of bases

2 circles

2(␲r2) or 2␲r2

Lateral Area and Surface Area of Cylinders


L = 2␲rh

Words

The surface area S of a cylinder is the lateral area L plus the area of the two bases (2␲r2).

Leave your answers in terms of ␲ if exact answers are required. If approximate answers are sufficient, find a decimal approximation.

h

S = L + 2B or S = 2␲rh +

EXAMPLE

circumference of base 2πr r

Symbols

Symbols

Exact vs. Approximate Answers

Model

The lateral area L of a cylinder with radius r and height h is the circumference of the base (2␲r) times the height h.

Words

area of bases πr 2

2␲r2

Surface Area of a Cylinder 1m

Find the lateral area and surface area of the cylinder. Round to the nearest tenth. Estimate S = (2 · 3 · 1 · 3) + (2 · 3 · 1) or 24 m2

Lateral Area

Surface Area

L = 2␲rh

S = L + 2␲r2

3m

= 6␲ + 2␲(1)2

= 2␲(1)(3) = 6␲

exact answer

= 8␲

exact answer

≈ 18.8 m2

approximate answer

≈ 25.1 m2

approximate answer

How does your estimate compare to your answer?

2. Find the lateral area and surface area of a cylinder with a radius of 7 centimeters and a height of 12 centimeters.

4 cm

FRUIT DRINKS Both containers hold about the same amount of pineapple juice. Does the box or the can have a greater surface area? Surface area of box Lateral Area Real-World Link A typical juice box is made of six layers of packaging. These layers include paper, foil, and special adhesives.

S= L + = Ph + = (2 · 7 + 2 · 4)(9) + = 254 cm2

Area of Bases 2B 2w 2(7)(4)

6 cm

7 cm

9 cm

9 cm Pineapple Juice

Surface area of can Lateral Area Area of Bases S= L + = 2␲rh + = 2␲(3)(9) +

2B 2␲r2 2␲(3)2

≈ 226 cm2

Since 254 cm2 > 226 cm2, the box has a greater surface area.

Source: dupont.com

3. CANNED FOODS Which can has a greater surface area: a tuna fish can with diameter 8 centimeters and height 4 centimeters or a soup can with a diameter 4 centimeters and height 8 centimeters?

Extra Examples at pre-alg.com Bob Mullenix


599

Examples 1–2 (pp. 598–599)

Draw a net of each solid shown or described. Then find the lateral area and surface area of each solid. Round to the nearest tenth, if necessary. 1.

2. 5 ft

5 ft

3.

10 cm

5 ft

14 in.

4 cm

3 cm

6 in.

5 cm

4. rectangular prism: length 3 cm, width 2 cm, height 1 cm 5. cylinder: radius 4 mm, height 1.6 mm Example 3 (p. 599)

HOMEWORK

HELP

For See Exercises Examples 7–10, 13, 14 1 11, 12, 15, 16 2 17–20 3

6. CRAFTS Desiree sews together pieces of plastic canvas to make tissue box covers. For which tissue box will she use more plastic canvas to cover the sides and the top? Explain.

A

Length (in.) 9

Width (in.) 4

Height (in.) 5

B

5

5

6

Box

Draw a net of each solid shown or described. Then find the lateral area and surface area of each solid. Round to the nearest tenth, if necessary. 7.

8.

9. 3.5 m

3 in. 7 in.

12 in. 3.5 m

9m

10 m

14 m 6 cm 8m

10.

10 m

10 m

13. 14. 15. 16.

11.

8.7 m

15 m

10 ft

12.

1 in.

20 ft 9 in.

10 m

cube: side length 7 ft rectangular prism: length 6.2 cm, width 4 cm, height 8.5 cm cylinder: radius 5 in., height 15 in. cylinder: diameter 4 m, height 20 m

17. PAINTING Hinto is planning to paint the walls of a bedroom 20 feet long, 15 feet wide, and 8 feet high. If 1 gallon of paint covers 400 square feet, how many gallons of paint should Hinto buy? AQUARIUMS A standard 20-gallon aquarium tank is a rectangular prism that holds approximately 4600 cubic inches of water. The bottom glass needs to be 24 inches by 12 inches to fit on the stand. 18. Find the height of the aquarium to the nearest inch. 19. Find the total amount of glass needed in square feet for the five faces. 20. An aquarium with an octagonal base has sides that are 9 inches wide and 16 inches high. The area of the base is 392.4 square inches. Do the bottom and sides of this tank have a greater surface area than the rectangular tank? Explain. 600 Chapter 11 Three-Dimensional Figures

21. PLUMBING Find the surface area (exterior and interior) of the PVC pipe shown. Round to the nearest tenth of an inch. EXTRA

IN IN

PRACTICE

IN


H.O.T. Problems

22. Find the surface area of the complex solid at the right. Use estimation to check the reasonableness of your answer. 23. OPEN ENDED Find the lateral and surface areas of a rectangular prism and a cylinder found in your home.

4 in.

4 in. 12 in.

24. CHALLENGE Suppose you double the length of the sides of a cube. How is the surface area affected? 25.

Writing in Math Refer to Example 3 on page 599. Explain why a company would prefer to sell juice in cans rather than in boxes if both contained the same amount of liquid. What other factors would the company have to analyze before choosing one package over the other? Explain.

26. How many 2-inch squares will completely cover a rectangular prism 10 inches long, 4 inches wide, and 6 inches high? A 40

B 62

C 240

28. The specifications of a cardboard box indicated that it was to have the same volume as a rectangular box 6 inches by 14 inches by 20 inches, but with less surface area. Which size box would NOT meet these requirements?

D 248

27. Find the amount of paper needed for the label on the can.

A 7 in. by 10 in. by 24 in.

£

B 7 in. by 12 in. by 20 in.

Ó { °

F 21.2 in2

C 10 in. by 12 in. by 14 in.

G 25.2 in2 H 29.2 in2

D 5 in. by 16 in. by 21 in.

Î °

J 42.4 in2

Find the volume of each solid. Round to the nearest tenth, if necessary. (Lessons 11-2 and 11-3)

29. rectangular pyramid: length 6 ft, width 5 ft, height 7 ft 30. cylinder: diameter 6 in., height 20 in.

PREREQUISITE SKILL Find each product. (Lesson 5-3) 31. 10.3(8)

32. 3.9(3.9)

33. 12.3(9.2)(6)

34. 5.9(12.1)(13.5)

1 35. _ · 2.6 2

1 36. _ · 82 · 90 2

1 _ 37. _ 61 2 2

1 1 _ 2 38. _ 10_

( )

2

( 4 )( 3 )


601

11-5

Surface Area: Pyramids and Cones

Main Ideas • Find surface areas of pyramids. • Find surface areas of cones.

You can use a concrete model to explore the surface area of a pyramid. Step 1

New Vocabulary slant height

Step 2 Step 3 Step 4

The net at the right is made of one square and four congruent isoceles triangles. Copy the net on a piece of paper, shading the base as shown. Use scissors to cut out the net. Fold on the dashed lines. Tape the sides together.

a. The resulting solid is a triangular pyramid. Measure each face of the pyramid. Then find the area of each face. b. Find the lateral area and the surface area of the pyramid.

Reading Math Slant Height The slant height of a pyramid is different from the altitude. Recall that the height of a pyramid is the distance from the vertex, perpendicular to the base.

Surface Areas of Pyramids The lateral faces of a pyramid are triangles that intersect at the vertex. The altitude or height of each lateral face is called the slant height. 3QUARE 0YRAMID

.ET OF 3QUARE 0YRAMID

VERTEX ALTITUDE

LATERAL FACE

LATERAL FACE

SLANT HEIGHT

BASE BASE SLANT HEIGHT

The surface area of a pyramid is the lateral area plus the area of the base.

EXAMPLE

Surface Area of a Pyramid

Find the surface area of the square pyramid. First find the lateral area. The lateral area of the pyramid is made up of four triangles. 1 Estimate L = 4(_)(6)(8) = 96 1 L=4 _ bh 2

2

() 1 = 4(_ (6)(8.2) 2) = 98.4 m2


8.2 m

Area of 4 triangles Replace b with 6 and h with 8.2. Simplify.

6m

Then find the surface area. The base of the pyramid is a square. Estimate

S = 100 + 36 or 136

S=L+B =L+

Write the formula.

s2

The area of a square is s2.

= 98.4 + 62 = 134.4

Substitution

m2

Simplify. Is your answer reasonable?

1. Find the surface area of the square pyramid with a base side length of 6 centimeters and a slant height of 18.4 centimeters.

ARCHITECTURE Use the information at the left to find how much stainless steel was used in making The Pyramid Arena. Find the lateral area only, since the bottom of the pyramid is not covered in glass. 1 Estimate L = 4(_)(600)(450) = 540,000 1 L=4 _ (bh)

Real-World Link The Pyramid Arena in Memphis, Tennessee, is a stainless steel square pyramid with a slant height of about 455 feet. Its square base is 591 feet on each side. Source: pyramidarena.com

2

(2) 1 = 4(_ (591)(455) 2) = 537,810

Formula for area of 4 triangles Replace b with 591 and h with 455. Simplify.

It took 537,810 square feet of stainless steel to cover the pyramid. Compare your answer to the estimate.

2. CANDLES A company produces a candle in the shape of a pyramid. The square base measures 5 inches on a side, and the slant height of the candle is 10 inches. Find the lateral area of the candle. Personal Tutor at pre-alg.com

Surface Areas of Cones You can also find surface areas of cones. The net of a cone shows the regions that make up the cone. Net of Cone

Model of Cone

ᐉ

ᐉ r

r

The lateral area of a cone with slant height is one-half the circumference 1 · 2␲r · or L = ␲r. The base of the cone of the base, 2␲r, times . So L = _ 2 2 is a circle with area ␲r .

Extra Examples at pre-alg.com SuperStock

The surface area of a cone

equals

the lateral area

plus

the area of the base.

S

=

␲r

+

␲r2

Lesson 11-5 Surface Area: Pyramids and Cones

603

Surface Area of a Cone Words

The surface area S of a cone with slant height and radius r is the lateral area plus the area of the base.

Model ᐉ r

Symbols S = L + B or S = ␲r + ␲r2

EXAMPLE

Surface Area of a Cone

Find the surface area of the cone. Round to the nearest tenth. S = ␲r +

␲r2

15 m

Formula for surface area of a cone

= ␲(10.6)(15) + ␲(10.6)2 Replace r with 10.6 and with 15. ≈ 852.5

10.6 m

Simplify.

The surface area of the cone is about 852.5 square meters.

1 3. Find the surface area of a cone with a radius of 3_ yards and a slant 2 height of 10 yards. Round to the nearest tenth.

Examples 1, 3 (p. 602–604)

Draw a net of each solid. Then find the lateral and surface area of each solid. 1.

2.

3. A ⫽ 6.9 m

6.3 ft

5 cm

2

6m 4m

13 cm

4m 4 ft

Example 2 (p. 603)

4m

4 ft

4. ARCHITECTURE The small tower of a historic house is shaped like a regular hexagonal pyramid as shown at the right. How much roofing will be needed to cover this tower? (Hint: Do not include the base of the pyramid.)

14 ft

8 ft

HOMEWORK

HELP

For See Exercises Examples 5–8, 12–13 1 15–18 2 9–11, 14 3

Draw a net of each solid. Then find the lateral and surface area of each solid. 5.

6.

7.

9m

6 in.

6.9 ft

8 ft

8 ft

8m


1

8m 1 5 2 in.

5 2 in.

8 ft

Find the surface area of each solid. Round to the nearest tenth, if necessary. 8.

9.

5.2 in.

10.

5.2 in. 6 in.

n °

10 cm

6 in. 6 in.

£È°È

5 cm

11. cone: radius 7.5 mm, slant height 14 mm 12. square pyramid: base side length 9 yd, slant height 8 yd 1 2 in , slant height 8 in. 13. triangular pyramid: base side length 6 in., base area 13_ 4

14. cone: diameter 19 cm, slant height 30 cm

15. HOTELS The Luxor Hotel in Las Vegas is a square pyramid made from glass with a base length of 646 feet and a slant height of about 476 feet. How much glass was used for the four sides?

Real-World Career Architect An architect uses math to combine geometric solids and understand their properties to form unified structures. For more information, go to pre-alg.com.

16. FUND-RAISING The cheerleaders are selling small megaphones decorated with the school crest. There are two sizes, as shown at the right. What is the difference in the amount of plastic used in these two sizes? Round to the nearest square inch. (Note that a megaphone is open at the bottom.)

8 in.

6.5 in.

4 in.

Style 8M

3.5 in.

Style 65M

ARCHITECTURE For Exercises 17 and 18, use the following information. A roofing company is preparing bids on two jobs involving cone-shaped roofs. Roofing material is usually sold in 100-square-foot squares. For each roof, find the lateral surface area to the nearest square foot. Then determine the squares of roofing materials that would be needed to cover each surface. 18.

17.

12 ft 23 ftt

9 fft 8 ft

EXTRA

PRACTICE

See pages 787, 804.


20.

5 cm

21. 5 in.


10.6 m 15 cm

A ⫽ 179.0 m2 8.3 m

H.O.T. Problems

12.3 in.

15.2 in.

22. OPEN ENDED Describe a situation in everyday life when a person might use the formulas for the surface area of a cone or a pyramid. Lesson 11-5 Surface Area: Pyramids and Cones

(t)Bob Daemmrich, (bl)John Elk III/Stock Boston, (br)John Elk III/Stock Boston

605

23. CHALLENGE A bar of lead in the shape of a rectangular prism 13 inches by 2 inches by 1 inch is melted and recast into 100 conical fishing sinkers. The sinkers have a diameter of 1 inch, a height of 1 inch, and a slant height of about 1.1 inches. Compare the total surface area of all the sinkers to the surface area of the original lead bar. How does the volume of the 100 sinkers compare to the volume of the bar? CHALLENGE Find the lateral and total surface area of each pyramid. 24.

25.

CM CM

M

IN

Writing in Math How is surface area used in architecture? Why do building contractors and architects need to know about surface areas?

28. What is the lateral area of a model of a building in the shape of a square pyramid if the slant height is 7 inches? A 17.5

M

29. Find the surface area of the ice cream cone covered by the wrapper. F 9.4 in2

in2

G 11.2 in2

B 35

in2

H 15.5 in2

C 70

in2

in2

J 20.0

IN

D 95 in2

D IN

IN

27.

26.

IN

Find the surface area of each solid. Round to the nearest tenth, if necessary. (Lesson 11-4) 30. rectangular prism: length 2 ft, width 1 ft, height 0.5 ft 31. cylinder: radius 4 cm, height 13.8 cm 32. Find the volume of a cone that has a height of 6 inches and radius of 2 inches. Round to the nearest tenth. (Lesson 11-3) 33. GEOMETRY Mrs. Morales used the parallelogram at the right as a pattern for a paving stone for her sidewalk. If m∠1 is 130°, find m∠2. (Lesson 10-4)

£ Ó

{ Î

34. Suppose ABC is reflected over the y-axis. If the coordinates of ABC are A(-2, -3), B(1, -1), and C(3, 2), what are the coordinates of A'B'C' ? (Lesson 10-3) 35. RETAIL Find the discount for a $45 shirt that is on sale for 20% off. (Lesson 6-8)

PREREQUISITE SKILL Solve each proportion. (Lesson 6-3) x 1 36. _ =_ 6

24

9 n 37. _ =_ 15


5

18 2.7 38. _ =_ 21

n

7.5 3.6 39. _ =_ 6.0

x

EXPLORE

11-6

Geometry Lab

Similar Solids A model car is an exact replica of a real car, but much smaller. The dimensions of the model and the original are proportional. Therefore, these two objects are similar solids. The number of times that you increase or decrease the linear dimensions is called the scale factor. You can use sugar cubes or centimeter blocks to investigate similar solids.

ACTIVITY 1 • If each edge of a sugar cube is 1 unit long, then each face is 1 square unit and the volume of the cube is 1 cubic unit. • Make a cube that has sides twice as long as the original cube.

1 unit 1 unit 1 unit

ANALYZE THE RESULTS 1. How many small cubes did you use? 2. What is the area of one face of the original cube? 3. What is the area of one face of the cube that you built? 4. What is the volume of the original cube? 5. What is the volume of the cube that you built?

ACTIVITY 2 Build a cube that has sides three times longer than a sugar cube or centimeter block.

ANALYZE THE RESULTS 6. How many small cubes did you use? 7. What is the area of one face of the cube? 8. What is the volume of the cube? 9. Copy and complete the table at the right. 10. What happens to the area of a face when the length of a side is doubled? tripled? 11. Considering the unit cube, if the scale factor is x, what is the area of one face? the surface area?

Scale Factor

Side Area of Volume Length a Base

1 2 3

12. What happens to the volume of a cube when the length of a side is doubled? tripled? 13. Consider a unit cube with side length s. Let the scale factor be x. Write an equation for the cube’s volume V. 14. MAKE A CONJECTURE What are the surface area and the volume of a cube if the sides are 4 times longer than the original cube? Explore 11-6 Geometry Lab: Similar Solids

607

11-6

Similar Solids

Main Ideas • Identify similar solids. • Solve problems involving similar solids.

1 The model train below is _ the size of the original train. 87

New Vocabulary similar solids

a. The model boxcar is shaped like a rectangular prism. If it is 8.5 inches long and 1 inch wide, what are the length and width of the original train boxcar to the nearest hundredth of a foot? b. A model tank car is 7 inches long and is shaped like a cylinder. What is the length of the original tank car? c. Make a conjecture about the radius of the original tank car compared to the model.

Identify Similar Solids The cubes below have the same shape. 6 or 3. We say that The ratio of their corresponding edge lengths is _ Review Vocabulary Similar Polygons polygons that have the same shape but not the same size (Lesson 9-6)

2

3 is the scale factor. The cubes are similar solids because they have the same shape and their corresponding linear measures are proportional.

EXAMPLE

6 cm 6 cm

2 cm

6 cm

2 cm

Identify Similar Solids

Determine whether each pair of solids is similar. a.

32 cm

40 32 _ _

1.25 cm 1.0 cm

40 cm

1

1.25

Write a proportion comparing radii and heights.

32(1.25) 1(40) Find the cross products. 40 = 40

Simplify.

The radii and heights are proportional, so the cylinders are similar. b. 14 in.

20 in.

7 14 _ _ 20

7 in.

12 in.

12

Write a proportion comparing corresponding edge lengths.

14(12) 20(7) Find the cross products. 168 ≠ 140

Simplify.

The corresponding measures are not proportional, so the pyramids are not similar. 608 Chapter 11 Three-Dimensional Figures Aaron Haupt

2 cm

1A. £ä

£x £n

ÓÈ

1B. £{ Þ`

Ç°Ó Þ`

££°Ó Þ`

Þ`

Use Similar Solids You can find missing measures if you know solids are similar.

EXAMPLE

Find Missing Measures

The square pyramids at the right are similar. Find the height of pyramid B.

Pyramid A

Pyramid B

18 m

Since the two pyramids are similar, the ratio of their corresponding linear measures are proportional.

h 14 m

height of pyramid A base length of pyramid A ___ = __

height of pyramid B base length of pyramid B 18 14 _ =_ Substitute the 28 h known values. 14h = 28(18) Find the cross products.

h = 36

28 m

Simplify.

The height of pyramid B is 36 m. *ÀÃ Scale Factors When the lengths of all dimensions of a solid are multiplied by a scale factor x, then the surface area is multiplied by x2 and the volume is multiplied by x3.

*ÀÃ

2. The triangular prisms are similar. Find the height of prism A.

n vÌ Ó{ vÌ

£ä vÌ

H

3 . The prisms below are similar with a scale factor of _ 2

Prism X Y

Surface Area

Volume

90

m2

54 m3

40

m2

m3

16

2m

3m 3m

6m

Prism X

4m

2m

Prism Y

Notice the pattern in the following ratios. surface area of prism X 90 9 __ =_ or _ 40 4 surface area of prism Y

volume of prism X 54 27 __ =_ or _ volume of prism Y

16

8

_9 = _3

2

3 27 _ = _

3

4

8

(2)

(2)

Lesson 11-6 Similar Solids

609

Ratios of Similar Solids Words

If two solids are similar with a scale factor a of _, then the surface areas have a ratio of

Models

b

Symbols

(_ab )2 and the volumes have a ratio of (_ab )3. surface area of Solid A a 2 __ = (_) b surface area of Solid B volume of Solid A a 3 __ _ =( ) b volume of Solid B

EXAMPLE

a

b

Solid A

Solid B

Find Surface Areas of Similar Solids

A cube has a surface area of 100 square centimeters. If the dimensions are doubled, what is the surface area of the new cube? 1 The cubes are similar and the scale factor of the side lengths _a is _ .

Therefore, surface areas of the cubes have a ratio of _a

()

2

or

b

2

b _1 2. Set up a proportion to find the surface area of the new cube. 2

()

surface area of original cube ___ = _a

( )2

Write a proportion.

()

Substitute the known values. Let S = the surface area of the new cube.

b surface area of new cube 100 1 2 _ = _ 2 S 100 1 _ _ = 4 S

12 2 = 12 · 12 or 14

4 · 100 = S · 1 Find the cross products. 400 = S

Multiply.

3. A cone has a surface area of 160 square inches. If the dimensions are 1 , what is the surface area of the new cone? reduced by _ 2

SPACE TRAVEL A small model of the NASA space capsule is built on a scale of 1 cm to 20 cm and has a volume of 155 cm3. What is the volume of the actual space capsule? Real-World Link Engineers need to know the volume of the actual space capsules in order to estimate air pressure. Source: space.about.com

Explore Plan Solve

1 You know the scale factor _a is _ and the volume of the model is 20 b 3 155 cm .

3 1 Since the volumes have a ratio of _a and _a = _ , replace a 20 b b 3 a _ with 1 and b with 20 in .

b volume of model a 3 __ _ = Write the ratio of volumes. b volume of capsule 1 3 = _ Replace a with 1 and b with 20. 20 1 =_ Simplify. 8000

( )

The volume of the capsule is 8000 times the volume of the model. 8000 · 155 cm3 = 1,240,000 cm3 610 Chapter 11 Three-Dimensional Figures NASA


Check

Use estimation to check the reasonableness of this answer. 8000 · 100 = 800,000 and 8000 · 200 = 1,600,000, so the answer must be between 800,000 and 1,600,000. The answer 1,240,000 cm3 is reasonable.

4. SPORTS Baseballs and softballs are similar in shape. The scale factor between a baseball and a softball is 1 inch to 1.3 inches, and the volume of a baseball is about 12.8 in3. What is the volume of a softball? Round to the nearest tenth. Personal Tutor at pre-alg.com

Example 1 (pp. 608–609)

Determine whether each pair of solids is similar. 2.

1. 1 in. 1 in.

6m

4m

4 in.

3 in.

8 in. 4 in. 2m

Example 2 (p. 609)

3m

Find the missing measure for each pair of similar solids. 3.

4.

45 ft 6 ft 250 ft

45 cm

30 cm

x

x

24 cm 75 cm

Example 3 (p. 610)

Example 4 (pp. 610–611)

HOMEWORK

For Exercises 9–12 13, 14 15, 16 17–20

HELP

See Examples 1 2 3 4

y

5. A pyramid has a surface area of 50 square feet. If the dimensions are tripled, what is the surface area of the new pyramid? ARCHITECTURE For Exercises 6–8, use the following information. A model for an office building is 60 centimeters long, 42 centimeters wide, and 350 centimeters high. On the model, 1 centimeter represents 1.5 meters. 6. How tall is the actual building in meters? 7. What is the scale factor between the model and the building? 8. Determine the volume of the building in cubic meters.


8 mm

10.

60 mm

9 ft 5 ft 9 ft

5 ft

10 mm

2.5 ft

10 ft

18 ft 4.5 ft

72 mm

11.

4m

8m 10 in.

6 in. 5 in. 5 in.

3m

6m

12.

10 m

5m

6 in. 6 in.


611

Find the missing measure for each pair of similar solids. 13.

14.

12 ft

21 m

4 ft 15.3 ft

x

x

y 6m

5m

15 m

15. A rectangular prism has a surface area of 300 square millimeters. If the dimensions are quadrupled, what is the surface area of the new prism? 16. A sphere has a surface area of 6400 square yards. If the dimensions are one-eighth the original size, what is the surface area of the new sphere?

Real-World Link At 146.5 meters high, the Great Pyramid stood as the tallest structure in the world for more than 4,000 years. Today it stands at 137 meters high. Source: pbs.org

HISTORY The Mankaure Pyramid in Egypt has a square base that is 110 meters on each side and a height of 68.8 meters. Suppose you want to construct a scale model of the pyramid using a scale of 4 meters to 2 centimeters. 17. What is the scale factor between the pyramid and its model? 18. How much greater is the volume of the actual pyramid than the volume of the model? 19. What is the volume of the model? Round to the nearest tenth. 20. If the slant height of the pyramid is 88.1 meters, how much material will you need to use for the model? Determine whether each pair of solids is sometimes, always, or never similar. Explain. 21. two cubes 22. two prisms 23. a cone and a cylinder 24. two spheres 25. The two cylinders are similar. What is the surface area of cylinder B if the volume of cylinder A is 75 cubic meters? Round to the nearest tenth.

x £x

EXTRA

PRACTIICE

Cylinder A

Cylinder B


H.O.T. Problems

26. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you determine whether two solids are similar. 27. FIND THE ERROR Fred and Cassandra are finding the ratio of the surface areas of a building given that the model of the building was built on a scale of 1 centimeter to 5 meters. Who is correct? Explain your reasoning. Fred

Cassandra

2 a2 _ 1 _ = 1 =_

2 1 a2 _ _ = 1 =_

b2

5002

250,000

b2

52

25

28. OPEN ENDED Draw and label two cones that are similar. Justify that they are similar. 29. NUMBER SENSE Describe what happens to the surface area of a cone if its radius and slant height are doubled. 612 Chapter 11 Three-Dimensional Figures CORBIS

30. CHALLENGE The dimensions of a triangular prism are decreased so that the 1 that of the original volume. Are the two volume of the new prism is _ 3 prisms similar? Explain. 31.

Writing in Math How can linear dimensions be used to identify similar solids? Illustrate your answer with a description of the ratios needed for two solids to be similar and an example of two solids that are not similar.

33. For the similar cones, find the ratio of the surface area of the larger cone to the smaller cone.

32. Which prism shown in the table is NOT similar to the other three? Prism

Length

Width

Height

A

4

3

2

B

6

4.5

3

C

5

4

2

D

28

21

14

A prism A

C prism C

B prism B

D prism D

£ä V

3 F _

£x V

15 H _

2 9 G_ 4

10 27 J _ 8

Find the surface area of each solid. Round to the nearest tenth, if necessary. (Lessons 11-4 and 11-5) 34.

13 in.

35.

4 ft

5 ft

36.

37.

M

22 m

10 in. 10 ft

6 ft

14 m

M

M

38. GARDENS Marty is designing 2 flower beds shaped like equilateral triangles. The lengths of each side of the flower beds are 8 feet and 20 feet, respectively. What is the ratio of the area of the larger flower bed to the smaller flower bed? (Lesson 10-6) 39. Angles J and K are complementary. Find m∠K if m∠J is 25°. (Lesson 10-1) Find each square root. (Lesson 9-1) 40. ±兹 256

41. ±兹 324

42. 兹 0.81

43. 兹 2.25

Algebra and Architecture Able to Leap Tall Buildings It is time to complete your project. Use the information you have

gathered about your building to prepare a report. Be sure to include information and facts about your building as well as a comparison of its size to some familiar item. Cross-Curricular Project at pre-alg.com


613

Precision and Accuracy All measurements taken in the real world are approximations. The greater care in which a measurement is taken, the more accurate it will be. In everyday language, precision and accuracy are used to mean the same thing. When measurement is involved, these two terms have different meanings.

Term

Definition

Example

precision

the degree of exactness in which a measurement is made

A measure of 12.355 grams is more precise than a measure of 12 grams.

accuracy

the degree of conformity of a measurement with the true value

Suppose the actual mass of an object is 12.355 grams. Then a measure of 12 grams is more accurate than a measure of 18 grams.

Reading to Learn 1. Describe in your own words the difference between accuracy and precision. 2. RESEARCH Use the Internet or other resources to find an instrument used in science that gives very precise measurements. Describe the precision of the instrument. 3. Use at least two different measuring instruments to measure the length, width, height, or weight of two objects in your home. Describe the measuring instruments that you used and explain which measurement was most precise. Choose the correct term or terms to determine the degree of precision needed in each measurement situation. 4. In a travel brochure, the length of a cruise ship is described in (millimeters, meters). 5. The weight of a bag of apples in a grocery store is given to the nearest (tenth of a pound, tenth of an ounce). 6. In a science experiment, the mass of one drop of solution is found to the nearest 0.01 (gram, kilogram). 7. A person making a jacket measures the fabric to the nearest (inch, eighth of an inch). 8. CONSTRUCTION A construction company is ordering cement to complete all the sidewalks in a new neighborhood. Would the precision or accuracy be more important in the completion of their order? Explain. 614 Chapter 11 Three-Dimensional Figures

CH

APTER


11


IDS ES RAM #ON

0Y

3URFACE !REA


6OLUME #H 0RI SM S #Y LIN DE RS

Key Vocabulary

Key Concepts Three-Dimensional Figures

(Lesson 11-1)

• Prisms, pyramids, cylinders, cones, and spheres are three-dimensional figures. • Prisms and pyramids are polyhedrons and are named by the shape of their bases.

base (p. 576) cone (p. 576) cylinder (p. 576) edge (p. 575) face (p. 575) lateral area (p. 597) lateral face (p. 597) nets (p. 597) plane (p. 575) polyhedron (p. 575)

prism (p. 576) pyramid (p. 576) similar solids (p. 608) slant height (p. 602) solid (p. 575) sphere (p. 590) surface area (p. 597) vertex (p. 575) volume (p. 583)

• Cylinders, cones, and spheres are not polyhedrons.

Volume


• rectangular prism: V = Bh or wh • cylinder: V = r2h • pyramid: V = _ Bh 1 3

• cone: V = _ r2h

Vocabulary Check

• sphere V = _ r3

Determine whether each statement is true or false. If false, replace the underlined word or phrase to make a true statement. 1. The surface area of a pyramid is the sum of the areas of its lateral faces.

1 3

4 3

Surface Area


• The surface area of a solid is the sum of the lateral area plus the area of the base(s). • rectangular prism: S = Ph + 2B • cylinder: S =

2r2

+ 2rh

• pyramid: S = L + B • cone: S = r + r2

Similar Solids

(Lesson 11-6)

• Solids are similar if they have the same shape and their corresponding linear measures are proportional. • The ratio of the surface areas of two similar solids is proportional to the square of the scale factor between them. • The ratio of the volumes of two similar solids is proportional to the cube of the scale factor between them.


2. Volume is the amount of space that a solid contains. 3. Two similar solids with a scale factor of _a b a3 have surface areas in the ratio of _ . 3 b

4. The edge of a pyramid is the length of an altitude of one of its lateral faces. 5. A triangular prism has two bases. 6. A solid with two bases that are parallel circles is called a cone. 7. Prisms and pyramids are named by the shapes of their bases. 8. Figures that have the same shape and corresponding linear measures that are proportional are called similar solids.


615

CH

A PT ER

11




(pp. 575–581)

Identify each solid. Name the bases, faces, edges, and vertices. 9. Q R T

J

S V

U

10.

M

K

X

W

C

L

There is one triangular base, so the solid is a triangular pyramid. base: KLM faces: JKL, JLM, JMK, KLM −− −− −− −− −−− −−− edges: JK, JL, JM, KL, LM, MK vertices: J, K, L, M

G D

H

F

11–2

Example 1 Identify the solid. Name the bases, faces, edges, and vertices.

J

Volume: Prisms and Cylinders

(pp. 583–588)


3.4 m

12.

Example 2 Find the volume of the rectangular prism. V = wh

1.9 mm

Volume of a rectangular prism

n

= 9 · 8 · 16 Substitution

6m 0.8 mm 0.5 mm

13.

14.

= 1152

Simplify.

The volume is 1152 cubic meters.

°

Example 3 Find the volume of the cylinder.

7 cm 5 cm

£È

11 cm

£Î°{ °

V = r 2h

Volume of a cylinder

2.3 mm

6.0 mm

= (2.3)2(6.0) Substitution 15. BEVERAGES A 12-ounce can of soda 3 inches high with a radius measures 4_ 4 1 inches. Find the amount of soda of 1_ 8 that can fit in the can. Round to the nearest tenth.


≈ 99.7

Simplify.

The volume is about 99.7 cubic millimeters.



11–3

Volume: Pyramids, Cones, and Spheres

(pp. 589–594)


17. 3 ft

Example 4 Find the volume of the cone. Round to the nearest tenth.

A 7.5 m2 5.1 m 8 in.

2 ft

2 ft 4 in.

18. STORAGE Mr. Owens built a conical storage shed with a base 14 feet in diameter and a height of 11 feet. What is the volume of the shed?

1 2 V=_ r h

3 _ = 1 ()(42)(8) 3

≈ 134.0

Volume of a cone Replace r with 4 and h with 8. Simplify.

The volume is about 134.0 cubic inches.

11–4

Surface Area: Prisms and Cylinders

(pp. 597–601)

Find the lateral area and surface area of each solid. Round to the nearest tenth, if necessary. 19.

MM

Example 5 Find the lateral area and surface area of the rectangular prism.

2 in.

MM

4 in 5 in. MM MM

20.

°x vÌ x vÌ

L = Ph

Write the formula.

= (2 + 2w)(h)

P = 2 + 2w

= (2 · 5 + 2 · 4)(2)

Substitution

= 36

Simplify.

The lateral area is 36 square inches. 21. TELEVISION Coaxial cable is used to transmit cable television programming. The cable is covered by rubber sheathing. A typical coaxial cable has a diameter of 3 inches. How much rubber sheathing is there in 100 feet of cable?

S = L + 2B = L + 2(w)

Write the formula. B = w

= 36 + 2(5 · 4) Substitution = 76

Simplify.

The surface area is 76 square inches.


617

CH

A PT ER

11 11–5


Surface Area: Pyramids and Cones

(pp. 602–606)


23.

18.1 cm

6 in. 14 cm

3 in.

S=L+B

vÌ

(2 ) 1 = 4(_ (9)(17) + 9 2)

vÌ

1 bh + s2 =4 _

= 387

25.

1

3 2 in.

The surface area is 387 square feet. £x°Î V

5 in.

n V

26. CONTAINERS Two kinds of perfumes are packaged in square pyramidal bottles. Perfume A is packaged in a bottle with a base 4 inches square and a slant height of 2.5 inches. Perfume B is packaged in a bottle with a base 2 inches square and has a slant height of 4.5 inches. Which bottle contains more glass?

11–6

£Ç vÌ

2

3 in.

24.

Example 6 Find the surface area of the square pyramid. Round to the nearest tenth.

Similar Solids

1.5 m

S=L+B

4m

= ␲r + ␲r2 = ␲(1.5)(4) + ␲(1.52) ≈ 25.9 The surface area is about 25.9 square meters.

(pp. 608–613)

Find the missing measure for each pair of similar solids. (Lesson 11-6) 27.

Example 7 Find the surface area of the cone. Round to the nearest tenth.

Example 8 Find the missing measure for the pair of similar solids.

x

x

28 in.

4 in. 9 in. 135 in.

15 in.

28.

12 m

8m x

6m

21 m

y

29. A prism has a surface area of 160 square feet. If the dimensions are _1 the original prism, what is the 4 surface area of the new prism? 618 Chapter 11 Three-Dimensional Figures

9 in.

15 in.

15 in.

Since the pyramids are similar, set up a proportion. 9 x _ =_ 15

28

28 · 9 = 15x 16.8 = x The slant height is 16.8 inches.

CH

A PT ER

11

Practice Test

1. WEDDING Draw a top-count view of the sketch of the wedding cake below.


10. Ó

x Ç

{

£

11. 2. MULTIPLE CHOICE The radius of a quarter is about 12 millimeters and the width is about 1.5 millimeters. How much metal is there in a roll of 40 quarters? A 169.6

mm3

B 2714.3

mm3

C 6785.8

mm3

D 27,143.4 mm3

Find the volume of each solid. Round to the nearest tenth, if necessary. 3. cylinder: radius 1.7 mm, height 8 mm 4. rectangular pyramid: length 14 in., width 8 in., height 5 in. 5. cube: length 9.2 cm 6. cone: diameter 26 ft, height 31 ft 7. BREAKFAST Find the maximum amount of milk that can fit in the cereal bowl, in the shape of a hemisphere (half of a sphere). Round to the nearest tenth.

FT


FT

12. n ° £Ó °

£ä °

£ä °

13. BUILDINGS The front entrance of The Louvre museum in Paris is in the shape of a square glass pyramid with a slant height of about 92 feet. Its square base is 116 feet on each side. How much glass was used in making the pyramid? 14. MULTIPLE CHOICE A square pyramid has a surface area of 25 square yards. If the dimensions are doubled, what is the surface area of the new cube? F 50 yd2 G 100 yd2

H 200 yd2 J 625 yd2

È °

8. FURNITURE Find the surface area of the ottoman that will be reupholstered, not including the bottom. Round to the nearest tenth. FT

£

n { °

15. Are the cylinders described in the table similar? Explain your reasoning. Cylinder

Diameter (mm)

Slant Height (mm)

A

24

21

B

16

14

16. MULTIPLE CHOICE A model of a new grocery store is 15 inches long, 9 inches wide, and 7 inches high. The scale is 50 feet to 3 inches. Find the length of the actual store. A B C D

45 ft 150 ft 250 ft 750 ft


619

CH

A PT ER


11


Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The wheel of a steamroller is shaped like a cylinder with a radius of 2 feet. The width of the wheel is 6 feet 6 inches. To the nearest square foot, what is the area of pavement with which the surface of the wheel will come into contact in one complete rotation? A 107

ft2

"

!

Ón

nä

#

.

,

Which procedure can be used to find the measure of ∠M? A Subtract 108 from 180. B Divide 108 by 2.

B 95 ft2 C 90

4. ABC is similar to LMN.

C Subtract 108 from 360.

ft2

D Divide 180 by 3.

D 82 ft2 5. The following statements are true about LMN. ,

Question 1 Unless specified otherwise, encourage [␲] on their calculators to students to use evaluate expressions involving ␲. Then follow any instructions regarding rounding.

2. A cardboard box is 36 inches long, 24 inches wide, and 18 inches tall. What is the volume of the box in cubic feet? F 5 ft3

-

.

• The measure of each angle is evenly divisible by 9.

G 6 ft3

• The measure of ∠L is less than 40°.

H 8 ft3

• The measure of ∠M is equal to the measure of ∠N.

J 9 ft3

• The measure of ∠M is twice the measure of ∠L.

3. GRIDDABLE What is the area, in square feet, of the basement floor shown below? Óä vÌ

Which choice fits all 4 statements for angles L, M, and N? F m ∠L = 45°

£n vÌ

H m ∠L = 45°

m ∠M = 63°

m ∠M = 72°

m ∠N = 72°

m ∠N = 63°

G m ∠L = 36°

J m ∠L = 18°

Ó{ vÌ

m ∠M = 72°

m ∠M = 81°

m ∠N = 72°

m ∠N = 81°

£{ vÌ




6. GRIDDABLE Kyle is using clay to build models of famous pyramids for his science fair project. How many cubic centimeters of clay are needed to build the pyramid shown below?

9. In the spreadsheet, a formula applied to the values in columns A and B results in the values in column C. What is the formula?

£Ó V

£x V £x V

7. If DEF is translated 3 units down and 5 units to the right, what are the coordinates of point E'? $

&

Ç % È x { Î Ó £

A (-2, 3)

B

C

1

4

0

4

2

5

1

3

3

6

2

2

4

7

3

1

A C=A-B B C = A - 2B

Y

ÇÈx {Î Ó£" £ Ó Î { x È Ç

A

C C=A+B D C = A + 2B 10. GRIDDABLE A concrete worker is making six cement steps. Each step is 4 inches high, 7 inches deep, and 20 inches wide. What volume of cement in cubic inches will be needed to make these steps?

£ Ó Î { x È ÇX

Pre-AP Record your answers on a sheet of paper. Show your work. C (2, 4)

11. A manufacturer ships its product in boxes that are 3 feet by 2 feet by 2 feet. The company needs to store some products in a warehouse space that is 32 feet long by 8 feet wide by 10 feet high.

B (1, -2) D (3, 2) 8. Malcolm earned some spending money by mowing lawns in his neighborhood. He spent $6.75 each for 3 movie tickets, and he spent $6.80 for lunch. Later he bought a CD for $13.65. This left him with $9.30. Which expression can be used to find how much money Malcolm earned? F 3(6.75) + 6.80 + 13.65 - 9.30

a. What is the greatest number of boxes the company can store in this space? (All the boxes must be stored in the same position.)

G 3(6.75) + 6.80 + 13.65 + 9.30

b. What is the total volume of the stored boxes?

H 3(6.75 + 6.80 + 13.65 + 9.30)

c. What is the volume of the storage space?

J 3(6.75 + 6.80 + 13.65 - 9.30)

d. How much storage space is not filled with boxes?


1

2

3

4

5

6

7

8

9

10

11

Go to Lesson...

11-4

11-2

10-8

9-6

10-1

11-3

10-3

1-1

7-2

11-2

11-2


621

Extending Algebra to Statistics and Polynomials Focus Use statistical procedures to describe data sets and make predictions. Use basic principles of algebra to analyze and represent proportional and nonproportional linear relationships.

CHAPTER 12 More Statistics and Probability Use statistical procedures to describe data. Evaluate predictions and conclusions based on statistical data. Apply concepts of theoretical and experimental probability to make predictions.

CHAPTER 13 Polynomials and Nonlinear Functions Understand that a function can be described in a variety of ways.

622 Unit 5 Extending Algebra to Statistics and Polynomials David R. Frazier/Photo Researchers. Inc.

Algebra and Agriculture Down on the Farm Did you know that consumers from around the world spend about $547 billion on food from U.S. farms and ranches? Each year U.S. agricultural exports generate more than $100 billion and provide jobs for nearly 1 million workers. In this project, you will be using statistics and functions to analyze farming or ranching in the United States. Log on to pre-alg.com to begin.

Unit 5 Extending Algebra to Statistics and Polynomials

623

12 •

Apply concepts of theoretical and experimental probability to make predictions.

•

Use statistical procedures to describe data.

•

Evaluate predictions and conclusions based on statistical data.

More Statistics and Probability

Key Vocabulary histogram (p. 644) measures of variation (p. 633) probability (p. 665) simulations (p. 688) stem-and-leaf plot (p. 626)

Real-World Link Government The number of electors for each state can be displayed in a stem-and-leaf plot, box-and-whisker plot, or histogram.

More Statistics and Probability Make this Foldable to help you study the topics of statistics and probability. Begin with a piece of notebook paper.

1 Fold lengthwise to the holes.

2 Cut along the top line and then cut 10 tabs.

3 Label the lesson numbers and titles as shown. 3TEM AND ,EAF 0LOTS -EASURES OF 6ARIATION "OX AND 7HISKER 0LOTS (ISTOGRAMS LAYS

!PPROPR ATE $ISP

'RAPHS -ISLEADING

ABILITY MPLE 0ROB 3I MES TING /UTCO D #OUN S ION AN

624 Chapter 12 More Statistics and Probability Larry Fisher/Masterfile

UTAT ATIONS 0ERMM #O BIN

TE %VENTS

#OMPOSI



Option 1


Find the mean, median, and mode for each data set. (Lesson 5-9) 1. 10, 15, 23 2. 3.2, 5.1, 6.5, 6.5 3. AQUARIUM The following table shows the number of visitors to the aquarium each month. Find the mean, median, and mode. (Lesson 5-9)

Example 1

Find the mean, median, and mode for 1.8, 1.9, 1.3, 1.9, 2.0, 1.5, 1.1, 2.1, 1.4, and 1.6. 16.6 or 1.66 mean: __ = _ 1.8 + 1.9 + ... + 1.6 10

median: 1.1, 1.3, 1.4, 1.5, 1.6, 1.8, 1.9, 1.9, 2.0, 2.1 1.6 + 1.8 _ = 1.7

Visitors to the Aquarium (thousands) 3

11

5

4

5

3

6

3

12

2

2

4

6·5 5. _ 4·3 9·8·7 7. _

3·2·1

2

mode: 1.9 Example 2

Simplify. (Lesson 5-3) 5·4 4. _ 4·3 ·3·2 6. 4_

5·4·3

. Simplify _ 3·2·1 9·8·7 3

4

9·8·7 9·8·7 _ =_ 3·2·1

8. SURVEYS Seven-eighths of students surveyed said they drink one glass of milk per day. Three-fourths of those drink a glass of milk with dinner. What fraction of students drink their glass of milk with dinner? (Lesson 5-3)

1

1

= 3 · 4 · 7 or 84

Example 3

(Lesson 5-3 and 5-7)

5 1 7 Find _ +_ -_ .

2 4 7 _ 13. GARDENING Hannah filled a -gallon 1 8 12

12

3

3 1 _ 10. _ × 5 ×_ 2 6 4 3 1 1 _ _ 12. + - _ 8

watering can by pouring _ of a gallon of 4 water into the can. How much water was already in the can? (Lesson 5-7)


3·2·1

Find each sum, difference, or product. 3 5 4 9. _ ×_ ×_ 8 5 9 5 1 1 _ _ 11. + -_

10

12

8

8

12

Simplify.

12

5 5 _ 7 1 7 _ 1 _ _ +_ -_ =_ · 2 +_ · 3 -_ ·2 12

12

2

8

3

15 14 2 =_ +_ -_ 24

27 =_

24

12

2

24

24 3 1 = 1_ or 1_ 8 24


625

12-1

Stem-and-Leaf Plots

Main Ideas • Display data in stemand-leaf plots.

The number of electors for each state, including the District of Columbia, is shown in the table below.

• Interpret data in stem-and-leaf plots.

Number of Electors AL:

New Vocabulary

9 CT:

AK: 3 DE:

7 ID:

4 LA:

9 MS: 6 NJ: 15 OK: 7 TN: 11 WV: 5

3 IL: 21 ME: 4 MO: 11 NM: 5 OR: 7 TX: 34 WI: 10

AZ: 10 DC: 3 IN: 11 MD: 10 MT: 3 NY: 31 PA: 21 UT: 5 WY: 3

stem-and-leaf plot stems leaves back-to-back stem-and-leaf plot

AR: 6 FL: 27 IA:

7 MA: 12 NE:

5 NC: 15 RI:

CA: 55 GA: 15 KS:

6 MI: 17 NV:

5 ND: 3 SC: 8 VA: 13

CO: 9 HI:

4 VT: 3

4 KY: 8 MN: 10 NH: 4 OH: 20 SD: 3 WA: 11


Write each number on a self-stick note. Then group the numbers: 0–9, 10–19, 20–29, 30–39, 40–49, 50–59. a. Is there an equal number of electors in each group? Explain. b. Name an advantage of displaying the data in groups.

Display Data In a stem-and-leaf plot, numerical data are listed in ascending or descending order. The greatest place value of the data is used for the stems. The next greatest place value forms the leaves.

Draw a Stem-and-Leaf Plot OLYMPICS The table shows the total points scored in the first beach volleyball match played for both teams of each country in the 2004 Olympics. Display the data for the men’s teams in a stem-and-leaf plot. Step 1

Find the least and the greatest number. Then identify the greatest place value digit in each number. In this case, tens. 42 61

The least number has 4 in the tens place.

The greatest number has 6 in the tens place.

Step 2 Draw a vertical line and write the stems from 4 to 6 to the left of the line. 626 Chapter 12 More Statistics and Probability

Beach Volleyball Scores Country

Men

Women

Greece

52

47

United States

61

42

Brazil

42

42

Canada

44

42

South Africa

60

17

Cuba

50

54

Germany

55

52

Australia

42

42

Switzerland

49

29

Norway

46

37

Source: athens2004.com

Stem 4 5 6

Step 3

Write the leaves to the right of the line, with the corresponding stem. For example, for 42, write 2 to the right of 4.

Step 4 Rearrange the leaves so they are ordered from least to greatest. Then include a key or an explanation.

Stem Leaf 4 24296 5 205 6 10 Stem Leaf 4 22469 5 025 6 01 5 | 2 = 52 points

1. OLYMPICS Display the data for the women’s teams in a stem-and-leaf plot. Real-World Link The first President was George Washington. He was 57 years old at the time of his inauguration. He served as President from 1789 to 1797 and earned $25,000 per year.

Interpret Stem-And-Leaf Plots It is often easier to interpret data when they are displayed in a stem-and-leaf plot instead of a table.

Interpret Data


PRESIDENTS The stem-and-leaf plot lists the ages of the U.S. Presidents at the time of their first inauguration. Stem Leaf 4 23667899 5 0011112244444555566677778 6 0 1 1 1 2 4 4 6 8 9 5 0 = 50 years Source: The World Almanac

a. In which interval do most of the ages occur? Most of the data occur in the 50–59 interval. b. What is the age difference between the youngest and oldest President? The youngest age is 42. The oldest age is 69. The difference between these ages is 69 - 42 or 27. c. What is the median age of a President at first inauguration? The median, or the number in the middle, is 55. Look Back To review mean, median, and mode, see Lesson 5–9.

Refer to the stem-and-leaf plot of men’s Olympic times in Example 1. 2A. In which interval(s) do most of the scores occur? 2B. What is the difference between the greatest and least scores? 2C. What is the median score ?

Two sets of data can be compared using a back-to-back stem-and-leaf plot. The back-to-back stem-and-leaf plot below shows the scores of two basketball teams for the games in one season. Falcons Stem Cardinals

The leaves for one set of data are on one side of the stem.


76554222 88854 100 1 8 = 81 points

6 7 8

24 022579 13466899 8 6 = 86 points

The leaves for the other set of data are on the other side of the stem.

Lesson 12-1 Stem-and-Leaf Plots

“George Washington”, 1796. Gilbert Stuart. Oil on canvas. Bequest of Mrs. Benjamin Ogle Tayloe. Collection of The Corcoran Gallery of

627

WEATHER The average monthly temperatures for Seattle, WA Stem Helena, MT 2 016 Helena, Montana, and Seattle, Washington, are 3 15 shown. 65311 630 6511 1 6 = 61˚

a. Which city has lower monthly temperatures? Explain. Helena; it experiences temperatures in the 20s and 30s.

4 5 6

45 36 178 4 5 = 45˚

b. Which city has more varied temperatures? Explain. The data for Helena are spread out, while the data for Seattle are clustered. So, Helena has the more varied temperatures.

GRADES The test grades of two math classes are shown.

3rd Period Stem 7th Period 88322 763100 32110 8 7 = 78%

3A. Which class had higher scores? Explain. 3B. Which class had more varied test scores? Explain.

7 8 9

3 12566899 022333356 7 3 = 73%


Example 1 (pp. 626–627)

Display each set of data in a stem-and-leaf plot. 1.

Average Life Span Animal

Years

Animal

Years

Asian Elephant

40

African Elephant

35

Horse

20

Red Fox

Moose

12

Cow

Animal Lion

7

15

Chipmunk

15

Years 6

Hippopotamus

41


2.

Summer Paralympic Games Participating Countries Year

‘60

‘64

‘68

‘72

‘76

‘80

‘84

‘88

‘92

‘96

‘00

‘04

Countries

23

22

29

44

42

42

42

61

82

103

128

136

Source: paralympic.org

Example 2 (p. 627)

Example 3 (p. 628)

SCHOOL For Exercises 3–5, use the test score data shown at the right. 3. Find the lowest and highest scores. 4. What is the median score? 5. Write a statement that describes the data. FOOD For Exercises 6 and 7, use the food data shown in the back-to-back stem-and-leaf plot. 6. What is the greatest number of fat grams in each sandwich? 7. In general, which type of sandwich has a lower amount of fat? Explain.

628 Chapter 12 More Statistics and Probability

Stem Leaf 5 6 7 8 9

09 4578 044556788 233578 01559 5 9 = 59%

Fat (g) in Sandwiches Chicken Stem Burgers 8 985533 0 80=8g

0 1 2 3

059 06 036 2 6 = 26 g

HOMEWORK

HELP


Display each set of data in a stem-and-leaf plot. 8.

9.

State Representatives Largest States State

2004-2005 Big 12 Women’s Softball University

Number

Wins

California

53

Baylor

51

Florida

25

Iowa State

18

Illinois

19

Kansas

31

Michigan

15

Missouri

44

New York

29

Nebraska

36

Ohio

18

Oklahoma

50

Pennsylvania

19

Oklahoma State

35

Texas

32

Texas

49

Texas A&M

47

Texas Tech

23

Source: www.house.gov

Source: big12sports.com

10.

Percent of Young Adults (18–24) in U.S. Living at Home Year

1990

1991

1992

1993

1994

1995

1996

Percent

52.8%

54.4%

54.1%

53.0%

52.9%

52.6%

53.5%

Year

1997

1998

1999

2000

2001

2002

2003

Percent

54.1%

53.1%

53.3%

52.1%

50.2%

50.7%

50.3%

Source: U.S. Bureau of the Census

11.

Approximate Number of Students per Computer in U.S. Public Schools Year Percent Year Percent

‘89–’90

‘90–’91

‘91–’92

‘92–’93

‘93–’94

‘94–’95

‘95–’96

22

20

18

16

14

11

10

‘96–’97

‘97–’98

‘98–’99

‘99–’00

‘00–’01

‘01–’02

‘02–’03

8

6

6

5

5

5

5


ANALYZE TABLES For Exercises 12–16, use the table shown. 12. Display the number of home runs in a stem-and-leaf plot. 13. What is the most home runs hit between 1995 and 2005? 14. How many of the season leaders hit fewer than 50 home runs? 15. What is the median number of home runs hit by a single season home run leader? 16. Write a sentence that describes the data.

National League Single Season Home Run Leaders, 1995-2005 Year 1995

Player Dante Bichette

Home Runs 40

1996

Andres Galarraga

47

1997

Larry Walker

49

1998

Mark McGwire

70

1999

Mark McGwire

65

2000

Sammy Sosa

50

2001

Barry Bonds

73

2002

Sammy Sosa

49

2003

Jim Thome

47

2004

Adrian Beltre

48

2005

Andruw Jones

51

Lesson 12-1 Stem-and-Leaf Plots

629

ANALYZE TABLES For Exercises 17–20, use the information shown in the back-to-back stem-and-leaf plot. NCAA Women’s Basketball Statistics Overall Games Won, 2004–2005 Big Ten Conference Stem Big East Conference 55 97620 8840 0 1 = 10

0 1 2

1 23347999 356 2 5 = 25

Source: espn.com

Real-World Link The Louisiana Tech women’s basketball team has the best winning percentage in Division I. Over a 28-year period, the team has 793 wins and 133 losses. Source: infoplease.com

1 7. 18. 19. 20.

What is the greatest number of games won by a Big Ten Conference team? What is the least number of games won by a Big East Conference team? How many teams are in the Big East Conference? Compare the median number of games won by each conference.

21. RESEARCH Use the Internet, a newspaper, or another reference source to gather data about a topic that interests you. Make a stem-and-leaf plot of the data. Write a sentence that interprets the data. Tell whether each statement is sometimes, always, or never true. 22. A back-to-back stem-and-leaf plot has two sets of data. 23. A basic stem-and-leaf plot has two keys. Display each set of data in a stem-and-leaf plot. 24.

EXTRA

PRACTICE


25.

Olympic Men’s 400-m Hurdles Time (s), 1900–2004 57.6 53.0 55.0 54.0 52.6 53.4 51.7 52.4 50.8 50.1 49.3 49.6 47.6 47.8 47.2 46.8

47.5 48.7 47.5 51.1 48.1 47.8 47.6

Heights (ft) of Tallest Buildings in Miami, Florida 789 435 625 559 487 792 510 480 460 484 450 500 520 764 559 492 474 484 Source: The World Almanac

Source: olympic.org

H.O.T. Problems

26. COLLECT THE DATA Display the foot lengths, in inches, of the students in your class in a stem-and-leaf plot. Then write a few sentences that analyze the data. CHALLENGE Suppose you have a frequency table and a stem-and-leaf plot that display the same data. 27. For which display is it easier to find the median? Explain. 28. For which display is it easier to find the mean? Explain. 29. For which display is it easier to find the mode? Explain. 30. FIND THE ERROR The stem-and-leaf plot shows the heights of students in an eighth-grade class. Janice says that the number of students in the class is 61 - 46, or 15. Kirk says that the number of students in the class is the number of leaves, or 25. Who is correct? Explain your reasoning. Height of Students Stem Leaf 4 66777799 5 11233466778889 6 0 0 1 4 6 = 46 in.

630 Chapter 12 More Statistics and Probability CORBIS

31.

Writing in Math Use the data about the Electoral College on page 626 to explain how stem-and-leaf plots can help you understand an election. Include a stem-and-leaf plot and an explanation telling how a presidential candidate might use the display.

32. The back-to-back stem-and-leaf plot shows the amount of protein in certain foods.

Which of the following is a true statement? A The median amount of protein in dairy products is 9 grams.

Amount of Protein (g) Dairy Products 98877522 0 6 6 2 = 26 grams

B The difference between the greatest and least amount of protein in dairy products is 28 grams.

Legumes, Nuts, Seeds 0 1 2 3

569 458

C The average amount of protein in legumes, nuts, and seeds is more than the average amount in dairy products.

9 3 9 = 39 grams

D The greatest amount of protein in legumes, nuts, and seeds is 93 grams.

33. Determine whether a cone with a height 14 centimeters and radius 8 centimeters is similar to a cone with a height of 12 centimeters and a radius of 6 centimeters. (Lesson 11-6) Find the surface area of each solid. Round to the nearest tenth, if necessary. (Lesson 11-5)

34.

35.

CM

36. M

MM CM MM

M

M

37. SHADOWS At the same time a 40-foot silo casts a 22-foot shadow, a fence casts a 3.3-foot shadow. Find the height of the fence. (Lesson 9-6) Express each decimal or fraction as a percent. Round to the nearest hundredth percent, if necessary. (Lesson 6-5) 38. 0.36

39. 2.47

40. 0.019

41. 0.0065

6 42. _

4 43. _

15 44. _

24 45. _

25

7

8

1500

46. BUILDINGS The perimeter of a rectangular window in an office building is 204 inches. The length is 6 inches less than twice the width. Find its length. (Lesson 3-8)

PREREQUISITE SKILL Find the median for each set of data. Round to the nearest tenth, if necessary. (Lesson 5-9) 47. 23, 45, 21, 35, 28 48. 78, 54, 50, 64, 39, 45 49. 0.4, 1.3, 0.8, 1.8 Lesson 12-1 Stem-and-Leaf Plots

631


EXTEND

12-1

Stem-and-Leaf Plots and Line Plots

You can use the sorting capability of a graphing calculator to help you construct stem-and-leaf plots and line plots of data.

ACTIVITY GOVERNMENT The map at the right shows the number of electoral votes for each state. Create a stem and leaf plot and a line plot of the data.

Electoral Votes by State 11

4 3

7

3

4 3

Step 1 First enter the data into list L1.

5

KEYSTROKES:

1 11 ENTER STAT 7 ENTER 55 ENTER . . . 27 ENTER

55

5

9

10

)

6 7

5 34

2 2nd [L1]

STAT

10

31

17

21 20 21 11 5 13 8 11 15 11 8 6 6 9 15 9 27

12 4 7 15 3 10 3

3

ENTER 4

Step 3 View the list on the calculator to create the plots. Use the arrow keys to scroll through the data. KEYSTROKES:

1

STAT

Stem-and-Leaf Plot

Line Plot ⫻ ⫻ ⫻ ⫻⫻⫻ ⫻⫻⫻ ⫻ ⫻⫻ ⫻ ⫻⫻⫻⫻⫻ ⫻⫻⫻ ⫻ ⫻⫻⫻⫻⫻⫻⫻⫻⫻ ⫻⫻⫻⫻⫻⫻⫻⫻⫻⫻⫻ ⫻ ⫻

Stem Leaf 0 • 1 2 3 5

7

5

Step 2 Sort the data to order the list from least to greatest. KEYSTROKES:

10

3

34

333333334444455555 666777788999 00001111235557 0117 14 5 0|3=3

5

10

15

⫻ ⫻⫻ 20

⫻ 25

ANALYZE THE RESULTS 1. The table below shows the areas of the lakes in Kentucky’s state parks. Make a stem-and-leaf plot and a line plot of the data. 225

1140

32

2300

788

28

43

9

750

12

1110

183

2. When is it helpful to use the calculator to create stem-and-leaf plots or line plots? 632 Chapter 12 More Statistics and Probability

⫻ 30

⫻

⫻ 55

12-2

Measures of Variation

Main Ideas • Find measures of variation. • Use measures of variation to interpret and compare data.

New Vocabulary measures of variation range quartiles lower quartile upper quartile interquartile range outlier

The race that attracts the largest audience in auto racing is the Daytona 500. The average speed of each winning car from 1994 to 2005 is shown. Car Driver

Speed (mph)

Sterling Marlin ('94)

157

Dale Jarrett ('00)

156

Sterling Marlin ('95)

142

Michael Waltrip ('01)

162

Dale Jarrett ('96)

154

Ward Burton ('02)

143

Jeff Gordon ('97)

148

Michael Waltrip ('03)

134

Dale Earnhardt ('98)

173

Dale Earnhardt, Jr. ('04)

156

Jeff Gordon ('99)

162

Jeff Gordon ('05)

135

Car Driver

Speed (mph)

Source: daytonainternationalspeedway.com

a. What is the fastest speed? What is the slowest speed? b. Find the difference between the fastest and slowest speeds. c. Write a sentence comparing the fastest winning average speed and the slowest winning average speed.

Measures of Variation In statistics, measures of variation are used to describe the distribution of the data. One measure of variation is the range. The range of a set of data is the difference between the greatest and the least values of the set. It describes how a set of data varies.

EXAMPLE

Range

Find the range of each set of data. a. {5, 11, 16, 8, 4, 7, 15, 6} The greatest value is 16, and the least value is 4. So, the range is 16 - 4, or 12.

b. Stem Leaf 5 44446689 6 1 7 7 6 | 1 = 61

The greatest value is 77, and the least value is 54. So, the range is 77 - 54, or 23.


1A. {24, 15, 20, 21, 11, 14, 12}

1B. Stem Leaf 1 0023678 2 25699 3 128 2 | 6 = 26

Lesson 12-2 Measures of Variation

633

In a set of data, the quartiles are the values that divide the data into four equal parts. Recall that the median of a set of data separates the set in half. lower half

median

33 35 40 40 41

upper half

43 44

The median of the lower half of a set of data is the lower quartile, or LQ.

46 50

68

The median of the upper half of a set of data is the upper quartile, or UQ.

The upper and lower quartiles can be used to find another measure of variation called the interquartile range. Data that is more than 1.5 times the value of the interquartile range beyond the quartiles are called outliers. Interquartile Range Words

The interquartile range is the range of the middle half of a set of data. It is the difference between the upper quartile and the lower quartile.

Symbols Interquartile range = UQ - LQ

Reading Math Statistics A small interquartile range means that the data in the middle of the set are close in value. A large interquartile range means that the data in the middle are spread out, or vary.

EXAMPLE

Interquartile Range and Outliers

Find the interquartile range and any outliers for {36, 30, 61, 21, 34, 27}. Step 1 List the data from least to greatest. Then find the median. 21

27

30

34

36

61

30 + 34

median = or 32 2

Step 2 Find the upper and lower quartiles. lower half

21 27 LQ

upper half

30

34 36

median

61

UQ

The interquartile range is 36 - 27, or 9. Step 3 Find the limits for the outliers. Multiply the interquartile range, 9, by 1.5.

9 × 1.5 = 13.5

Subtract 13.5 from the lower quartile.

27 - 13.5 = 13.5

Add 13.5 to the upper quartile.

36 + 13.5 = 49.5

The limits for the outliers are 13.5 and 49.5. There are no values less than 13.5. One value, 61, is greater than 49.5. So, 61 is the one outlier.

Find the interquartile range and any outliers for each set of data. 2A. {49, 6, 40, 62, 51, 35, 43} 2B. {42, 49, 53, 41, 44, 67, 61, 55} Personal Tutor at pre-alg.com


Use Measures of Variation You can use measures of variation to interpret and compare data.

TRAFFIC LAWS The maximum allowable speed limits for certain western and eastern states are listed in the stem-and-leaf plot.

Western States Stem Eastern States 5

5

55

6

555555555

5555555500 0 7 = 70 mph

7

000 6 5 = 65 mph

a. What is the median speed limit Source: infoplease.com for each region? The median speed limit for the western states is 75 miles per hour. The median speed limit for the eastern states is 65 miles per hour.

Real-World Link

b. Compare the western states’ range with the eastern states’ range. The range for the east is 70 - 65, or 5 mph, and the range for the west is 75 - 55, or 20 mph. So, the speed limits in the west vary more. Also, the speed limits in the west are generally higher than in the east.

In 1974, the national speed limit was 55 mph. Today, the speed limits for the 50 states range from 55 to 75 mph. Source: infoplease.com

TEMPERATURE The average monthly Tucson, AZ Stem temperatures of Tucson, Arizona, and Hot 4 Springs, New Mexico, are listed in the 99522 5 stem-and-leaf plot. 6 6 3A. What is the median temperature for each 41 7 city? 7541 8 3B. Compare Tucson’s range with Hot Springs’ 2 5 = 52°F range. Source: Weatherbase

Examples 1, 2

Hot Springs, NM 0279 3 029 389 0 4 0 = 40°F

Find the range, interquartile range, and any outliers for each set of data.

(pp. 633–634)

1. {82, 85, 98, 42, 76, 91}

2. Stem Leaf 7 23669 8 001 9 9 7 6 = 76

Example 3 (p. 635)

SCIENCE For Exercises 3–5, use the information in the table. 3. Which planet’s day length divides the data in half? 4. What is the median length of day for the planets? Name any outliers in the data. 5. Write a sentence describing how the lengths of days vary. Include a statement about any outliers in the data.

Extra Examples at pre-alg.com Eastcott Momatiuk/Getty Images

0LANET

,ENGTH OF $AY %ARTH HOURS

-ERCURY

6ENUS

%ARTH

-ARS

*UPITER

3ATURN

5RANUS

.EPTUNE

4HE LENGTHS ARE APPROXIMATE 3OURCE 4HE 7ORLD !LMANAC


635

HOMEWORK

HELP


Find the range, interquartile range, and any outliers for each set of data. 6. {65, 64, 73, 34, 15, 43, 92}

7. {9, 13, 25, 9, 1, 5, 6, 8}

8. {68°, 74°, 65°, 55°, 75°, 82°, 32°, 69°, 70°, 77°} 9. {$28, $12, $25, $23, $29, $24, $26, $31, $10, $29, $23} 10. Stem Leaf

11.

Stem Leaf

0 1225

4 0

1 3478999

5 01157778

2 66

6

2 6 = 26

779

5 7 = 57

ANALYZE TABLES For Exercises 12 and 13, use the data in the table. Average Temperature (°F) City

Feb.

July

Asheville, NC

39

73

Atlanta, GA

47

Birmingham, AL Fresno, CA

City

Feb.

July

Little Rock, AR

45

82

80

Louisville, KY

38

78

47

80

Oklahoma City, OK

42

82

51

81

Portland, OR

43

68

Houston, TX

55

84

Syracuse, NY

25

70

Indianapolis, IN

31

75

Tampa, FL

63

83

Jackson, MS

49

81

Washington, D.C.

34

77


12. Find the interquartile range for each month’s set of data. Are there any outliers in each set of data? 13. Which month has more consistent temperatures? Justify your answer. ANALYZE GRAPHS For Exercises 14 and 15, use the data in the stem-and-leaf plot. Total Points Scored by Winners 1960–2005 14. Find the range, median, Rose Bowl Stem Cotton Bowl upper quartile, lower quartile, interquartile 0 77 range, and any outliers for 8777777444430 1 0000234477799 each set of data. 8877743322111000 2 01344478889 15. Write a few sentences that 888874444 3 0001115555668888 compare the data. 65542221

4

156

5

5

7 1 = 17 points

2 4 = 24 points


EXTRA

PRACTICE


H.O.T. Problems

16. FITNESS Find the range, median, upper quartile, lower quartile, interquartile range, and any outliers for Mark’s exercise times. Day Exercise Time (h)

1

2

3

4

5

6

7

8

9

10

0.66

0.43

1.25

0.2

0.53

0.6

0.58

0.48

0.84

0.63

17. OPEN ENDED Write a list of at least twelve numbers that has an interquartile range of ten.


CHALLENGE Write a set of data that satisfies each condition. 18. 12 pieces of data, a median of 60, an interquartile range of 20 19. 12 pieces of data, a median of 60, an interquartile range of 50 20. Compare the measures of variation for each set of data in Exercises 19 and 20. What conclusions can be drawn about the sets of data? 21.

Writing in Math Use the data about average speeds of winning cars of the Daytona 500 on page 633 to explain why measures of variation are important in interpreting data. Include the median, range, interquartile range, and any outliers for the set of data and an explanation telling what they tell about the speeds of the winning cars.

22. About how many times greater is the range of mountain heights in Alaska than in Colorado? Height (ft) Mountains in Alaska and Colorado Colorado Alaska 14,163 14,410 20,320 14,238 14,433 14,309 16,550 14,530 14,831 14,083 14,264 14,197 17,400 16,237 14,070 14,269 14,196 14,150 15,885 14,573 16,390 14,165 14,420 14,246 15,638 14,730 16,286 14,286 14,265 14,361

C 10

B 8

D 18

F The heights of the mountains in Alaska vary by 6200 feet. G The heights of the mountains in Colorado are clustered around the median height. H The median height of a mountain in Alaska is 16,000 feet. J The heights of the mountains in Colorado tend to be less consistent than the heights of the mountains in Alaska.


A 4

23. Which sentence best describes the data shown in the table in Exercise 22?

24. Display the data set {$12, $15, $18, $21, $14, $37, $27, $9} in a stem-and-leaf plot. (Lesson 12-1) Determine whether each pair of solids is similar. (Lesson 11-6) 25. 26. 4 in.

6 in. 10 in.

5 in.

6 in.

8 cm

12 cm

3 in. 9.8 cm

14.7 cm

Find the volume of each cone described. Round to the nearest tenth. (Lesson 11-3) 27. radius 7 cm, height 9 cm 28. diameter 8.4 yd, height 6.5 yd 29. The circumference of a circle is 9.82 feet. Find the radius of the circle to the nearest tenth. (Lesson 10-7) 3 30. TRAVEL How long would it take a train traveling 80 _ miles per hour to go 4 3 _ 363 miles? (Lesson 5-4) 8

PREREQUISITE SKILL Order each set of decimals from least to greatest. (page 742) 31. 5.6, 5.3, 4.8, 4.3, 5.0, 4.9 32. 0.3, 1.4, 0.6, 1.5, 0.2, 0.8, 1.2


637

12-3

Box-and-Whisker Plots

Main Ideas • Display data in a boxand-whisker plot.

The table shows the average monthly temperatures for two cities. Average Monthly Temperature (°F)

• Interpret data in a box-and-whisker plot.

New Vocabulary box-and-whisker plot

J

F

M

A

M

J

J

A

S

O

N

D

Tampa, FL

61

63

67

72

78

82

83

83

82

76

69

63

Caribou, ME

10

13

25

38

52

61

66

63

54

43

31

16

Source: weather.com

a. Find the low, high, and the median temperature, and the upper and lower quartile for each city. b. Draw a number line extending from 0 to 85. Label every 5 units. c. About one-half inch above the number line, plot the data found in part a for Tampa using points. About three-fourths inch above the number line, plot the data for Caribou using points. d. Compare the average monthly temperatures.

Display Data A box-and-whisker plot divides a set of data into four parts using the median and quartiles. A box is drawn around the quartile values, and whiskers extend from each quartile to the extreme data points. median UQ

LQ lower extreme, or least value

upper extreme, or greatest value

Draw a Box-and-Whisker Plot GEOGRAPHY Display the data in a box-and-whisker plot. Atlantic Coast Coastline State

Amount (mi)

State Amount (mi)

State

Amount (mi)

State

Amount (mi)

DE

28

MD

31

NJ

130

RI

40

FL

580

MA

192

NY

127

SC

187

GA

100

NH

13

NC

301

VA

112

ME

228


Step 1 Find the least and greatest number. Then draw a number line that covers the range of the data. 0

50


100

150

200

250

300

350

400

450

500

550

600

Common Misconception You may think that the median always divides the box in half. However, the median may not divide the box in half because the data may be clustered toward one quartile.

Step 2 Find the median, the extremes, and the upper and lower quartiles. Since the data have an outlier, mark the greatest value that is not an outlier. Mark these points above the number line. LQ: 35.5

median: 127

UQ: 210

outlier: 580 greatest value that is not an outlier: 301

lower extreme: 13 0

50

100

150

200

250

300

350

400

450

500

550

600

Step 3 Draw a box and the whiskers. The box contains the UQ and the LQ.

0

50

100

150

The whiskers extend from each quartile to the extreme data points that are not outliers.

200

250

300

350

400

450

500

550

600

1. GEOGRAPHY The heights in feet of the most famous waterfalls in Africa and Asia are 406, 508, 630, 343, 480, 330, 726, 830, 330, 614, 1100, 885, 1137, and 890. Display the data in a box-and-whisker plot.

Reading Math Box-and-Whisker Plots If the length of the whisker or box is short, the values of the data in that part are concentrated. If the length of the whisker or box is long, the values of the data in that part are spread out.

Interpret Box-and-Whisker Plots Box-and-whisker plots separate data into four parts. The parts may differ in length. 25% of the data

25% of the data

25% of the data

25% of the data

Interpret Data EDUCATION Graduation rates for the 50 states are shown below.

50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 Source: postsecondary.org

a. Half of the states have a graduation rate under what percent? Half of the states have graduation rates under 72%. b. What does the box-and-whisker plot tell us about the data? The length of the plot is long. This tells us that the data are spread out. MUSIC The number of chart hits for the top female groups in the U.S. are shown. £ä £x Óä Óx Îä Îx 2A. About how many chart hits did at -ÕÀVi\ /« £ä v ÛiÀÞÌ}] Óääx least half of the top groups have? 2B. What does the box-and-whisker plot tell about the data?

{ä {x

Personal Tutor at pre-alg.com Lesson 12-3 Box-and-Whisker Plots

639

Double box-and-whisker plots can be used to compare two sets of data.

FITNESS Two fitness clubs are analyzing their daily attendance for the month of September. How does the daily attendance at SuperFit compare to the daily attendance at the Athletic Club? -Õ«iÀ Ì ÌiÌV ÕL {x xä xx

Èä Èx

Çä Çx nä nx

ä x £ää £äx ££ä

SuperFit had a daily attendance between 48 and 82. The Athletic Club had a daily attendance between 52 and 110. Thus, the daily attendance at the Athletic Club varies more than the daily attendance at SuperFit.

3. NUTRITION The amount of food energy in Calories for fruits and vegetables is displayed. How does the food energy of fruits compare to vegetables?

Example 1 (pp. 638–639)

6i}iÌ>LiÃ ÀÕÌÃ ä

xä £ää £xä Óää Óxä Îää Îxä {ää {xä

-ÕÀVi\ 1°-° i«Ì° v }ÀVÕÌÕÀi

Draw a box-and-whisker plot for each set of data. 1. 25, 30, 27, 35, 19, 23, 25, 22, 40, 34, 20 2. $15, $22, $29, $30, $32, $50, $26, $22, $36, $31 OLYMPICS For Exercises 3 and 4, use the data shown in the table. Winning Distances for Women’s Olympic High Jump Year

1928

1932

1936

1948

1952

1956

1960

1964

1968

Distance (cm)

159

165

160

168

167

176

185

190

182

Year

1972

1976

1980

1984

1988

1992

1996

2000

2004

Distance (cm)

192

193

197

202

203

202

205

201

206

Source: olympic.org

Example 2 (p. 639)

Example 3 (p. 640)

3. Make a box-and-whisker plot for the data. 4. Write a sentence describing what the length of the box-and-whisker plot tells about the winning distance for the women’s high jump. TRAVEL For Exercises 5 and 6, use the double box-and-whisker plot shown. Average Gas Mileage for Various Sedans and SUVs Sedans SUVs

15 17 19

21 23

25 27 29 31

33 35 37

39 41

43

Source: classifieds2000.com

5. Which types of vehicles tend to be less fuel-efficient? 6. Compare the most fuel-efficient SUV to the least fuel-efficient sedan. 640 Chapter 12 More Statistics and Probability


HOMEWORK

HELP

For See Exercises Examples 7–10 1 11–13 2 14 3

Draw a box-and-whisker plot for each set of data. 7. 65, 92, 74, 61, 55, 35, 88, 99, 97, 100, 96 8. 60, 104, 80, 68, 159, 90, 100, 69, 104, 99, 130, 60 9. 80, 72, 20, 40, 63, 51, 55, 78, 81, 73, 77, 65, 67, 68, 59 10. $95, $105, $85, $122, $165, $55, $100, $158, $174, $162 SCHOOL For Exercises 11–13, use the box-and-whisker plot shown. Math Quiz Scores

70

60

80

90

100

11. What was the highest quiz score? 12. What percent of the students scored between 80 and 96? 13. Based on the plot, how did the students’ scores vary? 14. ANALYZE GRAPHS The number of games won by the teams in each conference of the National Football League in 2004 is displayed below. Write a few sentences that compare the data. >Ì> ÌL> viÀiVi

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Real-World Link There are an estimated 500,000 earthquakes in the world each year. One hundred thousand of those can be felt and 100 of them cause damage. Source: neic.usgs.gov

Draw a box-and-whisker plot for each set of data. 15.

Magnitudes of Recent Major Earthquakes

16. Average Points Scored per Game for NBA Scoring Leaders 1984–2004

6.1

8.7

7.1

32.9

30.1

26.8

6.9

6.9

8.1

30.3

32.6

29.7

6.6

7.1

6.6

37.1

29.8

31.1

6.8

6.8

6.8

35.0

29.3

31.4

6.5

7.1

7.0

32.5

30.4

32.1

6.5

6.8

6.7

33.6

29.6

28.0

6.7

9.0

7.2

31.5

28.7

30.7

Source: neic.usgs.gov

Source: nba.com

ANALYZE GRAPHS The number of Nobel Physics Prizes won by countries is displayed below.

EXTRA

PRACTICE


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x

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Source: Top 10 of Everything, 2005

17. About how many prizes did at least half of the countries win? 18. Describe the data and shape of the box-and-whisker plot. Lesson 12-3 Box-and-Whisker Plots

Mark Downey/Getty Images

641

H.O.T. Problems

19. COLLECT THE DATA Display the time, in minutes, it takes students in your class to travel to school in a box-and-whisker plot. Then write a few sentences that analyze the data. 20. CHALLENGE Write a set of data that contains twelve values for which the box-and-whisker plot has no whiskers. NUMBER SENSE The data show the performance of a math class on a 50-point quiz: minimum: 28; lower quartile score: 30; median: 38; upper quartile score: 42; and maximum: 48 21. Suppose there are 13 students in the class. Give a range of scores that would satisfy all the data shown. 22. Suppose six students have scores ranging from 38 to 42. How many students might there be in the class? Explain your reasoning. 23.

Writing in Math Explain how box-and-whisker plots can help you interpret data. Include an advantage of displaying data in a box-andwhisker plot instead of in a table.

24. GRIDDABLE The box-and-whisker plot shows the average recorded wind speeds (in miles per hour) in U.S. cities. What was the average wind speed in miles per hour recorded in at least half of the cities? ÛiÀ>}i ,iVÀì` 7` -«ii`Ã

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25. What percent of the wind speeds range from 5.6 to 10.4 miles per hour? A 25%

B 50%

C 75%

D 100%

For Exercises 26 and 27, use the set of data {2.4, 2.1, 4.8, 2.7, 5.5, 1.4, 3.9}. 26. What is the range, interquartile range, and any outliers for the data? (Lesson 12-2) 27. Display the data in a stem-and-leaf plot. (Lesson 12-1) 28. ALGEBRA Solve -5a - 6 = 24. (Lesson 3-5)

Weekly Recreation Time

PREREQUISITE SKILL For Exercises 29 and 30, refer to the table shown. 29. How many people were surveyed? 30. How many people spend more than 7 hours a week on recreational activities? 642 Chapter 12 More Statistics and Probability

Time (h)

Tally

Frequency

0–3

|||

3

4–7

|||| |||

8

8–11

|||| ||||

9

||||

5

12–15


EXTEND

12-3


You can use a TI-83/84 Plus graphing calculator to create box-and-whisker plots.

EXAMPLE The table shows the ages of the students in two karate classes. Class A B

39 19

33 26

37 40

26 19

39 20

25 32

Age (years) 39 40 27 16 24 24

25 16

35 27

31 23

29 22

28 25

35 16

Make box-and-whisker plots for the ages in Class A and in Class B. Step 1 Enter the data.

Step 2 Format the graph.

• Clear any existing data.

• Turn on two statistical plots.

KEYSTROKES:

STAT

CLEAR

• Enter the Class A ages in L1 and the Class B ages in L2. KEYSTROKES:

Review entering a list on page 68.

KEYSTROKES:

Review statistical plots on page 68.

• For Plot 1, select the box-and-whisker plot and L1 as the Xlist. KEYSTROKES:

[L1] • Repeat for Plot 2, using L2 as the Xlist, to make a box-and-whisker plot for Class B.

Step 3 Graph the box-and-whisker plots. • Display the graph. KEYSTROKES:

Class A

ZOOM 9

Class B

Press TRACE . Move from one plot to the other using the up and down arrow keys. The right and left arrow keys allow you to find the least value, greatest value, and quartiles.

outlier

EXERCISES 1. What are the least, greatest, quartile, and median values for Classes A and B? 2. What is the interquartile range for Class A? Class B? 3. Are there any outliers? How does the graphing calculator show them? 4. a. Estimate the percent of Class A members who are high school students. b. Estimate the percent of Class B members who are high school students. 5. If you were a high school student, which class would you join? Explain. Other Calculator Keystrokes at pre-alg.com

Extend 12-3 Box-and-Whisker Plots

643

12-4

Histograms

Main Ideas • Display data in a histogram. • Interpret data in a histogram.

New Vocabulary histogram

The number of counties in each state in the United States is displayed in the table shown. This table is a frequency table.

Number of Counties in Each State Counties 1–25

a. What does each tally mark represent? b. What does the last column represent? c. What do you notice about the intervals that represent the counties?

Tally

Frequency

|||| |||| |||

13

26–50

|||| ||

51–75

|||| |||| ||

12

76–100

|||| |||| ||

12

101–125

7

||||

4

126–150

0

151–175

|

1

176–200

0

201–225

0

226–250

0

251–275

|

1


Display Data Another type of graph that can be used to display data is a histogram. A histogram uses bars to display numerical data that have been organized into equal intervals. Number of Counties in Each State

Number of States

15

There is no space between bars.

12

Because the intervals are equal, all of the bars have the same width.

9 6

Intervals with a frequency of 0 have no bar.

3 0

5 27 1– 25 0 25 6– 22 5 22 1– 20 0 20 6– 17 5 17 1– 15 0 15 6– 12 5

00 –1

12 1– 10

76

5 –7 51

0 –5 26

25 1–

Number of Counties

Draw a Histogram WATER PARKS The frequency table shows certain water park admission costs. Display the data in a histogram. Step 1 Draw and label a horizontal and vertical axis as shown. Include the title.

Water Park Admission Cost ($) 8–15

||||

Frequency 5

16–23

|||| ||

7

24–31

||||

4

32–39 40–47


Tally

0 ||

2

The symbol means there is a break in the scale. The scale from 0 to 7 has been omitted.

Water Park Admission Number of Parks

Break in Scale

Step 2 Show the intervals from the frequency table on the horizontal axis and an interval of 1 on the vertical axis. Step 3 For each cost interval, draw a bar whose height is given by the frequency.

8 6 4 2 0

8–15 16–23 24–31 32–39 40–47 Cost ($)

Age

1. RETAIL The table shows the number of people in different age groups entering a store during the first hour. Display the data in a histogram.

Tally

Frequency

||||

5

10–19

|||| |||

8

20–29

|||| |||| ||||

14

30–39

|||| |||| |||| |||

18

40–49

|||| |||| |||| ||||

20

50–59

|||| |||| |||

13

60–69

|||| |

0–9

6

Interpret Histograms A histogram gives a better visual display of data than a frequency table. Thus, it is easier to interpret data displayed in a histogram.

EXAMPLE

Interpret Data

SCHOOL Refer to the histogram at the right.

Since 30 students are 69–71 inches tall, and 10 students are 72–74 inches tall, 30 + 10 or 40 students are at least 69 inches tall.

100 90 80 70 60 50 40 30 20 10 0

57–59 60–62 63–65 66–68 69–71 72–74 Height (in.)

b. Is it possible to tell the height of the tallest student?

No, you can only tell that the tallest student is between 72 and 74 inches.

POPULATION Refer to the histogram at the right. 2A. How many cities in South Carolina have populations less than 150,000? 2B. Is it possible to find the median population? Explain.

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At Least Recall that at least means is greater than or equal to.

a. How many students are at least 69 inches tall?

Number of Students

Reading Math

Students’ Heights

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Personal Tutor at pre-alg.com Lesson 12-4 Histograms

645

OLYMPICS Use the histograms below to answer the question.

6 4 2 0

100-m Freestyle

8 6 4 2 0

80

75

70

–8

–7

–7

4

9

4

9

64

–6

9

4

4

9

4

Time (s)

65

–5

–5

–8

–7

69

–7

– 60

55

80

75

70

4

9

4

–6

–5

–5

– 65

60

55

50

Number of Women

100-m Backstroke

8

50

Number of Women

Olympic Women's Swimming, 1924–2004

Time (s)


Which event has more winning times less than 1 minute? The 100-meter freestyle has 2 + 7 or 9 athletes with a winning time less than 1 minute while the 100-meter backstroke has none.

3. OLYMPICS Use the histograms above to find the number of Olympic Games that were held from 1924 to 2004.

Display each set of data in a histogram. 1.

2.

Pet Survey Tally

Frequency

Score

1–3

|||| |||| |||| |||| |

21

95–100

4–6

|||| ||

7

89–94

|||| |||| ||

7–9

||

2

83–88

|||| ||||

9

0

77–82

|||| |

6

1

71–76

||||

4

10–12 13–15

Example 2 (p. 645)

Test Scores

Pets

|

ROLLER COASTERS For Exercises 3–6, use the histogram shown. 3. Describe the data. 4. Which interval has the most roller coasters? 5. Why is there a jagged line in the vertical axis? 6. How many states have no roller coasters? Explain.

Tally

Frequency 5

||||

12

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Example 1 (pp. 644–645)

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Example 3

ANALYZE GRAPHS For Exercises 7 and 8, use the histograms below.

(p. 646)

U.S. National Parks and Monuments U.S. National Parks

U.S. National Monuments

38

36

36

34

34

Number of States

Number of States

38

12 10 8 6 4

12 10 8 6 4

2

2

0

0 –1 12

1

3

–1 10

9 8–

3

7 6–

5 4–

2–

1

13

1 0–

– 12

–1 10

9 8–

7 6–

5 4–

3 2–

1 0–

Number of Parks

Number of Monuments


7. Are there more states with two or more national parks or two or more national monuments? 8. How many more states have either one or no national parks than either one or no national monuments?

HELP



Time (h)

Tally

Weekly Allowance Amount

Frequency

Tally

Frequency

0–3

||

2

$0–$5

|||| |||| |

11

4–6

|||

3

$6–$11

|||| ||||

9

7–9

11.

10.

Weekly Study Time

|||| |||

8

$12–$17

|||| |||

8

10–12

|||| |||| ||

12

$18–$23

|||

3

13–15

|||| ||||

10

$24–$29

||||

5

Touchdowns in a Season

12.

Goals in a Season

Amount

Tally

Frequency

Amount

Tally

Frequency

80–96

|||| ||||

10

65–69

|||| |

6

97–113

||||

5

70–74

|||| ||

7

114–130

||||

4

75–79

|||

3

131–147

||

2

80–84

0

85–89

|||

3

1

90–94

|

1

148–164 165–181

|

ANALYZE GRAPHS For Exercises 13–15, use the data in the histogram. 13. How many restaurants sell chicken sandwiches that cost under $3? 14. How many restaurants were surveyed? 15. What percent of the restaurants surveyed sell chicken sandwiches that cost between $2.00 and $2.49?

0

Cost of Chicken Sandwiches Number of Restaurants

HOMEWORK

6 5 4 3 2 1 0

$1.50– $2.00– $2.50– $3.00– $3.50– $1.99 $2.49 $2.99 $3.49 $3.99 Cost ($)

Lesson 12-4 Histograms

647

ANALYZE GRAPHS For Exercises 16–18, use the histograms below.

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Tall Buildings in Los Angeles, CA Number of Buildings

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14 12 10 8 6 4 2 0

400– 500– 600– 700– 800– 900– 1000– 499 599 699 799 899 999 1099 Height (ft)


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16. Which city has more buildings less than 500 feet tall? 17. Which city has a greater number of buildings at least 600 feet tall? 18. Compare the heights of the tall buildings in the two cities. 19. PETS The line plot shows the ages of pets students own. Create a histogram from the line plot.

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ANALYZE GRAPHS For Exercises 20 and 21, use the histogram below. 20. The histogram shows the -ÌÕìÌ ÌiÀiÌ 1Ãi "i >Þ number of minutes students spend on the Internet in one Óä day. Discuss all of the £n £È information that you can £{ collect from the histogram. £Ó 21. Are you able to find any £ä measures of central tendency n from the histogram? Explain È your reasoning. { -ÌÕìÌÃ

Frequency tables and histograms can help you analyze data. Visit pre-alg.com to continue work on your project.

EXTRA

PRACTIICE


H.O.T. Problems

22. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would draw a histogram.

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23. COLLECT THE DATA Conduct a survey of your classmates to determine the number of text messages each person sends or receives during a typical week. Then choose intervals, make a frequency table, and construct a histogram to represent the data. 24. REASONING Explain why there are no spaces between the bars of a histogram. 25. CHALLENGE Create a set of data that, when plotted on a histogram, has a gap between 40 and 50, three items in the 20–29 interval, and the median value in the 30–39 interval. Display your data in a histogram.


26.

Writing in Math How are histograms similar to frequency tables? Include an explanation describing how data are displayed in each.

The histogram shows the ages of the students in the drama club.

27. How old are the oldest students?

Number of Students

Drama Club Students 8 6 4 2 0

A 16–18 yr

C 17–18 yr

B 16–19 yr

D 18–19 yr

28. What is the total number of students in the drama club? 9 –1 18 7 –1 16 5 –1 14 3 –1 12

Age

F 20

H 24

G 22

J

26

PRESIDENTS For Exercises 29 and 30, use the data shown below. Ages of Past Presidents at Time of Death

44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 Source: The World Almanac

29. What percent of presidents died by the time they were 78 years old? (Lesson 12-3) 30. Find the range and interquartile range for the data. (Lesson 12-2) 31. GEOMETRY Square X has an area of 9 square feet. The sides of square Y are twice as long as the sides of square X. Find the area of square Y. (Lesson 10-6) 32. Find the distance between A(3, 7) and B(-2, 1). Round to the nearest tenth, if necessary. (Lesson 9-5) Convert each rate using dimensional analysis. (Lesson 6-1) 33. 36 cm/s =

34. 66 gal/h =

m/min 104,

35. Order 6.1 × 6100, 6.1 × greatest. (Lesson 4-7)

10-5,

0.0061, and 6.1 ×

PREREQUISITE SKILL An entertainment attendance survey with high school students was taken, and the results are shown in the Venn diagram. (Lesson 4-3) 36. How many students have attended a musical? 37. How many students have attended an opera? 38. How many students have attended both? 39. How many students have attended neither? 40. How many students participated in the survey?

10-2

qt/min

from least to

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Lesson 12-4 Histograms

649


EXTEND

12-4

Histograms

You can use a TI-83/84 Plus graphing calculator to make a histogram.

ACTIVITY PRESIDENTS The list shows the ages of the first 43 Presidents at the time of inauguration: 57, 61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46, 54, 49, 50, 47 55, 55, 54, 42, 51, 56, 55, 51, 54, 51, 60, 62, 43, 55, 56, 61, 52, 69, 64, 46, and 54. Make a histogram to show the age distribution. Step 1 Enter the data.

Step 2 Format the graph.

• Clear any existing data in list L1.

• Turn on the statistical plot.

KEYSTROKES:

STAT

ENTER

KEYSTROKES:

CLEAR

ENTER

• Enter the data in L1. KEYSTROKES:

2nd [STAT PLOT] ENTER

ENTER

• Select the histogram and L1 as the Xlist.

Review entering a list on page 68.

ENTER

KEYSTROKES:

2nd

[L1] ENTER Step 3 Graph the histogram. Set the viewing window so the x-axis goes from 40 to 75 in increments of 5, and the y-axis goes from -5 to 15 in increments of 1. So, [40, 75] scl: 5 by [-5, 15] scl: 1. Then graph. KEYSTROKES:

WINDOW 40 ENTER 75 ENTER 5 ENTER

-5 ENTER 15 ENTER 1 ENTER GRAPH

EXERCISES 1. Press TRACE . Find the frequency of each interval using the right and left arrow keys. 2. Discuss why the domain is from 40 to 69 for this data set. 3. How does the graphing calculator determine the size of the intervals? 4. At inauguration, how many presidents have been at least 45, but less than 65? 5. What percent of presidents falls in the interval of Exercise 4? 6. Can you tell from the histogram how many presidents were inaugurated at age 52? Explain. 7. Refer to Example 2 on page 627. How does the stem-and-leaf plot compare to the histogram you have graphed here? Which graph is easier to read? 650 Chapter 12 More Statistics and Probability

Other Calculator Keystrokes at pre-alg.com

12-5

Selecting an Appropriate Display Interactive lab pre-alg.com

Main Idea • Select an appropriate display for a set of data.

The table shows the women’s winning times, in minutes, for the Boston Marathon from 1980 to 2005. 154 146 149 142 149 154 144 145 144 144 145 144 143 145 141 145 147 146 143 143 146 143 140 145 144 145 Source: The World Almanac

a. Draw a graph to show the number of times each winning time occurs. b. Draw a graph to show the number of items divided into intervals. c. Draw a graph to show how the items are spread out.

Select Appropriate Displays Data can be visually represented in many different ways. Graphs display data to help readers make sense of the information. Some graphs that you already know are listed below. Statistical Graphs Display

Best Time to Use

Bar Graph Box-and-Whisker Plot

to display the frequency of data in categories to divide a set of data into four parts using the median and quartiles to compare parts of the data to the whole to compare the number of values in intervals to display numerical data that has been organized into equal intervals to show change over a period of time to display how many times each number occurs in data to list all the data in a condensed form to list data individually or by groups to display relationships among sets of data

Circle Graph Frequency Table Histogram Line Graph Line Plot

Review Vocabulary Venn diagram a diagram that is used to show the relationship among sets of numbers using overlapping circles in a rectangle (Lesson 4-4)

Stem-and-Leaf Plot Table Venn Diagram

Select an Appropriate Display a. SUPREME COURT The table shows the number of years of service of the longest serving justices on the Supreme Court. Select an appropriate type of display for this situation. Then make the display with or without using technology. Years of Service Frequency

18–21

22–25

26–29

30–33

34–37

13

7

8

8

5


(continued on the next page) Lesson 12-5 Selecting an Appropriate Display

651

The data can be represented in two ways. First, you can use a histogram showing the number in each interval. Second, you can show how each part is related to the whole by using a circle graph. 9EARS OF 3ERVICE OF ,ONGEST 3ERVING 3UPREME #OURT *USTICES

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b. MUSIC Out of 85 teens surveyed, 40 own a digital music player and 69 own a CD-player. Of those students who own either kind of player, 31 own both devices. Select an appropriate type of display for this situation. Then make the display with or without using technology.

Real-World Career Market Research Analyst A market research analyst uses math to analyze data to make recommendations about promotion, distribution, design, and pricing of products and services.

A Venn diagram would show the relationship among the four groups: teens who own a digital music player, teens who own a CD-player, teens who own both, and teens who own neither. }Ì> ÕÃV *>ÞiÀ

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1A. TELEVISION The ratings for the highest-rated TV show is shown for each season. Select an appropriate type of display for this situation and justify your selection. Then make the display with or without using technology. ‘94–‘95

‘95–’96

‘96–’97

‘97–’98

‘98–’99

‘99–’00

‘00–’01

‘01–’02

‘02–’03

‘03–’04

20.5

22.0

21.2

22.0

17.8

18.6

17.4

15.3

16.1

15.9

Source: Nielsen Media Research

1B. MEDICINE Of the 45 patients in a doctor’s office one week, 27 had a cough, 36 had a fever, and 23 had both symptoms. Select an appropriate type of display for this situation and justify your selection. Then make the display with or without using technology. Even though different graphs can generally display the same data, there is usually one graph that you can use to help you make meaningful conclusions about the data. When deciding which graph to use, always ask yourself what you want your data to show. 652 Chapter 12 More Statistics and Probability Michael Newman/PhotoEdit


Which graph best represents the data in the table if you want to show how the data are spread out? Average Precipitation (in.) in Jackson, MS Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

5.7

4.5

5.7

6.0

4.9

3.8

4.7

3.7

3.2

3.4

5.0

5.3

Source: National Climatic Data Center

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3 4 5 6

C

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Read the Test Item Determining how a set of data is spread out involves finding where the majority of the data is located. Solve the Test Item You can eliminate choice D because it is difficult to find where the majority of the data is located exactly. Even though choice B groups the data into intervals, the graph does not show the spread of the data. Compared with choice A, choice C shows exactly where the majority of the data is located by displaying the measures of variation in a graphical form. So, choice C is the answer.

2. Which type of graph best represents the data if you want to show the frequency of the data? F circle graph

Quiz Scores Score

5

4

3

2

1

Number of Students

11

5

7

1

4

G line plot H box-and-whisker plot J

line graph Personal Tutor at pre-alg.com Lesson 12-5 Selecting an Appropriate Display

653

Example 1

Select an appropriate type of display for each data set. Justify your choice.

(pp. 651–652)

1. the stock price of a company for the last thirty days 2. judges’ scores in a skateboarding competition

Example 2

3. MULTIPLE CHOICE In a survey of 20 freshman, 15 play in marching band, 8 play in the jazz band, and 5 play in both. Which graph best represents this situation? A C Band Frequency

(p. 653)

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HELP


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Select an appropriate type of display for each data set. Justify your choice. 4. 5. 6. 7.

number of cars sold in the last year by color compared to the total number of people entering a water park by age groups percent of voters who voted in the last presidential election in each state the relationship among the people who have at least one brother, at least one sister, have both, or have neither 8. number of pins 1–10 the bowling team knocked down on their first roll in every frame 9. the times of the middle 50% of runners who ran in the 100-meter dash Select an appropriate type of display for each data set. Then make the display with or without using technology. 10.

11.

Total Goals Scored in Men’s World Cup Final, 1930–2002 6 3 5

6 3 1

5 7 5

4 6 2


5 3 3

4 4

Percent of Graduates in Each State Taking SAT, 2004 10 53

32

60 20

10 64

85 5

16

5

9

8

29

70

5

28

52

7 66



27 85

5

11 10

14 87

6 49

7

12

73 8

67

73

76 68

8 40 80 83 56

74

71 52

19

72 62 7

12

Select an appropriate type of display for each data set. Then make the display with or without using technology. 12. 21 29 53 19 23

Real-World Link New York has 1088 libraries and branches, the most of any state.

13.

Ages of Employees 25 33 30 39 47

49 60 57 45 34

40 49 55 62 19

56 22 41 46 32

22 36 50 52 33

Number of Siblings

Frequency

0–1

||||| ||||| ||||| |||

2–3

||||| ||||| ||||| ||||| ||

4–5

||||| ||||| ||

6–7

||||| |||

14. LIBRARIES Select a graph that best represents the data below if it is divided into equal intervals. Then make the display with or without using technology. Number of Branches of Top U.S. Libraries 67 54 42 22

Source: Public Libraries Survey

84 84 78 59 62 37 32 12 19 34 24 25 29 21 9 21 24 51 22 22 21 19 33 41

36 37 19 14 22 50 25 22 26 16 17 32 13 32 26 22 18 20 14 10

Source: Public Libraries Survey

EXTRA

PRACTICE

See pages 789, 805.

15. ROLLER COASTERS Select a graph that best represents the data at the right if each interval is to be compared to the whole. Then make the display with or without using technology.


H.O.T. Problems

Speed (mph)

Number of Roller Coasters

70–79 80–89 90–99 100+

1 10 2 5

Source: rcdb.com

16. OPEN ENDED Give an example of data that can be represented using a Venn diagram. 17. CHALLENGE Draw a Venn diagram for the following situation. In a movie survey of 100 students, 53 students like comedies, 61 like mysteries, and 48 like action movies. Ten students like comedies and mysteries, but not action movies. Twelve students like mysteries and action movies but not comedies. Only 5 students like comedies and action movies, but not mysteries. Eighteen students like all three types of movies. 18. SELECT A TOOL Adriana took a survey of her class to find out the ages of the pets students own. The results are shown at the right. Which of the following graphs might Adriana use to display the results if she wants to give a general idea of how the data cluster together? Justify your selection(s). Then use the graph(s) to display the results with or without using technology. stem-and-leaf plot

19.

box-and-whisker plot

4 11 4 3 0 12

9 2 6 0 3 10

9 4 12 7 4 6

histogram

Writing in Math

How can you use different types of graphs to represent the same data? Include examples of three different types of graphs that represent the same data and an explanation of the similarities and differences among the graphs. Lesson 12-5 Selecting an Appropriate Display

Gail Mooney/CORBIS

1 7 12 10 12 3

655

20. Which graph best represents the data in the table?

U.S. Meat Consumption (millions of pounds)

1999

2000

2001

2002

2003

26.5

27.3

27.0

27.9

27.0

Source: U.S. Department of Agriculture

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21. BASKETBALL Display the data in a box-and-whisker plot and histogram.

175 179

Basketball Team Heights (cm)

170 181

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177 168

182 183

ÓääÓ

178 168

179 176


22. Find the volume of a sphere with a radius of 5.5 centimeters. (Lesson 11-3) 23. Find the circumference and area of a circle with a radius of 10 feet. Round to the nearest tenth. (Lesson 10-7) 24. BUILDINGS Suppose a tower casts a 186.6-foot shadow at the same time a nearby tourist casts a 1.8-foot shadow. If the tourist is 6 feet tall, how tall is the tower? (Lesson 9-6)

PREREQUISITE SKILL For Exercises 25 and 26, use the graph shown. (pages 759–760) 25. Which jeans cost the most? 26. How does the cost of Brand B compare to the cost of Brand C?


Cost ($)

Cost of Certain Jeans 45 40 35 30 25 0

A

B

C D Brand

E

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Spreadsheet Lab

EXTEND

12-5

Bar Graphs and Line Graphs ACTIVITY

Construct a line graph of the hours per person per year that Americans spend using media such as television and radio. Media Usage Year

1999

2000

2001

2002

2003

2004

2005

Hours Per Person*

3444

3505

3534

3599

3660

3732

3789

2006

2007

3811

3874

* some data projected

Source: Communications Industry Forecast & Report

Step 1 Enter the data in the spreadsheet.

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Step 2 Choose the line graph from the Chart menu. Then format the line graph.

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Notice that the vertical scale of the graph is automatically set to begin at a number that fits the data. Often the scale begins at a number other than 0. You must make sure that your graph is clearly labeled in order to avoid being misleading.

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£

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To construct a bar graph, enter the data in the spreadsheet. Then choose bar graph from the chart menu.

ANALYZE THE RESULTS 1. Construct a bar graph of the data at the right about the communications tools teens use most.

Communications Tools Teens Use Most Tool

2. What are advantages and disadvantages of using a spreadsheet program to construct the graph in Exercise 1? 3. What type of graph is most useful for displaying the data below about screen names? Explain your choice. Then use a spreadsheet to construct the graph. Number of Screen Names of Teens Who Instant Message Number

1

2

3

4

5

6

7 or more

Percent

52

22

7

3

2

1

11

Percent

Landline phone

52

Instant message

24

Cell phone

12

Text message

3

Email

5

Phone/written messages equally

4

Source: Pew Internet & American Life Project

Source: Pew Internet & American Life Project

Extend 12-5 Spreadsheet Lab: Bar Graphs and Line Graphs

657

CH

APTER

12


BASKETBALL For Exercises 1–3, use the following information. (Lesson 12-1) East High School’s basketball team scored the following number of points in the first ten games of the season:

8. AGES Display the data in the table in a histogram. (Lesson 12-4)

Points Scored Stem Leaf 4 38 5 8 6 1335 7

357

Age

Frequency

0–1

3

2–3

6

4–5

7

6–7

5

8–9

2

10–11

9

5|8 = 58 points

1. Find the lowest and highest scores. 2. What is the median number of points scored in a game?

SPORTS For Exercises 9–12, use the following information. (Lesson 12-4) The histogram shows prices for different brands of gym shoes.

3. Write a statement that describes the data.

4. MULTIPLE CHOICE The following shows the number of bottles of water that were sold during a 9-game football season. Find the upper quartile of the data. (Lesson 12-2) 95, 32, 55, 19, 39, 22, 20, 41, 26 A 41

C 48

B 42

D 56

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9. How many gym shoes sell for less than $30? 10. How many gym shoes were sampled?

MUSIC For Exercises 5–7, use the following information. (Lesson 12-3) The plot shows the number of millions of albums sold for the top male singers in the United States.

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5. What is the greatest number of albums sold by a male singer? 6. How many albums did at least half of the top singers sell? 7. What does the length of the box-and-whisker plot tell about the data? 658 Chapter 12 More Statistics and Probability

11. What fraction of gym shoes in the sample sell between $40 and $90? 12. Are you able to create a line plot from the data in the histogram? Explain. Select an appropriate type of display for each data set. Justify your answer. (Lesson 12-5) 13. number of blood donors who are type O compared to the total 14. number of people who voted in the last election by age groups 15. percent of people who use public transportation daily in each major city 16. number of students on the math team, number of students on the debate team, number of students on both teams

12-6

Misleading Graphs

Main Ideas

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• Evaluate predictions and conclusions based on data analysis.

The graphs below show the monthly sales for one year for a company.

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• Recognize when graphs are misleading.

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a. Do both graphs show the same information? b. Which graph suggests a dramatic increase in sales from May to June? c. Which graph suggests steady sales? d. How are the graphs similar? How are they different?

Misleading Graphs Two line graphs that represent the same data may look quite different. Consider the graphs above. Different vertical scales are used. So, each graph gives a different visual impression.

People (millions)

TRAVEL The graphs show the growth of the cruise industry. a. Why do the graphs look different? The vertical scales differ.

Reading Math Graphs A graph is also misleading if there is no title, there are no labels on either scale, and the vertical axis does not include zero.

Graph B; the size of the ship makes the increase appear more dramatic because both the height and width of the ship are increasing.

People (millions)

b. Which graph appears to show a greater increase in the growth of the cruise industry? Explain.

9 8 7 6 5 4 3 2 1 0

Graph A Cruise Industry Growth

1980

1990

2000

Year

6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

Graph B Cruise Industry Growth

1980

1990

2000

Year Source: cruise.org

Lesson 12-6 Misleading Graphs

659

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Labels and Scale When interpreting any graph, be sure to read the labels and the scales in addition to looking at the graph.

TRAVEL The graphs show domestic traveler spending in the United States.

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Reading Math

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1A. Why do the graphs look different? 1B. Which graph appears to show a greater increase in the growth traveler spending? Explain.

Evaluating Predictions and Conclusions When reading a statistical graph, you must interpret the information carefully and determine whether the inference made from data is valid.

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a. SCOOTERS The graph displays units sold during the life of a scooter company. According to the graph, the number of scooters did not increase as fast in the 2000s as they did from 1970-2000. Determine whether this statement is accurate. Justify your reasoning.

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No, the statement is not accurate. Îää The horizontal scale is inconsistent; Óää from 1970 to 2000, the interval is 10 ä years, but the interval is 1 year ½Çä ½nä ½ä ½ää ½ä£ ½äÓ ½äÎ ½ä{ ½äx from 2000 to 2005. Also, the graph 9i>ÀÃ only goes through 2005. You would have to wait until 2010 to know the number of scooters sold and compare the rate of change from 2000 to 2010 to the rate of increase from 1970 to 2000. b. Suppose the president of the company predicts the number of scooters sold in 2010 will be approximately 800,000. Determine whether this statement is accurate. Justify your reasoning. No, the statement is not accurate. Even though the number of scooters sold decreased from 2004 to 2005, this decrease may not necessarily continue for the next five years. The general trend from 2000 to 2005 is an increase and we could reasonably predict that this trend continues. 660 Chapter 12 More Statistics and Probability


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2. TRAVEL The graph displays the number of vehicles that go through the tollbooth. According to the heights of the bars in the graph, there were less than twice as many vehicles going through the tollbooth between 3 P.M. and 7 P.M. than between 7 P.M. and 11 P.M. Determine whether this statement is accurate. Justify your reasoning.

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Example 1

SCHOOL For Exercises 1 and 2, refer to the graphs below.

(pp. 659–660)

Alisa’s Science Grades Graph A

Alisa’s Science Grades Graph B 100 80

80

Grade

Grade

100

60

20 0

0 1

2 3 4 Grading Period

60 40

5

1

2 3 4 Grading Period

5

1. Explain why the graphs look different. 2. Which graph appears to show Alisa’s grades improving more? Explain. /iÝÌqiÃÃ>}}

4. COMMUNICATION The graph shows the number of area codes in the United States in two years. According to the graph, the number of area codes increased by about four times between 1995 and 2005. Determine whether this statement is accurate. Justify your reasoning.

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3. TECHNOLOGY The graph displays the number of students who received a text message on their cell phones in the past week. According to the graph, no sophomores received a text message. Determine whether this statement is accurate. Justify your reasoning.

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Example 2 (pp. 660–661)

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Source: AT&T and cnn.com

Lesson 12-6 Misleading Graphs

661

HELP

MOVIES For Exercises 5 and 6, refer to the graphs below.


Top Five All-Time Movies Graph A

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Amount Made ($ millions)

700

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600

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HOMEWORK

500 400

nää Èää {ää Óää

0

A

B

C Movie

D

ä

E

Ûi

Source: movieweb.com

5. Which graph gives the impression that the top all-time movie made far more money than any other top all-time movie? 6. Which graph shows that movie C made nearly as much money as the other top movies? JOBS For Exercises 7 and 8, refer to the graphs below. 1°-° 1i«ÞiÌ ,>Ìi À>«

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7. What causes the graphs to differ in their appearance? 8. Which graph appears to show that unemployment rates have increased rapidly since 2000? Explain your reasoning.


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ANALYZE GRAPHS For Exercises 9 and 10, *>VvV ÛÃ ÓääÎqÓää{ {x use the graph at the right showing the {ä number of wins after an 82-game Îx schedule. Îä 9. According to the graph, Dallas had Óx about 4 times as many wins as Óä Phoenix. Determine whether this ä statement is accurate. Justify your reasoning. /i> 10. San Jose won only 28 games during the 2002–2003 season. If they Source: nhl.com continue to improve each season, they should win about 96 games within the following two seasons. Determine whether this statement is accurate. Justify your reasoning.

PRACTICE


H.O.T. Problems

11. ANALYZE GRAPHS According to the graph, Broadway’s box-office sales in 1970–1971 were down from the previous two-year period shown. Determine whether this statement is accurate. Justify your reasoning. 12. OPEN ENDED Find an example of a misleading graph in a newspaper or magazine. Explain why it is misleading.

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EXTRA

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9i>À Source: The League of American Theatres and Producers, Inc.

CHALLENGE The table shows the number of U.S. households with basic cable television. 13. Draw a graph that shows a slow increase in the number of basic cable subscribers. 14. Redraw the graph so that it shows a rapid increase in the number of basic cable subscribers. 15.

Subscribers (millions)

2000

69.3

2001

72.9

2002

73.5

2003

73.4

A Sales in Month 4 were nearly 3 times that of the sales in Month 1. B Sales in Month 2 were about 1.5 times greater than sales in Month 1.

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Year

Writing in Math How can graphs be misleading? Source: Nielsen Media Research Give an example of a graph that is misleading and explain how to redraw the graph so it is not misleading.

16. Which sentence is a true statement about the data in the graph?

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C Sales for Month 2 were about half that for Month 4. D Sales for Month 3 were about two-thirds that for Month 4. £

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17. Select an appropriate display for the data in the table. Justify your choice. Then make the display. (Lessons 12-4 and 12-5)

18. Draw a box-and-whisker plot for {56°, 43°, 38°, 42°, 50°, 47°, 41°, 55°}. (Lesson 12-3) 19. Find the area of a triangle with a base of 6 feet and a height of 4.2 feet. (Lesson 10-6)

Books Read

Tally

Frequency

0–2

|||| ⁄ |||

8

3–5

||||

6–8

|||| ⁄ |||| ⁄

4 10

9–11

|||| ⁄

5

12–14

|||

3

PREREQUISITE SKILL Express each fraction as a percent. (Lesson 6-5) 3 20. _ 4

1 21. _ 5

2 22. _ 3

5 23. _ 6

3 24. _ 8

Lesson 12- 6 Misleading Graphs

663

Dealing with Bias In statistics, a sample is biased if it favors certain outcomes or parts of the population over others. While taking a random survey is the best way to eliminate bias or favoritism, there are still many ways in which a survey and its responses can be biased.

Voluntary Response Consider a survey where people call or write in. Those who take the time to voluntarily respond usually have strong opinions on an issue. This may result in bias.

Response Bias Some surveys are biased because either the people participating in the survey are influenced by the interviewer or the people do not give accurate responses.

Nonresponse Consider a survey where the selected individuals cannot be contacted or they refuse to cooperate. Since the survey does not include a portion of the population, bias results.

Poorly Worded Questions A survey is biased if it contains questions that are worded to influence people’s responses.

Reading to Learn Tell whether each situation may result in bias. Explain your reasoning. 1. Suppose a bakery wants to know what percent of households makes baked goods from scratch. A sample is taken of 300 households. An interviewer goes from door to door between 9 A.M. and 4 P.M. 2. A telephone survey of 500 urban households is taken. The interviewer asks, “Does anyone in your household use public transportation?”

3. A radio station is conducting a survey as to whether people want a law that prohibits the use of computers for downloading music files. The radio announcer gives a number to call to answer yes or no. Of the responses, 85% said they do not want this law. 4. An interviewer states, “Due to heavy traffic, should another lane be added to Main Street?” 664 Chapter 12 More Statistics and Probability David Young-Wolff/PhotoEdit

12-7 Main Ideas • Find the probability of simple events. • Use a sample to predict the actions of a larger group.

New Vocabulary outcomes simple event probability sample space theoretical probability experimental probability

Simple Probability

A popular word game is played using 100 letter tiles. The object of the game is to use the tiles to spell words, scoring as many points as possible. The table shows the distribution of the tiles.

Number of Tiles

Letter E

12

A, I

9

O

8

a. Write the ratio that compares the number of tiles labeled E to the total number of tiles.

N, R, T

6

D, L, S, U

4

b. What percent of the tiles are labeled E?

G

3

c. What fraction of tiles is this? d. Suppose a player chooses a tile. Is there a better chance of choosing a D or an N? Explain.

B, C, F, H, M, P, V, W, Y, blank J, K, Q, X, Z

2 1

Probability of Simple Events In the activity above, there are 27 possible tiles. These results are called outcomes. A simple event is one outcome or a collection of outcomes. For example, choosing an E-tile is a simple event. You can measure the chances of an event happening with probability. Each of the outcomes must be equally likely to happen. Probability Words Symbols

The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. number of favorable outcomes P(event) = ___ number of possible outcomes

The probability of an event is always between 0 and 1, inclusive. The closer a probability is to 1, the more likely it is to occur. equally likely to occur impossible to occur 0 0%


certain to occur 1 or 0.25 4

25%

1 or 0.50 2

3 or 0.75 4

50%

75%

1 100%

Find Probability

Suppose a number cube is rolled. What is the probability of rolling a prime number? There are 3 prime numbers on a number cube: 2, 3, and 5. There are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.

(continued on the next page)

Lesson 12-7 Simple Probability

665

of favorable outcomes P(prime) = number

number of possible outcomes 1 3 1 The probability of rolling a prime number is _ or 50%. = or _ 2 6 2

1. What is the probability of rolling a number greater than 1?

The set of all possible outcomes is called the sample space. For Example 1, the sample space was {1, 2, 3, 4, 5, 6}. When you toss a coin, the sample space is {heads, tails}.

EXAMPLE

Find Probability

Suppose two number cubes are rolled. Find the probability of rolling an even sum.

Reading Math P (even sum) P(even sum) is read as the probability of rolling an even sum.

Make a table showing the sample space when rolling two number cubes. There are 18 outcomes (shown in green) in which the sum is even. 18 1 So, P(even sum) = _ or _ . 36

2

This means there is a 50% chance of rolling an even sum.

+

1

2

3

4

5

6

1

(1, 1)

(1, 2)

(1, 3)

(1, 4)

(1, 5)

(1, 6)

2

(2, 1)

(2, 2)

(2, 3)

(2, 4)

(2, 5)

(2, 6)

3

(3, 1)

(3, 2)

(3, 3)

(3, 4)

(3, 5)

(3, 6)

4

(4, 1)

(4, 2)

(4, 3)

(4, 4)

(4, 5)

(4, 6)

5

(5, 1)

(5, 2)

(5, 3)

(5, 4)

(5, 5)

(5, 6)

6

(6, 1)

(6, 2)

(6, 3)

(6, 4)

(6, 5)

(6, 6)

2. Find the probability of rolling a difference of 1.

The probabilities in Examples 1 and 2 are called theoretical probabilities. Theoretical probability is what should occur in an experiment. Experimental probability is what actually occurs when repeating a probability experiment many times.

EXAMPLE

Find Experimental Probability

The table shows the results of an experiment in which a coin was tossed. Find the experimental probability of getting tails for this experiment.

Outcome

Tally

Frequency

Heads Tails

|||| |||| |||| |||| |||| |

14 11

number of times tails occur 11 11 ___ =_ or _ number of possible outcomes

14 + 11

25

11 The experimental probability of getting tails in this case is _ or 44%. 25

3. Find the experimental probability of tossing a coin and getting heads for the experiment above. 666 Chapter 12 More Statistics and Probability


Use a Sample to Make Predictions You can use an athlete’s past performance to predict whether she will get a hit or make a basket. You can also use the results of a survey to predict the actions of a larger group.

Make a Prediction HOBBIES The circle graph shows the results of a survey that asked teens, ages 13 to 19, what they would be doing if they were not online. Out of a group of 450 teens, predict how many would be listening to music.

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The total number of teens is 450. So, 450 is the base or whole. The percent is 26%. Look Back To review percent proportion, see Lesson 6-6.

Let n represent the part. What number is 26% of 450? Write the percent proportion. 26 n part → _ =_ ← percent whole → 450 100

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100 · n = 26 · 450 100n = 11700 n = 117

Simplify. Mentally divide each side by 100.

You can expect 117 teens to say that they would be listening to music.

4. Out of the 450 teens surveyed, how many would you expect to say they would be writing or drawing? Personal Tutor at pre-alg.com

Example 1 (pp. 665–666)

Example 2 (p. 666)

Ten cards are numbered 1 through 10, and one card is chosen at random. Determine the probability of each outcome. Express each probability as a fraction and as a percent. 1. P(5)

2. P(odd)

3. P(less than 3)

4. P(greater than 6)

For Exercises 5 and 6, refer to the table in Example 2 on page 666. Determine each probability. Express each probability as a fraction and as a percent. 5. P(sum of 2 or 6)

Example 3 (p. 666)

Example 4 (p. 667)

6. P(even or odd sum)

7. The table shows the results of an experiment in which a number cube was rolled. Find the experimental probability of rolling a 4. 8. FOOD Maresha took a sample from a package of jellybeans and found that 30% of the beans were red. Suppose there are 250 jellybeans in the package. How many can she expect to be red?

Number 1 2 3 4 5 6

Frequency || | |||| ||| ||| ||


667

HOMEWORK

HELP

For See Exercises Examples 9–19 1, 2 20, 21 3 22 4

A spinner like the one shown is used in a game. Determine the probability of each outcome if the spinner is equally likely to land on each section. Express each probability as a fraction and as a percent. 9. P(8)

12

2

11

3

10

5 9

8

10. P(red)

11. P(even)

12. P(prime)

13. P(greater than 5)

14. P(less than 2)

15. P(multiple of 3)

16. P(factor of 11)

17. P(not yellow)

18. What is the probability that a calendar is randomly turned to the month of January or April? 19. Find the probability that today is November 31. ANALYZE TABLES For Exercises 20 and 21, use the following information and the table shown. The table shows the approximate number of licensed automobile drivers in the United States in a recent year. An automobile company is conducting a telephone survey using a list of licensed drivers. 20. Find the probability that a driver will be 19 years old or younger. Express as a decimal rounded to the nearest hundredth and as a percent. 21. What is the probability that a randomly chosen driver will be 40–49 years old? Write as a decimal rounded to the nearest hundredth and as a percent. 22. ANALYZE GRAPHS Refer to the graph that shows the results of a survey that asked youth about what is important to their personal success. If 1200 youth were surveyed, how many would you expect to say friendships are a factor in personal success?

Age

Drivers (millions)

19 and under

10

20 –29

34

30 –39

40

40 – 49

40

50 – 59

30

60 – 69

18

70 and over

19

Total

191

Source: U.S. Department of Transportation

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A bag contains 2 red marbles, 4 blue marbles, 7 green marbles, and 5 yellow marbles. Suppose one marble is selected at random. Find the probability of each outcome. Express each probability as a fraction and as a percent. Round to the nearest tenth percent. 23. P(blue)

24. P(yellow)

25. P(not green)

26. P(purple)

27. P(red or blue)

28. P(blue or yellow)

29. P(not orange)

30. P(not blue and not red)

31. P(red or not green)


EXTRA

PRACTIICE


H.O.T. Problems

Suppose two spinners like the ones shown are spun. Find the probability of each outcome. (Hint: Make a table to show the sample space as in Example 2 on page 666.) 32. P(2, 7)

33. P(even, even)

34. P(sum of 9)

35. P(2, greater than 5)

1

2

5

6

4

3

8

7

36. OPEN ENDED Give an example of a real-world situation in which the probability of an event is 25%. 37. CHALLENGE In the English language, 13% of the letters used are E’s. Suppose you are guessing the letters in a two-letter word of a puzzle. Would you guess an E? Explain. 38.

Writing in Math

Use the information about the word game on page 665 to explain how probability can help you make predictions. Include the probability of choosing each letter tile in the game.

39. What is the probability of spinning an even number on the spinner shown? 1 A _

40. Two number cubes are rolled. Which 3 ? would result in a probability of _

2

4

6

F Rolling a sum of 6 or less.

5

2 C _

G Rolling a 5 on each number cube.

3 _ D 3 4

4 _ B 1 2

H Rolling a product that is even. J Rolling a 4 on at least one number cube.

41. STATISTICS Describe a situation that might cause a line graph to be misleading. (Lesson 12-6) 42. ENVIRONMENT What type of display would be appropriate for the data in the table if you wanted to display the frequency of the average fuel economy? Make the display. (Lesson 12-5) Average Fuel Economy for Light Vehicles Year

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

Miles per gallon

21.1

21.2

20.9

20.9

20.6

20.7

20.7

20.6

20.7

20.8

Source: Environmental Protection Agency

43. If (-a)2 = 144, what is the value of a? (Lesson 9-2)

PREREQUISITE SKILL Find each product. 44. 4 × 3 × 5

45. 10 × 9 × 7 × 3

46. 26 × 25 × 24

47. 19 × 3 × 11 × 9


669

12-8

Counting Outcomes

Main Ideas • Use tree diagrams or the Fundamental Counting Principle to count outcomes. • Use the Fundamental Counting Principle to find the probability of an event.

New Vocabulary tree diagram Fundamental Counting Principle

The basic model of a skateboard has 5 choices for decks and 3 choices for wheel sets, as shown at the right. How many different skateboards are possible?

Decks Alien Birdman Candy Radical Trickster

a. Write the names of each deck choice on 5 sticky notes of one color. Write the names of each type of wheel on 3 notes of another color.

Wheel Sets Eagle Cloud Red Hot

b. Choose one deck note and one wheel note. One possible skateboard is Alien, Eagle. c. Make a list of all the possible skateboards. d. How many different skateboard designs are possible?

Review Vocabulary Outcomes possible results of a probability experiment (Lesson 12-7)

Counting Outcomes To solve the skateboard problem above, you can look at a simpler problem. Suppose there are only three deck choices, Birdman, Alien, or Candy, and only two wheel choices, Eagle or Cloud. You can draw a tree diagram to represent the possible outcomes.

EXAMPLE

Use a Tree Diagram to Count Outcomes

How many different skateboards can be made from three deck choices and two wheel choices? You can draw a diagram to find the number of possible skateboards. List each deck choice.

Deck Birdman

Alien

Candy

Each wheel choice is paired with each deck choice.

Wheels

Outcome

Eagle

Birdman, Eagle

Cloud

Birdman, Cloud

Eagle

Alien, Eagle

Cloud

Alien, Cloud

Eagle

Candy, Eagle

Cloud

Candy, Cloud

There are 6 possible outcomes.

1. Draw a tree diagram to find the number of different outfits that can be made from 5 shirts and 4 pairs of pants? Personal Tutor at pre-alg.com


In Example 1, notice that the product of the number of decks and the number of types of wheels, 3 · 2, is the same as the number of outcomes, 6. The Fundamental Counting Principle relates the number of outcomes to the number of choices. Fundamental Counting Principle Words

If event M can occur in m outcomes and is followed by event N that can occur in n outcomes, then the event M followed by N can occur in m · n outcomes.

Example

If there are 5 possible decks and 3 possible sets of wheels, then there are 5 · 3 or 15 possible skateboards.

You can also use the Fundamental Counting Principle when there are more than two events.

EXAMPLE

Use the Fundamental Counting Principle

SKIING When you rent ski equipment at Bridger Peaks Ski Resort, you choose from 4 different types of ski boots, 5 lengths of skis, and 2 types of poles. How many different outfits are possible? Use the Fundamental Counting Principle.

Multiplying More than Two Factors Remember, when you multiply, you can change the order of the factors. For example, in 4 × 5 × 2 you can multiply 5 × 2 first, then multiply the product, 10, by 4 to get 40.

The number of types of boots

4

times

the number of lengths of skis

×

5

times

the number of types of poles

equals

the number of possible outcomes.

×

2

=

40

2. ROUTES When Shelly goes into her school, she can walk through 4 different doors. Once inside, she can go to her locker by using 4 different sets of stairs and then 3 different hallways. How many ways can Shelly get from outside the school to her locker?

Find the Probability of An Event When you know the number of outcomes, you can find the probability that an event will occur.

EXAMPLE

Find Probabilities

a. Jasmine is going to toss two coins. What is the probability that she will toss one head and one tail? First find the number of outcomes.

First Coin

Second Coin Outcomes

Heads

Heads

Tails

Heads

Tails

H, H

H, T

T, H

T, T

There are four possible outcomes. Extra Examples at pre-alg.com

Tails

(continued on the next page) Lesson 12-8 Counting Outcomes

671

Look at the tree diagram. There are two outcomes that have one head and one tail. of favorable outcomes P(one head, one tail) = number number of possible outcomes

2 1 or _ =_ 2

4

1 The probability that Jasmine will toss one head and one tail is _ . 2

b. What is the probability of winning a state lottery game where the winning number is made up of four digits from 0 to 9 chosen at random? First, find the number of possible outcomes. Use the Fundamental Counting Principle. choices for the 1st digit

10

times

choices for the 2nd digit

×

10

times

choices for the 3rd digit

×

10

times

choices for the 4th digit

equals

total number of outcomes

×

10

=

10,000

3B. What is the probability of randomly choosing a 5-letter password for an Internet Web site that consists of only vowels?

Example 1 (p. 670)

Example 2 (p. 671)

Examples 2, 3 (pp. 671–672)

1

,

,

*1,*

3A. Lamar is going to spin each spinner once. What is the probability that he will spin red and the number 9?

There are 10,000 possible outcomes. There is 1 winning number. So, the 1 . This probability can also probability of winning with one ticket is _ 10,000 be written as a decimal, 0.0001, or a percent, 0.01%.

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Draw a tree diagram to find the number of outcomes for each situation. 1. The spinner at the right is spun twice. 2. Two number cubes are rolled. 3. The spinner at the right is spun, and a number cube is rolled. Find the number of possible outcomes for each situation. 4. A coin is tossed, and a number cube is rolled. 5. Four coins are tossed. 6. A number from 1 to 50 and a color from 8 colors are chosen. FOOD SERVICE Hastings Cafeteria serves toast, a muffin, or a bagel with coffee, milk, or orange juice. 7. How many different breakfasts of one bread and one beverage are possible? 8. What is the probability that a customer chooses a bagel with orange juice if a bread and beverage are equally likely to be chosen?


HOMEWORK

HELP


Draw a tree diagram to find the number of outcomes for each situation. 9. Each spinner shown at the right is spun once. 1 2 10. Three coins are tossed. 3 6 11. A restaurant offers three types of pasta 5 4 with two types of sauce and a choice of meatball or sausage. 12. Andrew has a choice of a blue, yellow, white, or striped shirt with a choice of black, navy, or tan pants. Find the number of possible outcomes for each situation. 13. School sweatshirts come in four sizes and four colors. 14. A number cube is rolled twice. 15. Two coins are tossed and a number cube is rolled. 16. A car comes with two or four doors, a four- or six-cylinder engine, and a choice of six exterior colors. 17. A quiz has five true-false questions. 18. There are four answer choices for each of five multiple-choice questions on a quiz. Find the probability of each event. 19. Three coins are tossed. What is the probability of two heads and one tail? 20. Two six-sided number cubes are rolled. What is the probability of getting a 3 on exactly one of the number cubes? 21. An 8-sided die is rolled three times. What is the probability of getting three 7s? 22. What is the probability of winning a lottery game where the winning number is made up of five digits from 0 to 9 chosen at random? 23. SKATEBOARDS How many different deluxe skateboards are possible from 10 choices of decks, 8 choices of trucks (the axles that hold the wheels on), and 12 choices of wheels?

Real-World Link Skateboarding events have been part of the X (or Extreme) Games since they were first held in June of 1995. Source: infoplease.com

EXTRA

PRACTIICE


24. GAMES Suppose you play a game where each player rolls two number cubes and records the sum. The first player chooses whether to win with an even or an odd sum. Should the player choose even or odd? Explain your reasoning. ANALYZE TABLES The table shows Processor RAM Hard Drive Monitor Color the features you can choose to 80 GB 15 in. CRT red 3.0 GHz 256 MB customize a computer. gray 160 GB 17 in. CRT 3.2 GHz 512 MB 25. How many customized green 15 in. Flat 1 GB computers include the white 17 in. Flat 17-inch flat panel lime monitor? 26. How many customized computers include 512 MB of RAM and a 3.2gigahertz processor? 27. If the features were chosen at random, what is the probability of choosing a red computer with a 3.0-gigahertz processor and 160 gigabyte hard drive? Lesson 12-8 Counting Outcomes

Michael Zito/SportsChrome USA

673

H.O.T. Problems

28. OPEN ENDED Give an example of a real-world situation that would have twelve outcomes. 29. REASONING Compare and contrast using a tree diagram and using the Fundamental Counting Principle to find numbers of outcomes. 30. CHALLENGE Most states use a system to design motor vehicle license plates that allows for 6,760,000 different plates using a total of six digits and letters. Find a way to use the digits 0-9 and letters A-Z to produce this exact number of arrangements. 31.

Writing in Math Use the information about skateboards on page 670 to explain how you can count the number of skateboard designs that are available from a catalog. Include the relationship of the number of designs to the number of wheels and decks and how the number of skateboards would change if the number of types of decks was doubled.

32. A 4-character password uses the letters of the alphabet. Each letter can be used more than once, but the letter I is not used at all. How many different passwords are possible?

33. The spinner is spun twice. What is the probability that it will land on 2 after the first spin and on 5 after the second spin? 1 F _

64 1 G_ 16 1 H_ 8 5 J _ 8

A 100 B 13,800 C 303,600 D 390,625

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34. PROBABILITY What is the probability of randomly choosing the letter A from the letters in TEXARKANA? (Lesson 12-7) >Þ } /i«iÀ>ÌÕÀi

35. WEATHER Explain how the graph at the right may be misleading. 36. GEOMETERY Each dimension of a rectangular prism was doubled. How does the volume of the new solid compare to the volume of the original solid? (Lesson 11-2) Find the slope of the line that passes through each pair of points. (Lesson 7-5)

37. Q(-4, 4), R(3, 5)

38. A(2, 6), B(-1, 0)

39. X(8, -3), Y(-4, 1)

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(Lesson 12-6)

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Y

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PREREQUISITE SKILL Simplify. (Lesson 4-5) 6·5 40. _ 2·1

5·4·3 41. _ 3·2·1


8·7 42. _ 2·1

6·5·4·3 43. _ 4·3·2·1

X

Algebra Lab

EXTEND

12-8

Probability and Pascal’s Triangle

ACTIVITY Step 1

Copy and complete the tree diagram shown below listing all possible outcomes if you toss a penny and a dime.

Penny

Dime Outcomes

Heads

Tails

Heads

Tails

Heads

?

H, H

?

?

?

Step 2

Make another tree diagram showing the possible outcomes if you toss a penny, a nickel, and a dime.

Step 3

Make a third tree diagram to show the outcomes for tossing a penny, a nickel, a dime, and a quarter.

ANALYZE THE RESULTS 1. For tossing two coins, how many outcomes are there? How many have one head and one tail? 2. Find P(two heads), P(one head, one tail) and P(two tails). Do not simplify. 3. For tossing three coins, how many outcomes are there? How many have two heads and one tail? one head and two tails? 4. Find P(three heads), P(two heads, one tail), P(one head, two tails), and P(three tails). Do not simplify. 5. For tossing four coins, how many outcomes are there? How many have three heads and one tail? two heads and two tails? one head and three tails? 6. Find P(four heads), P(three heads, one tail), P(two heads, two tails), P(one head, three tails), and P(four tails). Do not simplify. Pascal was a French mathematician who lived in the 1600s. He is known for the triangle of numbers at the right, called Pascal’s triangle. 7. Examine the rows of Pascal’s triangle. Explain how the numbers in each row are related to tossing coins. (Hint: Row 2 relates to tossing

1 1 1 1 1

1 2

3 4

Row 0 Row 1 1 3

6

Row 2 1

4

Row 3 1 Row 4

Extend 12-8 Algebra Lab: Probability and Pascal’s Triangle Aaron Haupt

675

12-9

Permutations and Combinations

Main Ideas • Use permutations. • Use combinations.

Lenora, Robert, Ned, Paloma, and Patrick are running for president and treasurer of the class. How many pairs are possible for the two offices? Use the Fundamental Counting Principle.

New Vocabulary choices for choices for possible president times treasurer equals pairs

permutation combination

5

×

4

=

20

There are 20 possible pairs. How is the number of pairs different for five students running for two student council seats, where order is not important? a. Make a list of all possible pairs for class offices. (Note: Lenora-Robert is different than Robert-Lenora.) b. How does the Fundamental Counting Principle relate to the number of pairs you found? c. Make another list for student council seats. (Note: For this list, Lenora-Robert is the same as Robert-Lenora.) d. How does the answer in part a compare to the answer in part c?

Reading Math Permutation Permute means to change the order or arrangement of, especially to arrange the order in all possible ways.

Use Permutations An arrangement or listing in which order is important is called a permutation. The symbol P(5, 2) represents the number of permutations of 5 things taken 2 at a time, as in 5 students running for 2 offices. 5 choices for president 4 choices left for treasurer

P(5, 2) = 5 · 4 5 students

Choose 2.

Use a Permutation a. SWIMMING How many ways can six swimmers be arranged on a four-person relay team? On a relay team, the order of the swimmers is important. This arrangement is a permutation. 6 swimmers

Choose 4.

P(6, 4) = 6 · 5 · 4 · 3 = 360 676 Chapter 12 More Statistics and Probability

6 5 4 3

choices choices choices choices

for for for for

1st person 2nd person 3rd person 4th person

b. How many four-digit numbers can be made from the digits 1, 3, 5, and 7 if each digit is used only once? 4 3 2 1

P(4, 4) = 4 · 3 · 2 · 1 = 24

choices for the 1st digit choices remain for the 2nd digit choices remain for the 3rd digit choice remains for the 4th digit

1A. SOFTBALL How many ways can the first 3 batters of a batting order be arranged from a team of 12 players? 1B. How many 5-digit identification numbers can be made from the digits 3, 5, 6, 8, and 9 if each digit is used only once?

Use Combinations Sometimes order is not important. For example, pepperoni, mushrooms, and onions is the same as onions, pepperoni, and mushrooms when you order a pizza. An arrangement or listing where order is not important is called a combination.

Use a Combination a. SCHOOL COLORS How many ways can students choose two school colors from red, blue, white, and gold? Since order is not important, this arrangement is a combination. First, list all of the permutations of red, blue, white, and gold taken two at a time. Then cross off arrangements that are the same as another one.

RB

RW

RG

BR

BW

BG

WR

WB

WG

GR

GB

GW

RB and BR are not different in this case, so cross off one of them.

There are only six different arrangements. So, there are six ways to choose two colors from a list of four colors. b. FLOWERS How many ways can three flowers be chosen from tulips, daffodils, lilies, and roses? Real-World Link Pepperoni is America’s favorite pizza topping; 36% of all pizza orders contain pepperoni. Source: pizzaware.com

The arrangement is a combination because order is not important. TDL

TDR TLR

DLR DLT

TLD

TRD TRL

DRT DRL DTL DTR

LRT

LRD LTD

RTD

RTL

LTR

LDR LDT

RDL RDT RLT

First, list all of the permutations. Then cross off the arrangements that are the same.

RLD

There are 4 ways to choose three flowers from a list of four flowers.

2A. STUDENT COUNCIL In how many ways can you choose a committee of two students from Jimmy, Molly, Evita, Julián, Debra, and Bobbie? 2B. PIZZA How many ways can a customer choose 3 pizza toppings from pepperoni, onion, sausage, green pepper, and mushroom? Personal Tutor at pre-alg.com Lesson 12-9 Permutations and Combinations CORBIS

677

Reading Math Combination Notation C(4, 2) is read the number of combinations of 4 things taken 2 at a time.

You can find the number of combinations of items by dividing the number of permutations of the set of items by the number of ways each smaller set can be arranged. From 4 colors, take 2 at a time.

4 ·3 2·1

C(4, 2) = or 6 There are 2 · 1 ways to order 2 colors.

CHESS The students listed are playing in a chess tournament. If in the first round each player plays every other player once, what is the probability that the first match played involves Abigail? Checking Reasonableness of Results The number of combinations of 10 taken 2 at a time is less than the number of permutations of 10 taken 2 at a time because order does not matter in a combination. So, check that C(10, 2) < P(10, 2).

Explore Abigail playing Irene is the same as Irene playing Abigail, so this is a combination.

Lorenzo Abigail Irene Booker Mato

Kate Rashid Destiny William Mercedes

Plan

Find the combination of 10 people taken 2 at a time. This will give you the number of matches that take place during the first round. Then find how many of the matches involve Abigail.

Solve

·9 C(10, 2) = 10 or 45 There are 45 ways to choose 2 people to play. 2·1

Abigail plays each person once during the first round. If there are 9 other people, Abigail is involved in 9 games. So the probability that 9 1 or _ . Abigail plays in the first match is _ 45

Check

5

List all of the 2-player matches in the first round. Check to see if there are 45 matches.

3. CHESS Suppose Booker drops out of the chess tournament. What is the probability that the final first-round game involves Abigail and Mato?

Examples 1–2

Tell whether each situation is a permutation or combination. Then solve.

(pp. 676–677)

1. How many ways can 5 people be arranged in a line? 2. How many programs of 4 musical pieces can be made from 8 possible pieces? 3. How many ways can a 3-player team be chosen from 9 students? 4. How many ways can 6 different flowers be chosen from 12 different flowers?

Example 3

5. FOOD A pizza shop has 12 toppings from which to choose. If 3 toppings are chosen randomly for a pizza, what is the probability that it is topped with pepperoni, onion, and sausage?

(p. 678)



HOMEWORK

HELP


Tell whether each situation is a permutation or combination. Then solve. 6. How many ways can 6 cars line up for a race? 7. How many different flags can be made from the colors red, blue, green, and white if each flag has three vertical stripes? 8. How many ways can 4 shirts be chosen from 10 shirts to take on a trip? 9. How many ways can you buy 2 DVDs from a display of 15? 10. How many 3-digit numbers can you write using the digits 6, 7, and 8 exactly once in each number? 11. There are 12 paintings in a show. How many ways can the paintings take first, second, and third place? 12. How many 5-card hands can be dealt from a standard deck of 52 cards? 13. How many ways can you choose 3 flavors of ice cream from a choice of 14 flavors? 14. GEOMETRY Twelve points are marked on a circle. How many different line segments can be drawn between any two of the points? 15. HANDSHAKES Nine people gather for a meeting. Each person shakes hands with every other person exactly once. How many handshakes will take place? AMUSEMENT PARKS For Exercises 16–18, use the information at the left. 16. Suppose you only have time to ride eight of the coasters. How many ways are there to ride eight coasters if order is important? 17. How many ways are there to ride eight of the coasters if order is not important? 18. What is the probability that the first two of the eight coasters you ride are the Millennium Force and the Top Thrill Dragster? Suppose you chose to ride the two coasters at random.

Real-World Link Cedar Point Amusement Park in Ohio is known for its roller coasters. There are sixteen roller coasters at the park. Source: cedarpoint.com

EXTRA

PRACTICE


H.O.T. Problems

19. SPORTS In a best-of-three series, the first team to win 2 games wins the series. Two ways to win a best-of-three series are to win the first 2 games or lose the first game and win the next two games. How many ways are there to win a best-of-three series? a best-of-five series? a best-of-seven series? FLOWERS For Exercises 20–22, use the following information. Three roses are to be placed in a vase. The color choices are red, pink, white, yellow, and orange. 20. How many different 3-rose combinations can be made from the 5 roses? 21. What is the probability that 3 roses selected at random will include pink, white, and yellow? 22. What is the probability that 3 roses selected at random will not include red? TECHNOLOGY Suppose your cell phone uses a 4-digit personal identification number (PIN) to lock it from use. 23. How many PINs can you choose to lock your cell phone? 24. Use probability to explain how someone is unlikely to guess your PIN. 25. FIND THE ERROR Sindu thinks choosing five CDs from a collection of 30 to take to a party is a permutation. Amanda thinks it is a combination. Who is correct? Explain your reasoning. Lesson 12-9 Permutations and Combinations

Mark Burnett

679

26. OPEN ENDED Write a problem that can be solved by finding P(4, 3). 27. CHALLENGE Is the value of P(x, y) sometimes, always, or never greater than the value of C(x, y)? (Assume x ≠ 1, y ≠ 1, and x ≠ y.) 28. CHALLENGE Determine whether the following statement is always true. If it is not, provide a counterexample. The number of combinations of items is always greater than 1. 29.

Writing in Math Why is order sometimes important when determining outcomes? Give an example of a situation where order is important and one where order is not important.

31. How many different slates of officers could be made if a slate consists of one candidate for each office?

30. When the Rockets win a basketball game, the 12 players on the team give each other high-fives. How many distinct high-fives are given?

President

Secretary

Treasurer

A 24

Marina

Glenn

Ariel

B 36

Mirna

Sung

Sherita

C 66

Nestor

D 96

F 3

Westey

G 6

H 9

J 18

32. GAMES How many outcomes are possible for rolling three number cubes? (Lesson 12-8) 33. PROBABILITY Find the probability of choosing a girl’s name at random from 20 girls’ names and 50 boys’ names. Round to the nearest tenth percent. (Lesson 12-7) ANIMALS For Exercises 34–36, use the histogram. (Lesson 12-4) 34. How many years are there in each interval? 35. Which interval has the greatest number of animals? 36. How many of the animals in the histogram have a life span of more than 20 years?

14 12 10 8 6 4 2 0 00 –1 91 0 –9 81 0 –8 71 0 –7 61 0 –6 51 0 –5 41 0 –4 31 0 –3 21 0 –2 11 10 1–

Number of Animals

Life Spans of Animals

Years Source: The World Almanac

Solve each inequality. Then graph the solution on a number line. (Lesson 8-6) 37. 2a - 3 ≥ 9

38. 4c + 4 > 32

39. -2y + 3 < 9

PREREQUISITE SKILL Find each product. (Lesson 5-3) 1 _ 40. _ ·1 6

3

2 _ 41. _ ·3 3

6


1 _ 1 42. _ · 1 ·_ 3

3

3

3 2 _ 43. _ · _ ·1 8

7

6

EXPLORE

12-10


Probability Simulation

A random number generator can simulate a probability experiment. From the simulation, you can calculate experimental probabilities. Repeating a simulation may result in different probabilities since the numbers generated are different each time.

ACTIVITY Generate 30 random numbers from 1 to 6, simulating 30 rolls of a number cube. • Access the random number generator. • Enter 1 as a lower bound and 6 as an upper bound for 30 trials. KEYSTROKES:

51

6

30

A set of 30 numbers ranging from 1 to 6 appears. Use the right arrow key to see the next number in the set. Record all 30 numbers, as a column, on a separate sheet of paper.

ANALYZE THE RESULTS 1. Record how often each number on the number cube appeared. a. Find the experimental probability of each number. b. Compare the experimental probabilities with the theoretical probabilities. 2. Repeat the simulation of rolling a number cube 30 times. Record this second set of numbers in a column next to the first set of numbers. Each pair of 30 numbers represents a roll of two number cubes. Find the sum for each of the 30 pairs of rolls. a. Find the experimental probability of each sum. b. Compare the experimental probability with the theoretical probabilities. 3. Design an experiment to simulate 30 spins of a spinner that has equal sections colored red, white, and blue. a. Find the experimental probability of each color. b. Compare the experimental probabilities with the theoretical probabilities. 4. Suppose you play a game where there are three containers, each with ten balls numbered 0 to 9. Pick three numbers and then use the random number generator to simulate the game. Score 2 points if one number matches, 16 points if two numbers match, and 32 points if all three numbers match. (Note: numbers can appear more than once.) a. Play the game if the order of your numbers does not matter. Total your score for 10 simulations. b. Now play the game if the order of the numbers does matter. Total your score for 10 simulations. c. With which game rules did you score more points? Other Calculator Keystrokes at pre-alg.com

Explore 12-10 Probability Simulation

681

12-10

Probability of Composite Events

Main Ideas • Find the probability of independent and dependent events. • Find the probability of mutually exclusive events.

New Vocabulary composite events independent events dependent events mutually exclusive events

Place two red counters and two white counters in a paper bag. Then complete the following activity. Step 1 Without looking, remove a counter from the bag and record its color. Place the counter back in the bag. Step 2 Without looking, remove a second counter and record its color. The two colors are one trial. Place the counter back in the bag. Step 3 Repeat until you have 50 trials. Count and record the number of times you chose a red counter, followed by a white counter. a. What was your experimental probability for the red then white outcome? b. Would you expect the probability to be different if you did not place the first counter back in the bag? Explain your reasoning.

Probabilities of Independent and Dependent Events A composite event consists of two or more simple events. The activity above finds P(red and white), the probability of choosing a red counter, followed by a white counter. These are independent events. In independent events, the outcome of one event does not influence the outcome of a second event. Composite events may also be called compound events. 2 1 or _ P(red on 1st draw) = _

There are 4 counters and 2 of them are red.

2

4

You replaced the first counter. There are still 4 counters and 2 are white.

2 1 or _ P(white on 2nd draw) = _ 4

2

The probability of two independent events can be found by using multiplication. Probability of Two Independent Events Words

Reading Math Probability Notation Read P( A and B ) as the probability of A followed by B.

The probability of two independent events is found by multiplying the probability of the first event by the probability of the second event.

Symbols P( A and B) = P( A) · P(B) 1 1 1 Example P( red and white) = _ · _ or _


2

2

4

EXAMPLE

Probability of Independent Events

GAMES In some versions of the board game Parchisi, your piece returns to Start if you roll three doubles in a row. What is the probability of rolling three doubles in a row? The events are independent since each roll does not affect the outcome of the next roll. There are six ways to roll doubles, and there are 36 ways to roll two number 6 1 or _ . cubes. So, the probability of rolling doubles is _ 36

Real-World Link Forms of the game Parchisi (also known as Parchesi or Parcheesi) have been in existence since the 4th century a.d.

6

P(three doubles) = P(doubles on 1st roll) · P(doubles on 2nd roll) · P(doubles on 3rd roll)

_1

=

_1

·

6

·

6

_1 6

1 =_ 216

1 The probability of rolling three doubles in a row is _ . 216

Source: boardgames. about.com

1. CARDS Two cards are drawn from a deck of cards numbered 1–10. After a card is selected, it is returned to the deck. What is the probability of drawing an even card and then a card greater than 8?

If the outcome of one event affects the outcome of a second event, the events are called dependent events. In the opening activity, if you do not replace the first counter, the events are dependent events. Probability of Two Dependent Events Check for Reasonableness of Results Since at least one possibility is eliminated in a second event, the probability of 2 dependent events is greater than the probability of 2 similar independent events. In Example 2,

Words

If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs.

Symbols

P(A and B) = P(A) · P(B following A)

Example

1 2 1 P(red and white, without replacement) = _ · _ or _ 2

EXAMPLE

3

3

Probability of Dependent Events

P(quarter and dime) 3 =_ 20 (with replacement)

Reiko takes two coins at random from the 3 quarters, 5 dimes, and 2 nickels in her pocket. What is the probability that she chooses a quarter followed by a dime?

P(quarter and dime) 1 =_

3 _ ·5 P(quarter and dime) = _

6

(without replacement) 3 . _1 > 6

20

10 9 1 _ = 15 or _ 90 6

3 of 10 coins are quarters, 5 of 9 remaining coins are dimes.

2. FRUIT A bowl of apples contains 6 red delicious, 7 granny smith, and 3 macintosh. Find the probability of randomly choosing a macintosh and then a granny smith. Personal Tutor at pre-alg.com Lesson 12-10 Probability of Composite Events Matt Meadows

683

Mutually Exclusive Events If two events cannot happen at the same time, they are said to be mutually exclusive. For example, when you roll two numbers cubes, you cannot roll a sum that is both 5 and even. Second Number Cube

First Number Cube

⫹

1

2

3

4

5

6

1

2

3

4

5

6

7

2

3

4

5

6

7

8

3

4

5

6

7

8

9

4

5

6

7

8

9

10

5

6

7

8

9

10

11

6

7

8

9

10

11

12

The probability of two mutually exclusive events is found by adding. P(5 or even) = P(5) + P(even)

Look Back To review adding fractions with like denominators, see Lesson 5-5.

There are 18 even sums.

There are 4 sums of five.

18 4 +_ = _

36 36 22 11 = _ or _ 36 18

Probability of Mutually Exclusive Events Words

The probability of one or the other of two mutually exclusive events can be found by adding the probability of the first event to the probability of the second event.

Symbols

P(A or B) = P(A) + P(B)

Example

18 4 11 P(5 or even) = _ + _ or _

EXAMPLE

36

36

18

Probability of Mutually Exclusive Events

The spinner at the right is spun. What is the probability that the spinner will stop on blue or an even number? The events are mutually exclusive because the spinner cannot stop on both blue and an even number at the same time.

6

1

5

2 4

3

P(blue or even) = P(blue) + P(even) To review adding fractions with unlike denominators, see Lesson 5-7.

1 1 +_ =_

6 2 4 2 =_ or _ 6 3 2 The probability that the spinner will stop on blue or an even number is _ . 3

3. What is the probability that the spinner will stop on a number greater than 4 or red? 684 Chapter 12 More Statistics and Probability


Example 1 (p. 683)

Example 2 (p. 683)

Example 3 (p. 684)

A number cube is rolled and the spinner is spun. Find each probability. 1. P(an odd number and a B) 2. P(a composite number and a vowel)

2 1

D

A

C

B

A card is drawn from a deck of eight cards numbered from 1 to 8. The card is not replaced, and a second card is drawn. Find each probability. 3. P(5 and 2) 4. P(two odd numbers) 5. A card is drawn from a standard deck of 52 cards. What is the probability that it is a diamond or a club? 6. There are 3 books of poetry, 5 history books, and 4 books about animals on a shelf. If a book is chosen at random, what is the probability of choosing a book about history or animals? 7. GAMES Santos is playing a board game that involves rolling two number cubes. He needs to roll a sum of 5 or 8 to land on an open space. What is the probability that he will land on an open space?

HOMEWORK

For Exercises 8–11 12–15 16–19

HELP

See Examples 1 2 3

A number cube is rolled, and the spinner is spun. Find each probability. 8. P(3 and E) 9. P(an even number and A) 10. P(a prime number and a vowel) 11. P(an odd number and a consonant)

2 1

A

E D

B C

A bag contains 3 red marbles, 4 green marbles, 2 yellow marbles, and 5 blue marbles. Once a marble is drawn, it is not replaced. Find the probability of each outcome. 12. two yellow marbles in a row 13. two blue marbles in a row 14. a blue then a green marble 15. a yellow then a red marble A card is drawn from the cards shown. Find the probability of each outcome. 16. P(3 or multiple of 2) 17. P(4 or greater than 5) 18. P(odd or even) 19. P(2 or 6)

1 3 4

5 6 7 9

20. FAMILIES Each time a baby is born, the chance for either a boy or a girl is one-half. Find the probability that a family of four children has four boys. EXTRA

PRACTIICE


An eight-sided number cube numbered 1–8 is rolled. Find the probability of each outcome. 21. P(3 or even) 22. P(6 or prime) 23. P(0 or greater than 6) 24. P(greater than 5 or less than 5) Lesson 12-10 Probability of Composite Events

685

25. A bag contains six blue marbles and three red marbles. A marble is drawn, it is replaced, and another marble is drawn. What is the probability of drawing a red marble and a blue marble in either order? A bag contains 3 red marbles, 4 green marbles, 2 yellow marbles, and 5 blue marbles. Once a marble is drawn, it is not replaced. Find the probability of each outcome. 26. three green marbles in a row 27. a blue marble, a yellow marble, and then a red marble

Real-World Link People in the United States spend approximately 200 hours per year surfing on the Internet. Source: Statistical Abstract of the United States

ANALYZE GRAPHS For Exercises 28 and 29, use the graph. 28. What is the probability that a teen chosen at random has used the Internet for games and another teen chosen at random has used it for studying in the last 30 days? Write as a decimal rounded to the nearest hundredth.

/iiÃ 1Ãi v ÌiÀiÌ ÀÜÃi

x{¯

-ÌÕ`Þ À ,iÃi>ÀV vÀ -V

x£¯

1Ãi >

x£¯

*>Þ >iÃ

ÎÓ¯

*iÀViÌ -ÕÀVi\ ÜÜÜ°i`>v>Þ°À}

29. What is the probability that a teen chosen at random has used the Internet to browse and another teen chosen at random has used E-mail in the last 30 days? Write as a percent to the nearest whole percent.

H.O.T. Problems

30. OPEN ENDED Write an example of two mutually exclusive events. CHALLENGE In a bag containing 9 marbles, some are red, some are white, and some are blue. The probability of selecting a red marble, a white 1 . marble, and then a blue marble is _ 21 31. How many of each color are in the bag? 32. Explain why there is more than one correct answer in Exercise 31. 33. FIND THE ERROR Dale and Shannon are finding the probability of rolling a difference of 0 or 1 on two number cubes. Who is correct? Explain your reasoning. Dale

5 10 . _ 6 60 _ =_ or _ 36

36

1296

108

Shannon

10 6 16 + = or 49 36 36 36

34. CHALLENGE When two events are inclusive, they can happen at the same time. To find the probability of inclusive events, add the probabilities of the events and subtract the probability of both events happening. Find P(green or even) for the spinner shown at the right. 35.

1

5

2 4

3

Writing in Math Explain how composite events are related to simple events. Explain how the results of drawing two counters relate to the probability for drawing one counter and the difference between independent and dependent events.

686 Chapter 12 More Statistics and Probability Ryan McVay/Getty Images

6

36. A quarter is tossed and the spinner shown is spun. How do you find the probability of tossing heads and spinning an odd number? 3 1 ×_ A _ 2

5

3 1 +_ B _ 2

5

1 2 ×_ C _ 2

£ x

Ó {

Î

5

1 2 +_ D _ 2

5

37. A bag contains 3 green balls, 2 blue balls, 4 pink balls, and 1 yellow ball, all the same size. Rico chooses a ball at random. Then without replacing it, he chooses a second ball. What is the probability that Rico chooses a blue ball followed by a yellow ball? 1 F _ 50

1 G _

2 H _

45

3 J _ 20

45

38. What is the probability of rolling a number cube three times and getting numbers less than 3 each time? 1 A _ 216

1 B _

1 C_

27

27 D_

8

64

39. How many ways can a family of four be seated in a row of four chairs at the theater if the father sits in the aisle seat? (Lesson 12-9) 40. How many license plates can be made from 3 letters (A–Z) followed by 3 numbers (0-9)? (Lesson 12-9) 41. TESTING A quiz contains five true-false questions and five multiple choice questions, each of which has four answer choices. In how many ways can the quiz by answered if one answer is given for each question? (Lesson 12-8) In the figure, a b. Find the measure of each angle. (Lesson 10-1) 42. ∠1 44. ∠3 46. ∠5

43. ∠2 45. ∠4 47. ∠6

b

a

1 3

2

65˚ 6

4 5

In a right triangle, if a and b are the measures of the legs and c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. (Lesson 9-4) 48. a = 6, b = 8

49. a = 7, b = 40

50. a = 8, c = 15

52. BLOOD TYPES The distribution of blood types in a random survey is shown in the table. If there are 625 students at Ford Middle School, how many would you expect to have Type A blood? (Lesson 6-8) 53. Express 146 miles in 3 hours as a unit rate. Round to the nearest tenth, if necessary. (Lesson 6-1)

51. b = 63, c = 65 Distribution of Blood Types Type

Percent

A

40

B

9

AB

4

O

47

Lesson 12-10 Probability of Composite Events

687

Algebra Lab

EXTEND

12-10

Simulations

You can use a simulation to act out a situation so that you can see outcomes. For many problems, you can conduct a simulation of the outcomes by using items such as a number cube, a coin, or a spinner. The items or combination of items used should have the same number of outcomes as the number of possible outcomes of the situation.

ACTIVITY 1 A quiz has 10 true-false questions. The correct answers are T, F, F, T, T, T, F, F, T, F. You need to correctly answer 7 or more questions to pass the quiz. Is tossing a coin to decide your answers a good strategy for taking the quiz? Since two choices are available for each answer, tossing a coin is a reasonable activity to simulate guessing the answers. Step 1 Toss a coin and record the answer for each question. Write T(true) for tails and F(false) for heads. Step 2 Repeat the simulation three times. Step 3 Shade the cells with the correct answers. A sample for one simulation is shown in the table below. Answers

T

F

F

T T

T F F T

Simulation 1

F

T

T

F F F T T T

Number Correct F 2

F

Simulation 2 Simulation 3

ANALYZE THE RESULTS 1. Based on the simulations, is tossing a coin a good way to take the quiz? Explain. Use a simulation to act out the problem. 2. A restaurant includes prizes with children’s meals. Six different prizes are available. There is an equally likely chance of getting each prize each time. a. Use a number cube to simulate this problem. Let each number represent one of the prizes. Conduct a simulation until you have one of each number. b. Based on your simulation, how many meals must be purchased in order to get all six different prizes? 688 Chapter 12 More Statistics and Probability Aaron Haupt

ACTIVITY 2 Logan usually makes three out of every four free throws he attempts during a basketball game. Conduct the following experiment to simulate the probability of Logan’s making two free throws in a row. Step 1 Use red counters to represent a made basket and white counters to represent a missed basket. The probability Logan will make a free 3 throw is _ . So, use 3 red counters and 1 white counter. 4

Step 2 Conduct a simulation for 25 free throws. Step 3 Without looking, draw a counter from the bag and record its color. Replace the counter and draw a second counter. Step 4 Repeat 25 times and record the results of the simulation in a chart like the one shown below. Misses the first shot

Makes the first shot, misses the second

Makes both shots

ANALYZE THE RESULTS 3. Calculate the experimental probability that Logan makes two free throws in a row. 4. How do the results in Activity 2 compare to the theoretical probability that Logan will make two free throws in a row? (Hint: These are independent events.) 5. Trevor usually makes four out of every five free throws he attempts. Calculate the theoretical probability that Trevor will make two free throws in a row. 6. To simulate the probability of Trevor making two free throws in a row, Drew puts 50 red and blue marbles in a bag. How many red and how many blue marbles should Drew use? Explain your reasoning. 7. Conduct a simulation for this situation. Compare the theoretical probability with the experimental probability. 8. There are three gumball machines numbered 1, 2, and 3 in a video arcade. Each machine contains an equal number of gumballs, some orange, some red, and some green. Ali thinks that her chance of getting a green gumball is the same from each machine. She conducted an experiment in which she bought 20 gumballs from each of the three machines. She got four green gumballs from machine 1, eight from machine 2, and twelve from machine 3. Do these data support Ali’s hypothesis? Explain why or why not. Extend 12-10 Algebra Lab: Simulations

689

CH

APTER

12




3TEM AND ,EAF 0LOTS -EASURES OF 6ARIATION OTS "OX AND 7HISKER 0 (ISTOGRAMS LAYS

!PPROPRIATE $ISP

'RAPHS -ISLEADING

ABILITY MPLE 0ROB 3I MES TING /UTCO D #OUN S ION AN

Key Concepts

UTAT TIONS 0ERMM #O B NA

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#OMPOSI

Data Displays and Measures of Variation (Lessons 12-1 through 12-6) • A stem-and-leaf plot is most often used when displaying data in a condensed form. • The interquartile range is the range of the middle half of a set of data.

box-and-whisker plot (p. 638) combination (p. 677) composite events (p. 682) dependent events (p. 683) experimental probability (p. 666) histogram (p. 644) independent events (p. 682) interquartile range (p. 634) measures of variation (p. 633) mutually exclusive events (p. 684)

outcomes (p. 665) outlier (p. 634) permutation (p. 676) probability (p. 665) quartiles (p. 634) range (p. 633) sample space (p. 666) simple event (p. 665) stem-and-leaf plot (p. 626) theoretical probability (p. 666) tree diagram (p. 670)

• A box-and-whisker plot separates data into four parts. • A histogram displays data that have been organized into equal intervals.

Simple Probability and Counting Outcomes (Lessons 12-7 and 12-8) • The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. • The Fundamental Counting Principle states that an event M followed by an event N can occur in m × n ways if event M occurs in m ways and event N occurs in n ways.


(Lesson 12-9)

Vocabulary Check Choose the term that best matches each statement or phrase. Choose from the list above. 1. measures the chances of an event happening 2. two or more events that cannot occur at the same time 3. the difference between the greatest and the least values of the data set 4. what actually occurs when conducting a probability experiment

• Permutation: order is important.

5. one outcome or a collection of outcomes

• Combination: order is not important.

6. data that are more than 1.5 times the interquartile range from the quartiles


(Lesson 12-10)

• When the outcome of one event does not affect the outcome of a second event, these are called independent events. • When the outcome of one event does affect the outcome of a second event, these are called dependent events.


7. used to describe the distribution of the data 8. the set of all possible outcomes 9. what should occur in a probability experiment 10. the values that divide a set of data into four equal parts Vocabulary Review at pre-alg.com

Lesson-by-Lesson Review Stem-and-Leaf Plots

(pp. 626–631)

Display each set of data in a stem-andleaf plot. 11.

Example 1 Display the data below in a stem-and-leaf plot.

12. Height of Girls on Soccer Team (in.)

Lobster Length (mm)

Theater Attendance

59

60

65

57

56

110 128 118

59

62

61

146

58

62

61 55

112 145 124

140 119 140

124 129 123

142

120

114

13. RACING Frank’s and Shandra’s times for their last 8 races are shown below. Racing Times (min) Frank

Stem

89

0

12345 0

1 2

2|1 = 12 min

Shandra

Measures of Variation

76

80

77

77

77

79

84

80

76

78

69

79

66

84

85

The least number is 66, and the greatest number is 85. So, the stems are 6, 7, and 8. Write the leaves to the right of their corresponding stems in order from least to greatest. Stem 6 7 8

333445 11

1|3 = 13 min

In general, which runner has a faster time? Explain.

Leaf 69 566777899 00445

7|6 = 76 mm

(pp. 633–637)

14. {42, 45, 38, 27, 41, 39, 50}

Example 2 Find the range and interquartile range for the set of data {18, 11, 20, 26, 28, 15, 21, 20, 15, 23, 19}.

15. {7, 6, 1, 3, 4, 4, 5, 8, 11, 8, 5}

First, list the data from least to greatest.

Find the range and interquartile range for each set of data.

16. BIRTHDAY Every year on your birthday you record the high temperature for that day. The past eight years the highs were {58°, 64°, 72°, 62°, 74°, 80°, 65°, 70°}. Find the range and interquartile range for the set of data.

lower half

upper half

12–2

75

12–1

11, 15, 15, 18, 19, 20, 20, 21, 23, 26, 28 ↑ LQ

↑ median

↑ UQ

The range is 28 - 11, or 17. The interquartile range is 23 - 15, or 8.


691

CH

A PT ER

12 12–3



(pp. 638–642)

OLYMPICS For Exercises 17 and 18, use the box-and-whisker plot that shows the winning time for the men’s marathon in the Summer Olympic Games between 1928 and 2004.

Example 3 Use the box-and-whisker plot shown to find the percent of New York City marathons that were held on days that had a high temperature greater than 72.5°F. NYC Marathon High Temperatures (˚F)

£Ón £ÎÓ £ÎÈ £{ä £{{ £{n £xÓ £xÈ

17. Find the percent of winning marathon times that were under 131. 18. Write a sentence describing what the length of the box-and-whisker plot tells about the winning times for the men’s marathon.

12–4

Histograms

48

52

56

60

64

68

72

76

Each of the four parts represents 25% of the data, so 25% of the marathons had a high temperature greater than 72.5°F.

(pp. 644–649)

19. BOOK SURVEY The frequency table shows the results of a reading survey. Display the data in a histogram.

Example 4 Display the set of data in a histogram. Boys’ 50-Yard Dash Time (s)

Tally

Frequency

6.0–6.4

III

3

0–1

IIII IIII IIII

15

6.5–6.9

IIII I

6

2–3

IIII IIII IIII III

18

7.0–7.4

IIII III

8

4–5

IIII IIII II

12


Boys’ 50-Yard Dash 10 8 6 4

4

9 7.

5–

7.

7. 0– 7.

6.

5–

6.

4

9

2 0 –6 .4

Frequency 4 10 14 9 4 2

8 5

6. 0

Age 42–46 47–51 52–56 57–61 62–66 67–71

IIII III IIII

Number of Boys

20. U.S. PRESIDENTS The frequency table shows the ages at which 43 U.S. Presidents began their terms in office. Display the data in a histogram.

7.5–7.9 8.0–8.4

Time (s)

8.

Frequency

0–

Tally

8.

Books Read in a Month Books

80

Source: Chance



Selecting an Appropriate Display

(pp. 651–656)

Select an appropriate type of display for each set of data. Justify your choices. 21. average price of gasoline per gallon over the last 50 years 22. number of people watching a particular television program by age group 23. the scores for the top 25% of students who took a science test

12–6

Misleading Graphs

Example 5 The monthly profits for a small business are listed for the first half of 2005. Select an appropriate type of display for this situation and justify your selection. Jan. $1200

March $1375

April $1090

May $1405

June $1425

Since we are showing a change in profits over a period of time, the best display would be a line graph.

(pp. 659–663)

24. TICKET SALES Is the graph below misleading? Explain your reasoning.

Example 6 Explain why the graphs look different. 7iiÞ Ü>Vi À>«

Ticket Sales 300 250 200 150 100

Ü>Vi f®

Number of Tickets

Feb. $1150

A

B Class

C

Ó£ £n £x £Ó È Î ä

7iiÞ Ü>Vi À>« Ü>Vi f®

12–5

£ Ó Î { x È 9i>À

££ £ä n Ç È x ä

£ Ó Î { x È 9i>À

The vertical scales are different.

12–7

Simple Probability

(pp. 665–669)

MARBLES There are 2 blue marbles, 5 red marbles, and 8 green marbles in one bag. One marble is selected at random. Find the probability of each outcome. 25. P(red)

26. P(green)

27. P(blue or green) 28. P(not blue) 29. P(yellow)

30. P(green, red, or blue)

Example 7 Suppose a number cube is rolled. Find the probability of rolling a 5 or 6. number of favorable outcomes P(5 or 6) = ___ number of possible outcomes 1 2 = _ or _ 6 3 1 or The probability of rolling a 5 or 6 is _ 3 1 %. 33 _ 3


693

CH

A PT ER

12 12–8


Counting Outcomes

(pp. 670–674)

Find the number of possible outcomes for each situation. 31. Four coins are tossed. 32. A tennis shoe comes in men’s and women’s sizes; cross training, walking, and running styles; and blue, black, or white colors.

12–9


Example 8 A number cube is rolled three times. Find the number of possible outcomes. Outcomes on Outcomes on Outcomes on the first roll the second roll the third roll

×

6

6

33. How many ways can 3-person teams be chosen from 14 students?

=

216

There are 216 possible outcomes.

Example 9 How many ways can 8 horses place first, second, and third in a race? The order is important, so this is a permutation. 8 horses

34. How many 5-digit security codes are possible if each digit is a number from 0 to 9? 35. How many ways can you choose 2 team colors from a total of 7 colors?


6

(pp. 676–680)

Tell whether each situation is a permutation or a combination. Then solve.

12–10

×

Possible outcomes

Choose 3.

P(8, 3) = 8 · 7 · 6 = 336 There are 336 ways for 8 horses to place first, second, and third.

(pp. 682–687)

A card is drawn from a deck of eight cards numbered from 1 to 8. 36. The card is not replaced, and a second card is drawn. Find P(3 and 6). 37. The card is replaced, and a second card is drawn. Find P(4 and 2). 38. GOLFING A box contains 5 yellow balls, 10 green balls, 12 blue balls, and 7 red balls. What is the probability that the attendant will pick a red or blue ball?

Example 10 There are 3 red, 4 purple, and 2 green marbles in a bag. Find the probability of randomly drawing a purple marble and then a green marble without replacement. P(purple, then green) = P(purple on 1st draw) · P(green on 2nd draw) = 4 · 2 , or 1 9

8

9

The probability of drawing a purple marble and then a green marble is 1. 9


A PT ER

12

Practice Test

For Exercises 1–3, use the table shown.

Select an appropriate type of display for each data set. Justify your choice.

Students’ Heights (in.) 70

81

59

69

78

68

75

79

76

62

67

74

75

64

60

58

1. Display the data in a stem-and-leaf plot. 2. What is the median height of the students? 3. In which interval do more heights occur? For Exercises 4–6, use the stem-and-leaf plot shown.

Stem 7 8

Leaf 12237 004449

4. Find the range for the data. 9 5. Display the data in a box-and-whisker plot. 6. Find the interquartile range.

35666889 8|4 = $84

7. MULTIPLE CHOICE Refer to the box-andwhisker plot shown. Daily High Temperatures (˚F)

50

55

60

65

70

75

80

85

90

B 50%

C 75%

95

D 100%

Length of Bus Ride to School 0–9

Tally

Frequency

||||

4

10–19

|||| ||

7

20–29

|||| |

6

G 4

H 7


0–10 10–49 50–99 100–149 150–199 200–249 250–up

6 23 17 15 9 9 21 10

13

16 Percent

19

22

25

Source: Simmons

14. P(green)

Tell whether each situation is a permutation or combination. Then solve. 15. How many ways can 7 potted plants be arranged on a window sill? 16. A sand bucket contains 12 seashells. How many ways can you choose 3 of them? A card is drawn from the cards shown. Find the probability of each outcome.

9. MULTIPLE CHOICE Find the number of possible outcomes for a choice of fish, chicken, pork, or beef and a choice of green beans, asparagus, or mixed vegetables.

F 3

Average Weekly Travel Distance

13. P(red)

8. Display the data shown in a histogram. Time (min)

12. TRAVEL The distance adults drive each week is shown in the graph below. Is the graph misleading? Explain your reasoning.

There are 3 purple grapes, 5 red grapes, and 8 green grapes in a bowl. Suppose one grape is selected at random. Find each probability as a fraction.

What percent of the daily high temperatures range from 70° to 95°? A 25%

10. the relationship among people who have at least one dog, at least one cat, and those who have both 11. amount of money you spend on food compared to your total income

Distance (mi)

CH

J 12

1 5 4 3

17. P(5 or even) 18. P(greater than 9) 19. P(odd or even)

2 9 8

20. P(2 or greater than 5)


695

CH

A PT ER


12


Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The results of a random survey showed that 65 out of 90 people support the re-election of the mayor. How many people would you expect to vote for the mayor’s re-election if a total of 4,500 people turn out to vote? C 3,250 A 1,250 B 1,875 D 3,400

2. The bar graph shows the results of a survey on the favorite school sport among middle school students.

4. Between which two whole numbers is √40 located on a number line? C 8 and 9 A 6 and 7 B 7 and 8 D 9 and 10 5. Regina drove her car 252.8 miles on 8 gallons of gasoline. What was her gas mileage expressed as a unit rate? F 25.0 miles per gallon G 29.2 miles per gallon H 31.6 miles per gallon J 33.5 miles per gallon 6. GRIDDABLE What is the surface area, in square meters, of the rectangular prism?

È


>ÛÀÌi -V -«ÀÌ £ää

Ó

Çä

Î

Èä Îä Óä >ÃiL>

-VViÀ

ÌL>

>ÃiÌL>

Which statement best describes why a person reading the graph might get an incorrect idea about the favorite school sport of middle school students? F The title of the graph is misleading. G The graph does not include volleyball. H The intervals on the vertical scale are not consistent. J The vertical scale should show the number of votes for each sport.

3. GRIDDABLE By what factor would the volume of a rectangular prism increase if you multiply its dimensions by 2?


7. Juliet recorded the distance and the time she walked every day. What is the best estimate of how many minutes it will take her to walk 4 miles?

A 48 min B 52 min

Day

Distance (mi)

Time (min)

1

2

28

2

3

42

3

4

?

C 56 min D 64 min

Questions 1, 2, 4, 5, and 7 Eliminate the answer choices you know to be wrong. Then take your best guess from the choices that remain. If you can eliminate at least one answer choice, it is better to answer a question than to leave it blank.



8. The mass of an orchid seed is 0.0000035 gram. How is this length expressed in scientific notation? F 35.0 × 10-5 g G 3.5 × 10-6 g H 0.35 × 10-7 g J 3.5 × 106 g

12. The drawing shows the top-count view of a solid figure. Which drawing shows a threedimensional view of this solid figure? { Ó £

F

9. In the spinner below, the pointer stopped in the shaded region. What color should the shaded region be so that the probability of 3 ? the pointer landing on this color is _

H

ÀÌ

G

J

,

,

/ 7

,

A B C D

Ó Ó ä

ÀÌ

8

,

Î Ó ä

1 ,

ÀÌ

ÀÌ

13. A bag contains 4 red marbles, 3 blue marbles, and 2 white marbles. One marble is chosen without replacement. Then another marble is chosen. What is the probability that the first marble is red and the second marble is blue?

red white green black

7 A _

10. Sixteen pounds of ground beef will be divided into patties measuring one-quarter pound each. How many patties can be made? H 32 F 4 G 16 J 64

4 B _

1 C _

27

4 D _

27

6

9

Pre-AP Record your answers on a sheet of paper. Show your work. 14. The scores on a mathematics test are given in the table.

11. An office has 45 light fixtures. Each fixture uses two light bulbs and each bulb costs $0.89. Which expression could be used to find the total cost of replacing all of the bulbs in the office? A 45(0.98) B 2(45 + 0.89)

81

93

71

75

85

85

86

77

88

63

91

70

a. Display the data in a histogram. b. Which interval contains the greatest number of test scores? c. What percent of the class scored between 70 and 89?

45 C _ (0.89) 2

D 2(45)(0.89)


1

Go to Lesson...

12-7

2

3

4

12-6 11-2 9-1

5

6

7

8

9

10

11

12

13

6-1

11-4

7-1

4-7

12-7

5-4

1-2 11-1 12-10

14 12-4


697

13 •

Add, subtract, and multiply polynomials.

•

Use the properties and attributes of nonlinear functions.

Polynomials and Nonlinear Functions

Key Vocabulary cubic function (p. 722) degree (p. 702) nonlinear function (p. 720) polynomial (p. 701) quadratic function (p. 722)

Real-World Link Skydiving You have studied situations that can be modeled by linear functions. Many real-life situations, however, are not linear. The rate at which an object falls as a function of time can be modeled using a nonlinear function.

Polynomials Make this Foldable to help you organize your notes. Begin with a sheet of 11" by 17" paper.

1 Fold the short sides toward the middle.

3 Open. Cut along the second fold to make four tabs.


4 Fold each of the tabs as shown.

X

698 Chapter 13 Polynomials and Nonlinear Functions David Madison Sports Images



Option 1


Determine the number of monomials in each expression. (Lesson 4-2) 1. 2x3 2. a + 4 3. 8s - 5t

4. x2 + 3x - 1

1 5. _

6. 9x3 + 6x2 + 8x - 7

t

7. PHYSIOLOGY During a respiratory cycle, the volume of air in liters in the human lungs can be described by the function V(t) = 0.173t + 0.152t2 - 0.035t3 where t is the time in seconds. Find the number of monomials in the function. (Lesson 4-2)

Example 1

Determine the number of monomials in 2(x - 1) + 5. 2(x - 1) + 5 = 2x - 2 + 5 Distributive Property = 2x + 3 2x + 3 has two monomials.

Use the Distributive Property to write each expression as an equivalent algebraic expression. (Lesson 3-1) 8. 5(a + 4) 9. 2(3y - 8)

Example 2

10. -4(1 + 8n)

11. 6(x + 2y)

-10(n - 13)

12. (9b - 9c)3

13. 5(q - 2r + 3s)

14. ICE SKATING Admission to the skating rink is $8 and it costs $6 to rent skates. Write two equivalent expressions for the total cost for 14 people to go to the skating rink and rent ice skates. Then find the cost.

Simplify.

Use the Distributive Property to write -10(n - 13) as an equivalent algebraic expression. = -10[n + (-13)] Rewrite n - 13 as n + (-13). = -10 · n + (-10) · (-13) Distributive Property = -10n + 130

Simplify.

(Lesson 3-1)

Determine whether the graph of each equation is linear. (Lesson 7-6) 15. y = x - 2 16. y = x2 1 17. y = -_ x

18. y = 5

19. x = -9

1 20. y = _ x

2

21. PHYSICS When a sound travels through water, the distance y in meters that the sound travels in x seconds is given by the equation y = 1440x. Is the graph of this equation linear? (Lesson 7-6)

Example 3

Determine whether the graph of y = x3 is linear.

y

O

x

Plot points and graph the equation. The graph is not linear.

Chapter 13 Get Ready For Chapter 13

699

Prefixes and Polynomials Unlike a train, which runs on two tracks, a monorail runs on a single track. The prefix mono means one. A monomial is an algebraic expression with one term. Monomials

Not Monomials

5 2x y3

x+y 8n2 - n + 1 a3 + 4a2 + a - 6

You can determine the meaning of many words used in mathematics if you know what the prefixes mean. The words in the table below are used in mathematics and in everyday life. They contain the prefixes bi, tri, and poly. Prefix bi

Words and Models • bisect – to divide into two congruent parts • biannual – occurring twice a year • bicycle – a vehicle with two wheels X

tri

Z

Y

• triangle – a figure with three sides • triathlon – an athletic contest with three phases • trilogy – a series of three related literary works, such as films or books

A

C

poly

B

• polyhedron – a solid with many flat surfaces • polychrome – having many colors • polygon – a figure with many sides

Polyhedron

Reading to Learn 1. How are the words in each group of the table related? 2. What do the prefixes bi, tri, and poly mean? 3. Write the definition of binomial, trinomial, and polynomial. 4. Give an example of a binomial, a trinomial, and a polynomial. 5. RESEARCH Use the Internet or a dictionary to make a list of other words that have the prefixes bi, tri, and poly. Give the definition of each word. 700 Chapter 13 Polynomials and Nonliner Function Wilfried Krecichwost/The Image Bank/Getty Images

13-1

Polynomials

Main Ideas Heat index is a way to describe how hot it feels outside based on temperature and humidity.

• Identify and classify polynomials. • Find the degree of a polynomial.

Heat Index Humidity (%) 40 45 50

New Vocabulary polynomial binomial trinomial degree

80 79 80 81

Temperature (˚F) 90 93 95 96

100 110 115 120

Heat Index

A heat index is found using an expression similar to the one below. In this expression, x is the percent humidity, and y is the temperature. -42 + 2x + 10y - 0.2xy - 0.007x2 - 0.05y2 + 0.001x2y + 0.009xy2 0.000002x2y2 a. How many terms are in the expression for the heat index? b. What separates the terms of the expression?

Classify Polynomials Recall that a monomial is a number, a variable, or a product of numbers and/or variables. An algebraic expression that contains one or more monomials is called a polynomial. In a polynomial, there are no terms with variables in the denominator and no terms with variables under a radical sign. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial.

EXAMPLE

Polynomial Terms Examples

Monomial

Binomial

Trinomial

1

2

3

4 x 2y3

x+1 a - 5b c2 + d

a+b+c x2 + 2x + 1 3m3 - mn + 1

Classify Polynomials

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. a. 2x3 + 5x + 7

READING in the Content Area

This is a polynomial because it is the sum of three monomials. There are three terms, so it is a trinomial.

For strategies in reading this lesson, visit pre-alg.com.

1A. d 2 Extra Examples at pre-alg.com

1 b. t - 2 t

The expression is not a 1 has a polynomial because t2 variable in the denominator.

1B. x2 + y-2 - 3

Lesson 13-1 Polynomials

701

BrainPOP® pre-alg.com

Degrees of Polynomials The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant such as 6 or 10 is 0. The constant 0 has no degree. The degree of a polynomial is the same as that of the term with the greatest degree.

EXAMPLE

Degree of a Monomial or Polynomial

Find the degree of each polynomial. Degrees The degree of a is 1 because a = a1.

b. -4x2y

a. 5a

x2 has degree 2 and y has degree 1. The degree of -4x2y is 2 + 1 or 3.

The variable a has degree 1, so the degree of 5a is 1. c. x2 + 3x - 2 term x2 3x 2

d. a2 + ab2 + b4

degree 2 1 0

term a2 ab2 b4

The greatest degree is 2. So the degree of x2 + 3x - 2 is 2.

degree 2 1 + 2 or 3 4

The greatest degree is 4. So the degree of a2 + ab2 + b4 is 4.

2A. 3y2

2B. 5x3y2z

2C. q3 + 4 - r + s2

2D. 5x2y4 - 4

ECOLOGY In the early 1900s, the deer population of the Kaibab Plateau in Arizona was affected by hunters and by the food supply. The population from 1905 to 1930 can be approximated by the polynomial -0.13x5 + 3.13x4 + 4000, where x is the number of years since 1900. Evaluate the expression to find the size of the deer population in 1917. Real-World Career Ecologist An ecologist studies the relationships between organisms and their environment.


1917 - 1900 = 17 = x -0.13(17)5

+

3.13(17)4

x is the number of years since 1900.

+ 4000 Substitution.

= 80,839.32

The deer population in 1917 was approximately 80,839.

3. GEOMETRY The volume of a certain rectangular prism is given by the equation y = x3+ 4x2 - 3x - 24. If x = 5, find the volume of the prism. Personal Tutor at pre-alg.com

702 Chapter 13 Polynomials and Nonlinear Functions Raymond Gehman/CORBIS

Simplify.

Example 1 (p. 701)

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. 2. 1x - x

1. -7 Example 2 (p. 702)

Example 3 (p. 702)

HOMEWORK

HELP


3. x2 + xy2 - y2

Find the degree of each polynomial. 4. 4b2

5. -ac3

6. d2f5

7. 3x + 5

8. d2 + c4

9. 8x3y2

10. PHYSIOLOGY During a five-second respiratory cycle, the volume of air in liters in the human lungs can be described by the function y = 0.179t + 0.1522t2 - 0.0374t3, where t is the time in seconds. Evaluate the polynomial to find the volume of air in the lungs 3 seconds into a cycle.

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. 11. 16

1 2 12. -_ w

13. x2 - √7x

14. 11a2 + 4

15. √ 15c

2 16. 8 - _

17. 12 - n + n4

1 1 18. x2 - _ x+_

3

k

2

3

Find the degree of each polynomial. 19. 3

20. 56

21. 12c3

22. xyz2

23. 2 - 8n

24. g5 + 5h

25. x2 + 3x + 2

26. d2 + c4d2 x

GEOMETRY For Exercises 27 and 28, refer to the square at the right with a side length of x units. 27. Write a polynomial expression for the area of the small yellow rectangle. 28. Evaluate the expression to find the area of the yellow rectangle if x = 5 and y = 3.

x

y

GARDENING For Exercises 29 and 30, use the information below and the diagram at the right. Joe wants to place apple trees along the perimeter of his garden. 29. Write a polynomial that represents the perimeter of the garden in feet. 30. Evaluate the polynomial to find the perimeter of the garden if x = 2, y = 3, and z = 5.

EXTRA

PRACTICE


X Z

Z

YX

X

RESEARCH For Exercises 31–33, use the following information. Sarah’s grandparents have put $70 into a savings account every year on her birthday since she was born. The account earns interest at a rate of r. Research interest rates at a local bank. Assume that the interest rate doesn’t change. 31. Write an expression for the account balance on Sarah’s fifth birthday. 32. What is the degree of the polynomial? 33. Using some interest rate, how much money would be in Sarah’s account? Lesson 13-1 Polynomials

703

34. FIND THE ERROR Carlos and Tanisha are finding the degree of 5x + y2. Who is correct? Explain your reasoning.

H.O.T. Problems

Carlos 5x has degree 1. Y2 has degree 2. 2 5x + y has degree 1 + 2 or 3.

Tanisha 5x has degree 1. y2 has degree 2. 5x + y2 has degree 2.

Determine whether each statement is always, sometimes, or never true. Explain. 35. A trinomial has a degree of 3. 36. An integer is a monomial. 37. CHALLENGE Find the degree of ax + 3 + xx - 2b3 + bx + 2. 38.

Writing in Math Explain how to find the degree of a polynomial. Illustrate your explanation by creating a monomial that has a degree of 3 and a polynomial that has a degree of 3.

39. Choose the expression that is NOT a binomial. A x2 - 1

C m3 + n3

B a+b

D 7x + 2x

41. Write a polynomial to represent the value of q quarters, d dimes, and n nickels in dollars. A 25q + 10d + 5n B 0.25q + 0.1d + 0.05n

40. What is the degree of F 1

G 2

4x3

H 3

+ xy -

y2?

J 4

C 0.25q × 0.1d × 0.05n D (q + 25) + (d + 10) + (n + 5)

A number cube is rolled. Determine whether each event is mutually exclusive or inclusive. Then find the probability. (Lesson 12-10) 42. P(odd or greater than 3) 43. P(5 or even) 44. HOMEWORK Solana is designing a flag using stripes of the colors red, green, blue, yellow, and purple. If she uses each color only once, from how many color patterns does she have to choose? (Lesson 12-9) ALGEBRA Evaluate each expression if a = -2, b = -6, and c = 14. (Lesson 2-3) 45. a - c

46. b - a

47. a - b

48. c - a + b

49. b - a + c

50. a - b - c

PREREQUISITE SKILL Rewrite each expression using parentheses so that the terms having variables of the same power are grouped together. (Lesson 1-4) 51. (x + 4) + 2x

52. 3x2 - 1 + x2

53. (6n + 2) + (3n + 5)

54. (a + 2b) + (3a + b)

704 Chapter 13 Polynomials and Nonlinear Functions

EXTEND

Algebra Lab

13-1

Modeling Polynomials with Algebra Tiles represents the integer 1, X

In a set of algebra tiles,

£

variable x, and

represents x2. Red tiles are used to represent -1,

X

Ó

represents the

-x, and -x2. X

£

X

Ó

You can use these tiles to model monomials.

X

X X

X

X

X

X

You can also use algebra tiles to model polynomials. The polynomial 2x2 - 3x + 4 is modeled below. £

X

Ó

X

Ó

X

X

X

£

£ £

Ó

ÓX ÎX {

ANALYZE THE RESULTS Use algebra tiles to model each polynomial. 1. -3x2 3.

4x2

2. 5x + 3 4. 2x2 + 2x - 3

-x

5. Explain how you can tell whether an expression is a monomial, binomial, or trinomial by looking at the algebra tiles. 6. Name the polynomial modeled below. £ £ X

Ó

X

X

X

£ £ £

7. Explain how you would find the degree of a polynomial using algebra tiles. Extend 13-1 Algebra Lab: Modeling Polynomials with Algebra Tiles

705

13-2

Adding Polynomials

Main Idea • Add polynomials.

The polynomials 2x2 - 3x + 4 and -x2 + x - 2 are modeled below. £

X

Ó

X

Ó

£

X X X

X £

£

Ó

X £

£

Ó

Ó

ÓX ÎX {

X X Ó

Follow these steps to add the polynomials. Step 1 Combine the tiles that have the same shape. Step 2 When a positive tile is paired with a negative tile that is the same shape, the result is called a zero pair. Remove any zero pairs.

Algebra Tiles

X

Ó

X

Tiles that are the same shape and size represent like terms.

Ó

Ó

X

Ó

Ó

ÓX X ®

X X X

£

£

£

£

£

£

X

Î X X ®

{ Ó®

a. Write the polynomial for the tiles that remain. b. Find the sum of x2 + 4x + 2 and 7x2 - 2x + 3 by using algebra tiles. c. Compare and contrast finding the sum of polynomials with finding the sum of integers.

Add Polynomials Monomials that contain the same variables to the same power are like terms. Terms that differ only by their coefficient are called like terms. Like Terms 2x and 7x -x2y and 5x2y

Unlike Terms -6a and 7b 4ab2 and 4a2b

You can add polynomials by combining like terms.

EXAMPLE

Add Polynomials

Find each sum. a. (3x + 5) + (2x + 1) Method 1 Add vertically. 3x + 5 (+) 2x + 1 Align like terms. _ 5x + 6 Add.


Method 2 Add horizontally. (3x + 5) + (2x + 1)

Associative and Commutative = (3x + 2x) + (5 + 1) Properties

= 5x + 6 Extra Examples at pre-alg.com

b. (3x2 + x - 7) + (-x2 + 3x + 5) Negative Signs • When a monomial has a negative sign, the coefficient is a negative number. -4x → coefficient is -4. -b → coefficient is -1. • When a term in a polynomial is subtracted, “add its opposite” by making the coefficient negative. x - 2y → x + (-2y)

Method 1 3x2 + x - 7 2 + 3x + 5 Align like terms. (+)-x _______________ 2x2 + 4x - 2 Add.

Method 2 (3x2 + x - 7) + (-x2 + 3x + 5)

Write the expression.

= (3x2 - x2) + (x + 3x) + (-7 + 5)

Group like terms.

= 2x2 + 4x - 2

Simplify.

c. (x2 + xy + 2y2) + (6x2 - y2) x2 + xy + 2y2 (+) 6x2 - y2 _________________ 7x2 + xy + y2

Leave a space because there is no other term like xy.

Find each sum. 1A. (x + 3) + (2x + 5)

1B. (13x - 7y) + 3y

1C. (z2 + 4zw - w2) + (3z2 - w2)

GEOMETRY The lengths of the sides of golden rectangles are in the ratio 1:1.62. So, the length of a golden rectangle is approximately 1.62 times greater than its width.

x 1.62x

a. Find a formula for the perimeter of a golden rectangle. P = 2 + 2w

Formula for the perimeter of a rectangle

P = 2(1.62x) + 2x

Replace with 1.62x and w with x.

P = 3.24x + 2x or 5.24x Simplify. The formula is P = 5.24x, where x is the measure of the width. b. Find the perimeter of a golden rectangle if its width is 8.3 centimeters. perimeter = 5.24x

Perimeter of a golden rectangle

= 5.24(8.3) or 43.492 Replace x with 8.3 and simplify. Real-World Link The ancient Greeks often incorporated the golden ratio into their art and architecture. Source: mcn.net

The perimeter of the golden rectangle is 43.492 centimeters.

GAMES Stephan paid a flat fee of $15 to join a fantasy football league. It costs $0.15 every time he makes a trade. He also subscribed to the expert tip advice for a $5 fee plus $0.45 every time he asks for advice. 2A. Find the formula that gives the total amount that Stephan will pay during the fantasy football season. 2B. How much will Stephan pay if he makes 17 trades over the course of the season and asks for advice 8 times? Personal Tutor at pre-alg.com Lesson 13-2 Adding Polynomials

Antonio M. Rosario/Getty Images

707

Example 1 (pp. 706–707)

Find each sum. 1.

2.

4x + 5 (+) -x -3 ___________

3. (2x2 + 5x) + (9 - 7x) Example 2 (p. 707)

3a2 - 9a + 6 2 (+)4a -2 ______________

4. (3x2 - 2x + 1) + (x2 + 5x - 3) ÓX Ç

5. GEOMETRY Find the perimeter of the rectangle at the right. X Xx

HOMEWORK

HELP


Ó

X X X X X

X

Ó

X X X X X

X X X X X X X £ £ £ £

£ £ £ £

£ £ £ £

£ £ £ £

£ £ £ £

£

£

£

£

£

£ £ £ £

£ £ £ £ £ £

Find each sum. 6.

-5x + 4 (+) 8x - 1 __________

7.

8.

10x2 + 5xy + 7y2 (+) x2 - 3y2 _________________

9. 4a3 + a2 + 8a - 8 (+) 2a2 +6 ________________

7b - 5 (+)-9b +8 ___________

10. (3x + 9) + (x + 5)

11. (4x + 3) + (x - 1)

12. (6y - 5r) + (2y + 7r)

13. (8m - 2n) + (3m + n)

14.

(5x2

+ 6x + 4) +

(2x2

+ 3x + 1)

16. (x2 + y) + (4x2 + xy)

15. (-2x2 + x -5) + (x2 - 3x + 2) 17. (3a2 + b2) + (3a + b2)

ELECTRICITY For Exercises 18–20, refer to the information below. Terry and Rondell are charged the same rate per kilowatt hour for electricity. This month, Terry’s bill showed that she had used 770 kilowatt hours and had been charged an additional $6.50 for taxes and fees. Rondell’s bill showed that he had used 825 kilowatt hours, was also charged $6.50 for taxes and fees, but had received a $24 credit. Let x represent the rate per kilowatt hour that the company charges for electricity. 18. Write a polynomial expression to represent Terry’s bill for the month. 19. Write a polynomial expression to represent Rondell’s bill for the month. 20. Write an expression to represent the total combined bill of Terry and Rondell. PAYCHECKS For Exercises 21–23, refer to the information below. Jared and Will are paid the same hourly rate. At the end of the week, Jared’s paycheck showed that he worked 23 hours and had $12 deducted for taxes. Will worked 19 hours during the same week and had $10 deducted for taxes. Let x represent the hourly pay. 21. Write a polynomial expression to represent Jared’s pay for the week. 22. Write a polynomial expression to represent Will’s pay for the week. 23. Write a polynomial expression to represent the total weekly pay for Jared and Will. 708 Chapter 13 Polynomials and Nonlinear Functions

EXTRA

PRACTIICE

Find each sum. Then evaluate if a = -3, b = 4, and c = 2.

See pages 791, 806.

24. (3a + 5b) + (2a - 9b)

25. (-2a + 6) + (7b + 3c)

26. (3a + 5b - 4c) + (2a - 3b + 7c) + (-a + 4b - 2c)


27. (a2 + 7b2) + (5 - 3b2) + (2a2 - 7)

H.O.T. Problems

28. OPEN ENDED Write two binomials that have only one pair of like terms. 29. FIND THE ERROR Hai says that 7xyz and 2zyx are like terms. Devin says they are not. Who is correct? Explain. 30. CHALLENGE In the figure at the right, x2 is the area of the larger square, and y2 is the area of each of the two smaller squares. What is the perimeter of the whole rectangle? Explain. 31.

Ó

Y

Ó

Y

Ó

Writing in Math

Explain how algebra tiles can be used to add polynomials. Include a description of how algebra tiles represent like terms and zero pairs.

32. Which expression represents the perimeter of the triangle?

ÓX ÎY

X

X ÓY

{X ÓY

A 5x + 6y

33. Hannah makes x dollars per hour working at the grocery store. She makes y dollars per hour working at the library. One week she worked 9 hours at the grocery store and 12 hours at the library. Which expression represents her total earnings for that week? F 9 + x + 12 + y

B 3x + 7y

G (9 + 12)(x + y)

C 6x + 7y

H 9(12)xy

D 7x + 3y

J 9x + 12y

Find the degree of each polynomial. (Lesson 13-1) 34. a3b

35. 3x - 5y + z2

36. c2 - 7c3y4

A card is drawn from a standard deck of 52 playing cards. Find each probability. (Lesson 12-10) 37. P(2 or jack)

38. P(10 or red)

39. P(ace or black 7)

PREREQUISITE SKILL Rewrite each expression as an addition expression by using the additive inverse. (Lesson 2-3) 40. 15c - 26

41. x2 - 7

42. 1 - 2x

43. 6b - 3a2

44. (n + rt) - r 2

45. (s + t) - 2s Lesson 13-2 Adding Polynomials

709

13-3

Subtracting Polynomials

Main Idea • Subtract polynomials.

At the North Pole, buoy stations drift with the ice in the Arctic Ocean. The table shows the latitudes of two North Pole buoys. Station Latitude 1 89° 35.4'N = 89 degrees 35.4 minutes 5

85° 27.3'N = 85 degrees 27.3 minutes

a. What is the difference in degrees and the difference in minutes between the two stations? b. Explain how you can find the difference in latitude between any two locations, given the degrees and minutes.

Subtract Polynomials When you subtract measurements, you subtract like units. Consider the subtraction of the latitude measurements above. 89 degrees 35.4 minutes (-) 85 degrees 27.3 minutes 4 degrees 8.1 minutes

89 degrees - 85 degrees

35.4 minutes - 27.3 minutes

Similarly, when you subtract polynomials, you subtract like terms. 5x2 + 14x - 9 (-) x2 + 8x + 2 4x2 + 6x - 11

5x2 - 1x2 = 4x2

- 9 - 2 = -11

14x - 8x = 6x


Subtract Polynomials

Find each difference. a. (5x + 9) - (3x + 6) 5x + 9 (-) 3x + 6 Align like terms. 2x + 3 Subtract.

1A. (4y + 2) - (2y + 2) 710 Chapter 13 Polynomials and Nonlinear Functions

b. (4a2 + 7a + 4) - (3a2 + 2) 4a2 + 7a + 4 (-) 3a2 + 2 Align like terms. a2 + 7a + 2 Subtract.

1B. (3b2 + 14b + 1) - (b2 + 7b) Extra Examples at pre-alg.com

You can also subtract a polynomial by adding its additive inverse. To find the additive inverse of a polynomial, multiply the entire polynomial by -1. Polynomial

EXAMPLE

Multiply by -1

Additive Inverse

t

-1(t )

-t

x+3

-1(x + 3)

-x - 3

-a2 + b2 - c

-1(-a2 + b2 - c)

a2 - b2 + c

Subtract Using the Additive Inverse

Find each difference. a. (3x + 8) - (5x + 1) The additive inverse of 5x + 1 is (-1)(5x + 1) or -5x - 1. (3x + 8) - (5x + 1) = (3x + 8) + (-5x - 1) To subtract (5x + 1), add (-5x - 1). = (3x - 5x) + (8 - 1)

Group the like terms.

= -2x + 7

Simplify.

b. (4x2 + y2) - (-3xy + y2) The additive inverse of -3xy + y2 is (-1)(-3xy + y2) or 3xy - y2. 4x2 (-)

+ y2 -3xy + y2

(+)

+ y2 3xy - y2 Align the like terms and add the additive inverse.

4x2 + 3xy + 0

Zeros It can be helpful to add zeros as placeholders when a term in one polynomial does not have a corresponding like term in another polynomial. 4x2 + 0x y + y2 (+) 0x 2 + 3x y - y2 ________________

4x2

2A. (3y - 9) - (2y + 7)

2B. (5z2 + w2) - (-4zw + 3w2)

SHIPPING The cost for shipping a package that weighs x pounds from Charlotte to Chicago is shown in the table at the right. How much more does the Atlas Service charge for shipping the package?

Shipping Company

Cost ($)

Atlas Service

4x + 2.80

Bell Service

3x + 1.25

difference in cost = cost of Atlas Service - cost of Bell Service = (4x + 2.80) - (3x + 1.25)

Substitution

= (4x + 2.80) + (-3x - 1.25)

Add additive inverse.

= (4x - 3x) + (2.80 - 1.25)

Group like terms.

= x + 1.55

Simplify.

The Atlas Service charges x + 1.55 dollars more.

3. PHONES Superior Cell charges $30 for 500 minutes and $0.25 for every additional minute. National Cell charges $25 for 500 minutes and $0.35 for every additional minute. What is the difference between the costs? Personal Tutor at pre-alg.com Lesson 13-3 Subtracting Polynomials

711

Example 1 (p. 710)

Example 2 (p. 711)

Example 3 (p. 711)

Find each difference. 1. r2 + 5r (-)r2 + r

2.

3x2 + 5x + 4 (-) x2 -1

3. (9x + 5) - (4x + 3)

4. (2x + 4) - (-x + 5)

5. (3x2 + x) - (8 - 2x)

6. (6a2 - 3a + 9) - (7a2 + 5a - 1)

7. ARCHITECTURE The perimeter of the auditorium shown is 9x + 2 units. Find the length of the missing side.

¶ ÎÝ ³ £

ÎÝ ³ £

Ý

HOMEWORK

HELP


Find each difference. 8.

9.

8k + 9 (-) k + 2

-n2 + 1n (-) n2 - 5n

10.

5a2 + 9a - 12 (-) -3a2 + 5a - 7

11.

6y2 - 5y + 3 (-) 5y2 + 2y - 7

12.

5x2 - 4xy (-) - 3xy + 2y2

13.

9w2 +7 2 (-)-6w + 2w - 3

14. (3x + 4) - (x + 2)

15. (7x + 5) - (3x + 2)

16. (2y + 5) - (y + 8)

17. (3t - 2) - (5t - 4)

18. (2x + 3y) - (x - y)

19. (a2 + 6b2) - (-2a2 + 4b2)

20. (x2 + 6x) - (3x2 + 7)

21. (9n2 - 8) - (n + 4)

22. (6x2 + 3x + 9) - (2x2 + 8x + 1)

23. (3x2 - 5xy + 7y2) - (x2 - 3xy + 4y2)

24. 3x2z2 + x2 + z2 - (0.5x2 z2 + 3x2 + z2)

(

)

25. 4a + b3 - 1a - 3b 3 3

26. GEOMETRY Alyssa plans to trim a picture to fit into a frame. The area of the picture is 2x2 + 11x + 12 square units, but the area inside the frame is only 2x2 + 5x + 2 square units. How many square units of the picture will Alyssa have to trim so that it will fit into the frame?

EXTRA

PRACTICE

27. TEMPERATURE The lowest recorded temperature in the state of Texas occurred in 1933. Three years later, the state recorded its highest temperature. The difference between these two record temperatures is 46°F more than the sum of the temperatures. The sum of the two temperatures is 97°. What is the record high temperature and record low temperature in Texas?


28. LONGITUDE The longitude of Weather Station 1 is 162°16’36” and the longitude of Weather Station 5 is 68°8’2”. Find the difference in longitude between the two stations.


H.O.T. Problems

29. OPEN ENDED Write two polynomials whose difference is x2 + 2x - 4. 30. CHALLENGE Suppose A and B represent polynomials. If A + B = 3x2 + 2x - 2 and A - B = -x2 + 4x - 8, find A and B. 31. SELECT A TOOL Backyard Pools sells and installs pools and decks. Their most popular seller is a rectangular pool surrounded by a rectangular deck. The area of the pool is given by the equation y = x2 + 2x - 15. If the area of the entire pool-deck combo is represented by the function y = x2 + 7x, what techniques could be used to find the area of the deck? Justify your selections and then use your techniques to solve the problem. draw a model

32.

paper/pencil

calculator

Writing in Math Use the information about global positioning on page 710 to explain how subtracting polynomials is similar to subtracting measurements.

33. What is (5x - 7) - (3x - 4)? A 2x - 3

C 2x - 11

B 2x + 3

D 2x + 11

34. What is the additive inverse of -4h2 - hk - k 2?

35. Mario’s Pizza Palace charges $10 for a large pizza and $1.25 for each topping. Luigi’s Grill charges $12 for a large pizza plus $1.50 for each topping. Find the polynomial that represents how much more a pizza with t toppings would cost at Luigi’s than at Mario’s.

F 4h2 - hk - k 2

H -4h2 + hk + k 2

A 2t + 0.25

C 2 - 0.25t

G 4h2 + hk + k 2

J

-4h2 + hk - k 2

B 2 + 0.25t

D 2t - 0.25

Find each sum. (Lesson 13-2) 36. (2x - 3) + (x - 1)

37. (11x + 2y) + (x - 5y)

38. (5x2 - 7x + 9) + (3x2 + 4x - 6)

39. (4t - t2) + (8t + 2)

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. (Lesson 13-1) 1 40. 2 5a

41. x2 + 9

42. c2 - d3 + cd

43. Make a stem-and-leaf plot for the set of data shown below. (Lesson 12-1) 72, 64, 68, 66, 70, 89, 91, 54, 59, 71, 71, 85 ALGEBRA Solve each equation. Check your solution. (Lesson 3-3) 44. t - 18 = 24

45. 30 = 3 + y

46. -7 = x + 11

PREREQUISITE SKILLS Simplify each expression. (Lesson 4-1) 47. x(3x)

48. (2y)(4y)

49. (t2)(6t)

50. (w2)(-3w)

Lesson 13-3 Subtracting Polynomials

713

CH

APTER

13

Mid-Chapter Quiz Lessons 13–1 through 13–3

Find the degree of each polynomial. (Lesson 13-1) 1. cd3 2. a - 4a2 3. x2y + 7x2 - 21

13. MULTIPLE CHOICE The perimeter of the triangle is 8x + 3y centimeters. X Y CM

LANDSCAPING For Exercises 4–6, use the information and the diagram below. (Lesson 13-1) Lee wants to put a fence around the perimeter of his vegetable garden.

X Y CM

Find the length of the third side. (Lessons 13-2 and 13-3)

A 3x + 2y cm B 2x + 3y cm

xy

C 5x + y cm D 4x + 2y cm

x

x

Find each difference. (Lesson 13 -3) y

y z

4. Write a polynomial that represents the perimeter of the garden in feet. 5. What is the degree of the polynomial? 6. Evaluate the polynomial to find the perimeter of the garden if x = 3 feet, y = 5 feet, and z = 2 feet.

Find each sum. (Lesson 13-2) 7. (2x - 8) + (x - 7) 8. (3r + 6s) + (5r - 9s) 9. (x2 + 4x + 2) + (7x2 - 2x + 3)

GEOMETRY For Exercises 10–12, refer to the triangle. (Lesson 13-2)

14. 15. 16. 17.

(4x + 5) - (2x + 3) (5d2 - 3) - (2d2 - 7) (9x - 4y) - (12x - 9y) (2y2 - 4) - (y2 + 3y - 3)

DIRT BIKES For Exercises 18 and 19, use the following information. (Lesson 13-3) Josh and Marc are brothers who race dirt bikes. When practicing together Marc gives his younger brother, Josh, a 50-meter head start. After t seconds Josh is a distance 5t + 50 from the starting line and Marc is a distance 6t from the starting line. 18. How far ahead of Marc is Josh after t seconds? 19. Does Marc ever catch up with Josh? If so, after how many seconds?

20. MULTIPLE CHOICE The perimeter of the isosceles trapezoid shown is 16x + 1 units. 5x + 2

(2x ⫺ 30)˚ x˚

2x - 3

2x - 3

(x ⫺ 14)˚

10. Find the sum of the measures of the angles. 11. The sum of the measures of the angles in any triangle is 180º. Find the value of x. 12. Find the measure of each angle. 714 Chapter 13 Polynomials and Nonlinear Functions

Find the length of the missing base of the trapezoid. (Lesson 13-3) F 5x + 7 units G 7x + 5 units

H 9x - 4 J 7x - 3

EXPLORE

13-4

Algebra Lab

Modeling Multiplication Recall that algebra tiles are named based on their area. The area of each tile is the product of the width and length.

X

X

X X

X

These algebra tiles can be placed together to form a rectangle whose length and width each represent a polynomial. The area of the rectangle is the product of the polynomials.

ACTIVITY Use algebra tiles to find x(x + 2). Step 1 Make a rectangle with a width of x and a length of x + 2. Use algebra tiles to mark off the dimensions on a product mat.

X

£ £

X

Step 2 Using the marks as a guide, fill in the rectangle with algebra tiles. Step 3 The area of the rectangle is x2 + x + x. In simplest form, the area is x2 + 2x. Therefore, x(x + 2) = x2 + 2x.

X

X X

ANALYZE THE RESULTS Use algebra tiles to determine whether each statement is true or false. 1. x(x + 1) = x2 + 1

2. x(2x + 3) = 2x2 + 3x

3. (x + 2)2x = 2x2 + 4x

4. 2x(3x + 1) = 6x2 + x

Find each product using algebra tiles. 5. x(x + 5)

6. (2x + 1)x

7. (2x + 4)2x

8. 3x(2x + 1)

9. There is a square garden plot that measures x feet on a side. a. Suppose you double the length of the plot and increase the width by 3 feet. Write two expressions for the area of the new plot. b. If the original plot was 10 feet on a side, what is the area of the new plot? 10. Write a multiplication sentence that is represented by the model at the right.

X

X X X

X

X X X

X X X

Lesson 13-4 Algebra Lab: Modeling Multiplication

715

13-4

Multiplying a Polynomial by a Monomial

Main Idea • Multiply a polynomial by a monomial.

The Grande Arche office building in Paris, France, looks like a hollowedout prism.

2w ⫺ 52

a. Write an expression that represents the area of the rectangular region outlined on the photo. b. Recall that 2(4 + 1) = 2(4) + 2(1) by the Distributive Property. Use this property to simplify the expression you wrote in part a.

w

c. The Grande Arche is approximately w feet deep. Explain how you can write a polynomial to represent the volume of the hollowed-out region of the building. Then write the polynomial. XÎ

Multiply a Polynomial and a Monomial You can model the multiplication of a polynomial and a monomial by using algebra tiles.

Review Vocabulary Distributive Property The property of numbers that allows you to find the product of some number, n, and a sum of numbers, (s1 + s2 + … + sn), by taking the sum of n multiplied by every element in Sn. Example: n · (b + c + d) = nb + nc + nd

4HIS MODEL HAS A LENGTH OF X AND A WIDTH OF X

X

Ó

X X X

X

Ó

X X X

ÓX

The model shows the product of 2x and x + 3. The rectangular arrangement contains 2 x2-tiles and 6 x-tiles. So, the product of 2x and x + 3 is 2x2 + 6x. In general, the Distributive Property can be used to multiply a polynomial and a monomial.

EXAMPLE

Product of a Monomial and a Polynomial

Find each product. a. 4(5x + 1) 4(5x + 1) = 4(5x) + 4(1) Distributive Property = 20x + 4

Simplify.

b. (2x - 6)(3x) (2x - 6)(3x) = 2x(3x) - 6(3x) = 6x2 - 18

1A. (4z - 2)7 716 Chapter 13 Polynomials and Nonlinear Functions Dallas & John Heaton/CORBIS

Distributive Property Simplify.

1B. (3w)(14w + 6) Extra Examples at pre-alg.com

EXAMPLE

Product of a Monomial and a Polynomial

Find 3a(a2 + 2ab - 4b2). 3a(a2 + 2ab - 4b2) = 3a(a2) + 3a(2ab) - 3a(4b2) =

3a3

+

6a2b

-

12ab2

Simplify.

2A. -5x(3x2 - 2xy + 3y2)

Many standardized tests provide a Formula & Conversion Sheet. Make sure to use this sheet if it is available.


2B. (-3d4 + 2d3c + 11c2)14c

The width and length of a rectangle are 12b2df 3 and 4bd2f 5 respectively. What is the rectangle’s area? A 144b3d3f 8

C 16b2d2f 5

B 48b3d3f 8

D 3bdf 2

Read the Test Item The area of a rectangle is given by the formula A = w · . The width is 12b2df 3 and the length is 4bd2f 5. Solve the Test Item A=w· A = 12b2df 3 · 4bd2f 5 A = 12 · 4 ·

b2 · b

·d·

Substitution

d2 · f 3 · f 5

A = 12 · 4 · b2+1 · d1+2 · f 3+5 A = 48 ·

b3 · d3 · f 8

Use the Commutative Property. Product of Powers Simplify.

The answer is B. CHECK

12b2df 3 · 4bd2f 5 = 48b3d3f 8

The area of the rectangle is 48b3d3f 8.

3. GEOMETRY The area of the base of a rectangular solid is x2 + 4x - 20. If the height of the solid is 4x, what is its volume?

! X Ó { X Óä

H {X

F 4x3 + 16x2 - 80x units3 G 4x3 + 8x - 20 units3 H x2 - 20 units3 J 4x3 + 4x - 20 units3 Personal Tutor at pre-alg.com Lesson 13- 4 Multiplying a Polynomial by a Monomial

717

Examples 1, 2 (pp. 716–717)

Find each product. 1. (5y - 4)3 4.

Example 3 (p. 717)

HOMEWORK

HELP


t(4t2

- 7t + 8)

2. -5(3x2 - 7x)

3. a(a + 4)

5. (3x - 7y + 11)4x

6. a(h + j - k)

7. MULTIPLE CHOICE The perimeter of a tennis court is 228 feet. The length of the court is 6 feet more than twice the width. What are the dimensions of the tennis court? A 36 feet wide by 78 feet long

C 12 feet wide by 60 feet long

B 24 feet wide by 72 feet long

D 12 feet wide by 48 feet long

Find each product. 8. 7(2n + 5)

9. (1 + 4b)6

10. t(t - 9)

11. (x + 5)x

12. -a(7a + 6)

13. y(3 + 2y)

14. 4n(10 + 2n)

15. -3x(6x - 4)

16.

3y(y2

- 2)

1 7. ab(a2 + 7)

18. 5x(x + y)

19. 4m(m2 - m)

20. 7(-2x2 + 5x - 11)

21. -3y(6 - 9y + 4y2)

22. 4c(c3 + 7c - 10)

23. 6x2(-2x3 + 8x + 1)

24. BASKETBALL The dimensions of high school basketball courts are different from the dimensions of college basketball Measure courts, as shown in the table. Use the Perimeter information in the table to find the length Width and width of each court. Length 25. BOXES A box manufactured in the Netherlands was large enough to hold 43,000 liters of water. It was made from one large sheet of cardboard. If x is 1.2 meters and y is 0.1 meter, use the information in the diagram to write a 2x ⫹ 2y polynomial that represents the surface area of the box and find the total amount of cardboard used to make the box. (Hint: (2x + 2y)(6x - 2y) = 12x2 + 8xy - 4y2)

Basketball Courts High College (ft) School (ft) 268 288 w 2w - 16

w (2w - 16) + 10

x x

6x ⫺ 2y 2x

Simplify each expression. EXTRA

PRACTIICE


26. 6(y2 + 3) - (y2 + y)

27. (y3 + y + 2) - 4(y2 + y)

28. -1(6d3 + 4) + (8d3 + 3)

29. (x4 + 3x2 + 2) + -3(x2 + 1)

2

Solve each equation. 30. 30 = 6(-2w + 3)


31. 3(2a - 12) = 3a - 45

H.O.T. Problems

32. OPEN ENDED Write a polynomial and a monomial, each having a degree no greater than 1. What is their product? 33. REASONING Determine whether the following statement is true or false. Explain your reasoning or give a counterexample. If you change the order in which you multiply a polynomial and a monomial, the product will be different. 34. CHALLENGE You have seen how algebra tiles can be used to connect multiplying a polynomial by a monomial and the Distributive Property. Draw a model and write a sentence to show how to multiply two binomials: (a + b)(c + d). 35.

Writing in Math

Explain how the Distributive Property is used to multiply a polynomial by a monomial. Include an example showing the steps used to multiply a polynomial and monomial.

36. Seth is making a rectangular dog pen. He needs the length of the pen to be 5 feet less than 4 times the width. Write an expression for the area A of the dog pen.

37. The area of the rectangle is 252 square centimeters. What is its length? F 18 cm G 16 cm

A 10x - 10 ft2

H 14 cm

B 4x2 - 5x ft2

J 10 cm

x cm 2x ⫺ 10 cm

C 4x - 5 ft2 D x2 - 5x ft2

Find each sum or difference. (Lessons 13-2 and 13-3) 38. (2x - 1) + 5x

39. (9a + 3a2) + (a + 4)

40. (9x + 8y) - (x - 3y)

41. (13n2 + 6n + 5) - (6n2 + 5)

42. STATISTICS Describe two ways that a graph of sales of several brands of cereal could be misleading. (Lesson 12-6)

PREREQUISITE SKILL Complete each table to find the coordinates of three points through which the graph of each function passes. (Lesson 7-2) 43. y = 4x 44. y = 2x2 - 3 45. y = x3 + 1 x

4x

(x, y)

x

2x2 - 3

(x, y)

x

0

0

0

1

1

1

2

2

2

x3 + 1

(x, y)

Lesson 13- 4 Multiplying a Polynomial by a Monomial

719

13-5

Linear and Nonlinear Functions

Main Idea • Determine whether a function is linear or nonlinear.

New Vocabulary nonlinear function quadratic function cubic function

The sum of the lengths of three sides of a new deck is 40 feet. Suppose x represents the width of the deck. Then the length of the deck is 40 - 2x.

x

x 40 2x

a. Write an expression to represent the area of the deck. b. Find the area of the deck for widths of 6, 8, 10, 12, and 14 feet. c. Graph the points whose ordered pairs are (width, area). Do the points fall along a straight line? Explain.

Nonlinear Functions In Lesson 7-2, you learned that linear functions have graphs that are straight lines. These graphs represent constant rates of change. Nonlinear functions are functions that do not have constant rates of change. Therefore, their graphs are not straight lines.

EXAMPLE

Identify Functions Using Graphs

Determine whether each graph represents a linear or nonlinear function. Explain. a.

b.

y

y

y x2 1 x

O

Functions The degree of the independent variable determines the shape of the graph of a function.

x y 2

The graph is a curve, not a straight line, so it represents a nonlinear function.

1A.

Y

This graph is also a curve, so it represents a nonlinear function.

1B.

Y x xÈ X

Y

Y Ó X Î "


x

O

X

"

X

Review Vocabulary constant rate of change a consistent ratio of vertical change to horizontal change; Example: 5 Slope = _ (Lesson 8-4) 6

Recall that the equation for a linear function can be written in the form y = mx + b, where m represents the constant rate of change. Therefore, you can determine whether a function is linear by looking at its equation.

EXAMPLE

Identify Functions Using Equations

Determine whether each equation represents a linear or nonlinear function.

_

3 b. y = x

a. y = 10x This is linear because it can be written as y = 10x + 0.

1 2A. y = _ x

This is nonlinear because x is in the denominator and the equation cannot be written in the form y = mx + b.

2B. y = 2x2

3

The tables represent the functions in Example 2. Compare the rates of change. Linear Nonlinear x 1 2 3 4

+1 +1 +1

y = 10x 10 20 30 40

_3

y= x 3 1.5 1 0.75

x + 10

1 2 3 4

+1

+ 10

+1

+ 10

+1

The rate of change is constant.

- 1.5 - 0.5 - 0.25

The rate of change is not constant.

A nonlinear function does not increase or decrease at the same rate. You can check this by using a table.

EXAMPLE

Identify Functions Using Tables

Determine whether each table represents a linear or nonlinear function. Explain. a.

x 10 15 20 25

+5 +5 +5

y 120 100 80 60

b. - 20

+2

- 20

+2

- 20

+2

As x increases by 5, y decreases by 20. So this is a linear function.

3A.

x 1 2 3 4

y 6 9 12 15

x 2 4 6 8

y 4 16 36 64

+ 12 + 20 + 28

As x increases by 2, y increases

by a greater amount each time. So this is a nonlinear function.

3B.

x 1 2 3 4

y 1 4 9 16

Lesson 13-5 Linear and Nonlinear Functions

721

Some nonlinear functions are given special names.

Reading Math Cubic Cubic means threedimensional. A cubic function has a variable raised to the third power.

Nonlinear Functions A quadratic function is a function that can be described by an equation of the form y = ax2 + bx + c, where a ≠ 0. A cubic function is a function that can be described by an equation of the form y = ax3 + bx2 + cx + d, where a ≠ 0. An exponential function is a function that can be described by an equation of the form y = ax + c, where a ≠ 0 and a ≠ 1. An inverse variation function is a function that can be described by an equation k of the form y = _ x , where k ≠ 0.

Examples of these and other nonlinear functions are shown below. Nonlinear Functions

Quadratic

Cubic

Exponential

y

y

y

Inverse Variation y 1

y x3 O

y x2

x

EXAMPLE The trend in farm income can be modeled with a nonlinear function. Visit pre-alg.com to continue work on your project.

y x O

x y 2x

x

O

x

O

Describe a Linear Function

Which of the following is a linear function? a. y = 7x3 + 2

The independent variable has an exponent of 3.

b. y = (x - 1)5x = 5x2 - 5x


c. 4x + 3y = 12


d. -2x2 + 6y = 8 The independent variable has an exponent of 2. The linear function is c. Linear functions are always functions of the first degree.

4. The chart shows the price of oil futures per barrel over the last 12 months. Do these data represent a linear or nonlinear function? To check your answer, graph the data. Month

1

2

3

4

5

6

7

8

9

10

11

12

Price ($)

50.25

51.72

43.41

44.76

46.78

48.31

53.62

46.72

52.81

56.79

52.44

58.15



Determine whether each graph, equation, or table represents a linear or nonlinear function. Explain. Example 1

1.

2.

y

y

(p. 720)

x

O

x

O

Example 2 (p. 721)

Example 3

5 3. y = _ x

5.

(p. 721)

Example 4 (p. 722)

HOMEWORK

HELP

For See Exercises Examples 8–13 1 14–19 2, 4 20–23 3

x -4 -2 0 2

4. xy = 12 6.

y 13 0 4 0

x 8 9 10 11

y 19 22 25 28

7. The chart below shows the corresponding width for the possible different lengths of a rectangle with a fixed area of 20 ft2. Do these data represent a linear function? To check your answer, draw a graph. Length

0

1

2

3

4

5

10

15

20

Corresponding Width

-

20

10

6_

5

4

2

1 1_ 3

1

2 3

Determine whether each graph represents a linear or nonlinear function. Explain. 8.

9.

y

O

O

11.

x

y

x

O

x

12.

y

O

10.

y

x

13.

y

O

x

y

O


x

723

Determine whether each equation or table represents a linear or nonlinear function. Explain. 14. y = 0.9x

15. y = x3 + 2

3x 16. y = _

17. 2x + 3y = 12

18. y = 4x

19. xy = -6

Real-World Link

x 9 11 13 15

y -2 -8 -14 -20

21.

x 4 5 6 7

y 1 4 9 16

24. PATENTS The table shows the years in which the first six million patents were issued. Is the number of patents issued a linear function of time? Explain.

Abraham Lincoln is the only U.S. President to hold a patent. He invented a device to lift boats over shallow waters.

Year 1911 1936 1961 1976 1991 1999

Source: historyplace.com

Number of Patents Issued 1 million 2 million 3 million 4 million 5 million 6 million

Source: New York Times

22.

x -4 -2 0 2

y 12 0 4 0

23.

x -10 -9 -8 -7

y 20 18 16 14

25. BASEBALL The graph shows the average price of a baseball ticket. Would you describe the change in price as a linear function? Explain. Û}° >ÃiL> /ViÌ *ÀVi fÓä°ää Û}° *ÀVi

20.

4

f£x°ää f£ä°ää fx°ää

ä{ Óä

ää Óä

Ç £

{ £

£

£

fä°ää 9i>À -ÕÀVi\ Ìi>>ÀiÌ}°V

EXTRA

PRACTICE


H.O.T. Problems

26. MONEY Adam puts $15 into his savings account every month. Suzanne tries to double the amount of money in her bank account every month. Which person’s monthly balance represents a linear function? Explain why the other person’s balance is best represented by a nonlinear function. 27. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data that cannot be represented by a linear function. Explain your reasoning. 28. OPEN ENDED Use newspapers, magazines, or the Internet to find realworld examples of nonlinear situations. 29. Which One Doesn’t Belong? Identify the equation that doesn’t belong with the other three. 1 y=_ x-x 3

y2 = 3x

5x + y = 6

x = 5y

30. CHALLENGE Are all straight lines graphs of linear functions? Explain. 31.

Writing in Math How can you determine whether a function is linear? Include a list of ways in which a function can be represented and an explanation of how each representation can be used to identify the function as linear or nonlinear.

724 Chapter 13 Polynomials and Nonlinear Functions Mercury Archives/Getty Images

33. Which equation represents a nonlinear function if a > 1?

32. Which equation represents a linear function? 1 A y=_ x

C x2 - 1 = y

F y = ax x G y=_ a H y = ax

D y = x(x + 4)

J y=a+x

2

B 3xy = 12

34. Which graph represents a cubic equation? A

C

Y

Y

X

"

X

"

B

D

Y

Y

X

"

X

"

Find each product. (Lesson 13-4) 35. t(4 + 9t)

36. 5n(-1 + 3n)

37. (a - 2b)ab

Find each difference. (Lesson 13-3) 38. (2x + 7) - (x - 1)

39. (4x + y) - (5x + y)

40. (6a - a2) - (8a + 3)

41. GEOMETRY Classify a triangle with an angle of 125° as acute, obtuse, or right. (Lesson 9-3) Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. (Lesson 5-1) 2 42. _ 5

7 43. 3 _ 10

5 44. -_ 9

PREREQUISITE SKILL Use a table to graph each line. (Lesson 7-2) 46. y = -x 47. y = x - 4 48. y = 2x + 2

7 45. -1_ 9

1 49. y = -_ x+3 2


725

13-6

Graphing Quadratic and Cubic Functions

Main Ideas

• Graph cubic functions.

You can find the area of a square A by squaring the length of a side s. This relationship can be represented in different ways.

Area

equals

length of a side squared.

⎧ ⎨ ⎩

⎧ ⎨ ⎩

⎧ ⎨ ⎩

Words = Equation

=

s2

A

s s

Table

Graph

s

s2

(s, A)

0

02 0

(0, 0)

1

12

1

(1, 1)

2

22 4

(2, 4)

A A s2

Area

• Graph quadratic functions.

s O

a. The volume of a cube V equals the cube of the length of an edge a. Write a formula to represent the volume of a cube as a function of edge length.

Side

a a

a

b. Graph the volume as a function of edge length. (Hint: Use values of a like 0, 0.5, 1, 1.5, 2, and so on.) c. Would it be reasonable to use negative numbers for x-values in this situation? Explain.

Quadratic Functions In Lesson 13-5, you saw that functions can be represented using words, graphs, equations, and tables. This allows you to graph quadratic functions such as A = s2 using an equation or a table of values.

EXAMPLE

Graph Quadratic Functions

Graph each function. a. y = 2x2 Graphing It is often helpful to substitute decimal values of x in order to graph points that are closer together.

Make a table of values, plot the ordered pairs, and connect the points with a curve. y

x

2x 2

(x, y)

-1.5

2(-1.5)2 = 4.5

(-1.5, 4.5)

-1

2(-1)2

=2

0

2(0)2 = 0

(0, 0)

1

2(1)2 = 2

(1, 2)

1.5

2(1.5)2

= 4.5


y 2x 2

(-1, 2)

(1.5, 4.5)

O

x


b. y = - x2 + 3 -x 2 +

3

(x, y)

-2

-(-2)2

+ 3 = -1

(-2, -1)

-1

-(-1)2

+3=2

x

y

(-1, 2)

0

-(0)2 + 3 = 3

(0, 3)

1

-(1)2 + 3 = 2

(1, 2)

2

-(2)2

+ 3 = -1

y x 2 3

O

x

(2, -1)

Graph each function. 1A. y = 1x2

1B. y = 2x2 - 2

4

1C. y = -x2

BASEBALL Juan threw a baseball into the air. The equation that gives the ball’s height in meters h as a function of time t is h = -4.9t 2 + 16t + 1.4. Graph this equation and interpret your graph. What values for x and y are unreasonable? How high is the ball after 3 seconds? The function is h = -4.9t 2 + 16t + 1.4. Since the function has a degree of 2, it is nonlinear. Graph h = -4.9t 2 + 16t + 1.4. t 0

h = -4.9t 2 + 16 t + 1.4 1.4 = -4.9(0)2 + 16(0) + 1.4 -4.9(1)2

(t, h) (0, 1.4)

1

12.5 =

+ 16(1) + 1.4

(1, 12.5)

2

13.8 = -4.9(2)2 + 16(2) + 1.4

(2, 13.8)

3

5.3 = -4.9(3)2 + 16(3) + 1.4

(3, 5.3)

4

-13 =

-4.9(4)2

+ 16(4) + 1.4

(4, -13)

H £{ Ó] £Î°n® £Ó £] £Ó°x® £ä n È Î] x°Î® { Ó ä] £°{® " Î Ó £ Ó Î { xT

The graph shows the height of the ball that Juan threw over a period of about 3.3 seconds. The height of the ball at 0 seconds is 1.4 meters so Juan released the ball 1.4 meters off the ground. The ball reached its maximum height of 14.46 meters at about 1.63 seconds. The ball reached a height of 0 meters (or it landed on the ground) at about 3.4 seconds. Unreasonable values for x would be any negative numbers because time cannot be negative. Negative values for y are also unreasonable because the ball is not going beneath the ground. The height of the ball after three seconds is 5.3 meters.

2. FRAMES Mei is building a picture frame, and she wants the length to be 2 the width. Graph the 3 equation that gives the area of the framed picture for different widths and lengths. Give the area of the picture if the width is 6 inches.

Ó Ý Î

Ý

Personal Tutor at pre-alg.com Lesson 13-6 Graphing Quadratic and Cubic Functions

727

Cubic Functions You can also graph cubic functions such as the formula for the volume of a cube by making a table of values.

EXAMPLE

Graph Cubic Functions

Graph each function. a. y = x3 y = x3

x

-1.5 (-1.5)3 ≈ - 3.4

(-1.5, -3.4)

-1

(-1)3

(-1, -1)

0

(0)3

=0

1

(1)3

=1

1.5

y

(x, y)

= -1

(1.5)3 ≈ 3.4

y x3

O

(0, 0)

x

(1, 1) (1.5, 3.4)

b. y = x3 - 1 y = x3 - 1

x

-1.5 (-1.5)3 - 1 ≈ - 4.4 -1

(-1)3

- 1 = -2

(-1.5, -4.4)

0

(0)3 - 1 = -1

1

(1)3 -1 = 0

(1, 0)

1.5

(1.5)3 -1 ≈ 2.4

(1.5, 2.4)

y = -4x 3

x

Example 1

Example 2 (p. 727)

Example 3 (p. 728)

O

(-1, -2)

x

(0, -1)

3A. y = -4x3

(pp. 726–727)

y

(x, y)

y x3 1

3B. y = x3 + 4 (x, y)

y = x3 + 4

x

-2

-2

-1

-1

0

0

1

1

2

2

(x, y)

Graph each function. 1. y = x2

2. y = -2x2

3. y = x2 + 1

GEOMETRY A cube has edges measuring a units. 4. Write a quadratic equation for the surface area S of the cube. 5. Graph the surface area as a function of a. (Hint: Use values of a like 0, 0.5, 1, 1.5, 2, and so on.) 6. Describe the graph. Graph each function. 7. y = -x3


8. y = 0.5x3

9. y = x3 - 2

HOMEWORK

HELP

For See Exercises Examples 10–13, 1 20, 21 14–19 3 22–25 2

Graph each function. 10. y = 3x2

11. y = 0.5x2

12. y = -x2

13. y = -0.5x2

14. y = -2x3

15. y = 3x3

16. y = 2x3

17. y = 0.1x3

18. y = x3 + 1

19. y = 1 x3 + 2

20. y = x2 - 3

21. y = 1 x2 + 1

3

CONSTRUCTION For Exercises 22 and 23, use the information below and the figure at the right. A dog trainer is building a dog pen with a 100-foot roll of chain link fence. 22. Write and graph an equation to represent the area A of the pen. 23. What should the dimensions of the dog pen be to enclose the maximum area inside the fence? (Hint: Find the coordinates of the maximum point of the graph.)

2

x ft

50 x ft

GEOMETRY Write the function for each of the following. Then graph it. 24. the volume V of a rectangular prism as a function of a fixed height of 2 units and a square base with length s units 25. the volume V of a cylinder as a function of a fixed height of 0.2 unit and radius r units Graph each pair of equations on the same coordinate plane. Describe their similarities and differences. 26. y = x2 y = 3x2 EXTRA

PRACTICE


H.O.T. Problems

27. y = 0.5x3 y = 2x3

28. y = 2x2 y = -2x2

29. y = x3 y = x3 - 3

30. BOOKS The distance in inches that a book falls from a shelf is equal to sixteen times the time t squared, with the time given in seconds. Graph this function and determine how far the book will fall in 2.5 seconds. 31. OPEN ENDED Write a quadratic function and explain how to graph it. 32. REASONING Graph y = x2 and y = x3 in the first quadrant on the same coordinate plane. Explain which graph shows faster growth. 33. NUMBER SENSE Are the y-values for the quadratic equation y = x(x + 3) sometimes, always, or never negative? Explain your reasoning. CHALLENGE The graph of quadratic functions may have one maximum or one minimum point. The maximum point of a graph is the point with the greatest y-value coordinate. The minimum point is the point with the least y-value coordinate. Graph each equation. Find the coordinates of each point. 34. the maximum point of the graph of y = -x2 + 7 35. the minimum point of the graph of y = x2 - 6 36.

Writing in Math

How are functions, formulas, tables, and graphs related? Include an explanation of how to make a graph by using a rule and an explanation of how to write a rule by using a graph. Lesson 13-6 Graphing Quadratic and Cubic Functions

729

38. For a certain frozen pizza, the demand y can be modeled by the formula y = -10x2 + 60x + 180. What is a good estimate of the price that will result in the most demand?

y

O x

F $0

y 400

G $2

350 300

H $3 J $8 A y=

4x2

y 10x 2 60x 180

250 200 150 100

B y = -4x2 C y=

Monthly Sales ($ thousands)

37. Which equation represents the graph?

50

4x3

0

1

D y = -4x3

Water Vapor Pressure Vapor Pressure (kPa)

39. SCIENCE The graph shows how vapor pressure increases as the temperature increases. Is this relationship linear or nonlinear? Explain. (Lesson 13-5)

100 80 60 40 20 0

Find each product. (Lesson 13-4) 40. (2x - 4)5

41. n(n + 6)

2 3 4 5 6 7 8 x Cost ($)

20 40 60 80 100 Temperature (oC)

42. 3y(8 - 7y)

43. Draw a box-and-whisker plot for the following set of data. 19, 15, 15, 18, 22, 16, 17, 16, 20, 17, 19, 16 (Lesson 12-3) Write an equation in slope-intercept form for the line passing through each pair of points. (Lesson 7-7) 44. (3, 6) and (0, 9)

45. (2, 5) and (-1, -7)

46. (-4, -3) and (8, 6)

Express each percent as a decimal. (Lesson 6-5) 47. 77%

48. 8%

49. 421%

50. 3.56%

Down on the Farm It is time to complete your project. Use the information and data you have gathered to prepare a Web page about farming or ranching in the United States. Be sure to include at least five graphs or tables that show statistics about farming or ranching and at least one scatter plot that shows a farming or ranching statistic over time, from which you can make predictions. Cross-Curricular Projects at pre-alg.com


EXTEND

13-6


The Family of Quadratic Functions

A quadratic function can be described by an equation of the form ax2 + bx + c, where a ≠ 0. The graph of a quadratic function is called a parabola. The parent function of the family of quadratic functions is y = x2.

EXAMPLE Graph y = x2 and y = x2 + 4 and describe how they are related. Step 1 Enter the function y = x2.

Step 2 Enter the function y = x2 + 4.

• Enter y = x2 as Y1.

• Enter y = x2 + 4 as Y2.

KEYSTROKES:

X,T,␪,n

ENTER

KEYSTROKES:

X,T,␪,n

4 ENTER

Step 3 Graph both quadratic functions on the same screen. • Display the graph. KEYSTROKES:

ZOOM 6

The first function graphed is Y1 or y = x2. The second is Y2 or y = x2 + 4. Press TRACE and move along each function by using the right and left y x2 arrow keys. Move from one function to another by using the up and down arrow keys.

y x2 4

The graphs are similar in that they are both parabolas. However, the graph of y = x2 has its vertex at (0, 0), whereas the graph of y = x2 + 4 has its vertex at (0, 4).

EXERCISES 1. Graph y = x2, y = x2 - 5, and y = x2 - 3 on the same screen and draw the parabolas on grid paper. Compare and contrast the three parabolas. 2. MAKE A CONJECTURE How does adding or subtracting a constant c affect the graph of a quadratic function? 3. The three parabolas at the right are graphed in the standard viewing window and have the same shape as the graph of y = x2. Write an equation for each, beginning with the lowest parabola. 4. Clear all functions from the menu. Enter y = 0.4x2 as Y1, 2 2 y = x as Y2, and y = 3x as Y3. Graph the functions in the standard viewing window on the same screen. Then draw the graphs on the same coordinate grid. How does the shape of the parabola change as the coefficient of x2 increases? Other Calculator Keystrokes at pre-alg.com

Extend 13-6 The Family of Quadratic Functions

731

CH

APTER

13




X

Key Concepts Polynomials

binomial (p. 701) cubic function (p. 722) degree (p. 702) nonlinear function (p. 720) polynomial (p. 701) quadratic function (p. 722) trinomial (p. 701)

(Lesson 13-1)

• A polynomial is an algebraic expression that contains one or more monomials. • A binomial has two terms and a trinomial has three terms. • The degree of a monomial is the sum of the exponents of its variables.

Operations on Polynomials

(Lessons 13-2

through 13-4)

Vocabulary Check Choose the correct term to complete each sentence. 1. A (binomial, trinomial) is the sum or difference of three monomials. 2. Monomials that contain the same variables with the same (power, sign) are like terms.

• To add polynomials, add like terms.

3. The function y = 2x3 is an example if a (cubic, quadratic) function.

• To subtract polynomials, subtract like terms or add the additive inverse.

4. The equation y = x2 + 5x + 1 is an example of a (cubic, quadratic) function.

• To multiply a polynomial and a monomial, use the Distributive Property.

5. The terms x2 and 4x2 are examples of (binomials, like terms).


6. The equation y = 4x3 + x2 + 2 is an example of a (quadratic, cubic) function.

(Lesson 13-5)

• Nonlinear functions do not have constant rates of change.

Graphing Quadratic and Cubic Functions (Lesson 13-6) • Quadratic and cubic functions can be graphed by plotting points.


7. The graph of a quadratic function is a (straight line, curve). 8. To multiply a polynomial and a monomial, use the (Distributive, Commutative) Property.





Polynomials

(pp. 701–704)

Determine whether each expression is a polynomial. If so, find its degree and classify it as a monomial, binomial, or trinomial. 6 10. _ 9. c2 + 3 a +b 11.

12. x2 + xy2 - y4

√n

13. MEDICINE Doctors can study a patient’s heart by injecting dye in a vein near the heart. In a normal heart, the amount of dye in the bloodstream after t seconds is given by -0.006t4 + 0.14t3 - 0.53t2 + 1.79t. Find the degree of the polynomial.

13–2

Adding Polynomials

The expression is the difference of two monomials. So it is a binomial. x3 has degree 3, and -2xy has degree 1 + 1 or 2. So, the degree of x3 - 2xy is 3.

(pp. 706–709)

Example 2 Find (5x2 - 8x + 2) + (x2 + 6x).

Find each sum. 14. (3b + 8) + (5b - 5) 15. (-3y2 + 2) + (4y2 - y - 3) 16. GEOMETRY The areas of two rectangles are equal to x2 + 4x - 5 and 3x2 + 6x + 12. Find the sum of the areas of the two rectangles.

13–3

Example 1 State whether x3 - 2xy is a monomial, binomial, or a trinomial. Then find the degree.

Subtracting Polynomials

5x2 - 8x + 2 Align like terms. (+) x2 + 6x 2 6x - 2x + 2 Add. The sum is 6x2 - 2x + 2.

(pp. 710–713)

Find each difference. 17. (a2 + 15) - (3a2 - 10) 18. (x + 8) - (2x + 7) 19. SHIPPING The cost of shipping a box from Seattle to Portland via company X is 3x + 22. The cost of shipping a package the same distance via company Y is 2x + 45. What is the difference in the two prices?

Example 3 Find (4x2 + 7x + 4) (x2 + 2x + 1). 4x2 + 7x + 4 (-) x2 + 2x + 1 Align like terms. 3x2 + 5x + 3 Subtract. The difference is 3x2 + 5x + 3.


733

CH

A PT ER

13 13–4


Multiplying a Polynomial by a Monomial Find each product. 20. 5(4t - 2)

(pp. 716–719)

Example 4 Find -3x(x + 8y). 21. (2x + 3y)7

22. 6(2x2 + xy + 3y2) 23. 3y(-y2 - 8y + 4)

-3x(x + 8y) = -3x(x) + (-3x)(8y) Distributive Property.

=

24. GEOMETRY The length of a rectangle is x2 + 3x + 4. The width is 6x. What is the area?

13–5


x 26. y = _

y

2

x

y

y

3

4

6 9

Graphing Quadratic and Cubic Functions

b. y = x + 12

Linear; can be written as y = mx + b.

c.

Nonlinear; graph is not a straight line

Y

31. y = 33. y =

-x3

-2

32. y =

x3

34. y =

2x2

+1 +4

35. MANUFACTURING The Tube Factory is experimenting with cylinder designs that have a fixed height of 8 inches and a variable radius. Write an equation for the volume of the possible cylinders. Graph this equation.


"

X

(pp. 726–730)

Graph each function. 30. y = -3x2 29. y = x2 + 2 x3

is constant

25 22 19 16

28. SCHOOLS The equation y = 3254x2 + 0.0013x + 1427 represents a school district’s spending on students over the last five years. Is this equation linear? Explain.

13–6

Simplify.

Example 5 Determine whether each graph, equation, or table represents a linear or nonlinear function. Linear; rate of change a. x 7 8 9 10

27. x -6 -4 -2 0 O

- 24xy

(pp. 720–725)

Determine whether each graph, equation, or table represents a linear or nonlinear function. Explain. 25.

-3x2

Example 6 Graph y = -x2 + 3. x

(x, y)

y x 2 3

y

-2 (-2, -1) -1 (-1, 2) 0 (0, 3) 1 (1, 2) 2 (2, -1)

O

x

CH

A PT ER

13

Practice Test

Determine whether each expression is a polynomial. If it is, classify it as a monomial, binomial, or trinomial. 1. 3x3 - 2x + 7 3 4 p 3. _

5 2. 6 + _ m

Find each product. 15. x(3x - 5) 16. -5a(a2 - b2) 17. 6p(-2p2 + 3p - 4)

4. n2 + 8

5

Find the degree of each polynomial. 5. 5ab3

6. w5 - 3w3y4 + 1


7. METEOROLOGY The summer simmer index measures the discomfort level due to temperature and humidity. Meteorologists calculate this value by using a polynomial similar to 1.98x2 - 115.93x + 0.01xy 0.63y + 6.33. The variable x is the temperature in degrees Fahrenheit and y is the relative humidity expressed as a whole number. What is the discomfort level when it is 88°F and the humidity is 75%?

8. 9. 10. 11. 12. 13.

O x

(5y + 8) + (-2y + 3) (5a - 2b) + (-4a + 5b) (-3m3 + 5m - 9) + (7m3 - 2m2 + 4) (6p + 5) - (3p - 8) (5w - 3x) - (6w + 4x) (-2s2 + 4s - 7) - (6s2 - 7s - 9)

x 1 3 5 7

y 10 7 3 -2

20. 5x - 6y = 2 Graph each function. 21. y = 3x2

Find each sum or difference.

19.

y

1 3 22. y = _ x

23. y = -2x2 + 2

2

GEOMETRY For Exercises 24 and 25, refer to the rectangle below. 3x ⫺ 9 in. 12 ⫺ 2x in.

14. MULTIPLE CHOICE Every month for the last year, Sarah added x dollars to her jar of mad money which contained y dollars at the beginning of the year. She is going to take _1 of the total amount in the jar for spending 4 money on a school trip. Which expression represents the amount of money that will be left in the jar? 3 y + 9x A _ 4 _ B 1 y + 3x 4

1 C _ y - 12x 4

3 D _ y - 12x 4


24. Write an expression for the perimeter of the rectangle. 25. Find the value of x if the perimeter is 14 inches. 26. MULTIPLE CHOICE The length of a garden is equal to 5 less than four times its width. The perimeter of the garden is 40 feet. Find the length of the garden. F 1 ft

H 10 ft

G 5 ft

J

15 ft


735

CH

A PT ER


13


5. What is the total area, in square yards, of the yard?


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1. One machine makes plastic containers at a rate of 360 containers per hour. A newer machine makes the same containers at a rate of 10 containers per minute. If both machines run for four hours, how many containers will they make? A 3840 C 2440 B 3280 D 1480 2. A spinner and a number cube are used in a game. The spinner has an equal chance of landing on 1 of 5 letters: A, B, C, D, or E. The faces of the cube are numbered from 1 to 6. What is the probability of a player spinning a vowel and rolling an odd number? 1 1 H F 2 1 G 5

6 1 J 8

3. GRIDDABLE A quality control engineer determines that the probability of a carton of eggs containing a cracked egg is about 0.005. Based on this finding, how many cartons would you expect to contain a cracked egg in a shipment of 2,000 cartons? ’ is the 4. Suppose A’B result of a dilation by 2 with the center at the origin. Which are the coordinates of A? A (2, 1)

Y !

"g "

X

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F 200 yd G 100 yd

H 50 yd J 75 yd

6. Which rational number is the least? A 5.1 C 5.05 11 1 B D 5 2

25

7. Connor sold 4 fewer tickets to the band concert than Miguel sold. Kylie sold 3 times as many tickets as Connor. If the number of tickets Miguel sold is represented by m, which of these expressions represents the number of tickets that Kylie sold? F m-4 G 4 - 3m H 3m - 4 J 3(m - 4)

8. GRIDDABLE A piggy bank contains 6 quarters, 5 dimes, 10 nickels, and 4 pennies. Suppose Gina picks a coin at random. Then without replacing the first coin, she picks a second coin at random. What is the probability that she will pick a dime and then a quarter? Express the probability as a decimal.

1 B 2, _ 2

C (1, 2)

2

1 D _ , 2


Question 8 If you have time at the end of a test, go back to check your calculations and answers. If the test allows you to use a calculator, use it to check your calculations.



9. In the sequence below, which expression can be used to find the value of the term in the nth position? Position

Value of Term

1

0.5

2

1.5

3

2.5

4

3.5

5

4.5

n

?

A n - 0.5 n B 2

12. Which function includes all of the ordered pairs in the table? x

-2

-1

1

2

3

y

4

2

-2

-4

-6

A y = -x + 2 B y = 2x

C y = -2x D y=2-x

13. Students taste-tested three brands of instant hot cereal and chose their favorite brand. Which of these statements is NOT supported by the data in the table?

C 2n n D 4

Hot Cereal Brand X

10. GRIDDABLE Brooke wants to fill her new aquarium two-thirds full of water. The aquarium dimensions are 20 inches by 1 inches. What volume of 20 inches by 8 2 water, in cubic inches, is needed? Round to the nearest tenth.

Y

Z

Girls

12

5

10

Boys

10

15

5

F Twice as many girls as boys chose Brand Z. G The total number of students who chose Brand X is 22. H Three times as many boys chose Brand Y as Brand Z. J Half of the students who chose Brand Z were boys.

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n Ó °

Pre-AP

11. The table shows the number of sandwiches sold during twenty lunchtimes. Which measure of central tendency best represents the data?

Record your answers on a sheet of paper. Show your work. 14. Each post has a diameter of 12 inches. The heights of the posts are 6 feet, 5 feet, 4 feet, 3 feet, and 2 feet. a. What is the total volume of all five posts? Use = 3.14. b. The posts are made of a material whose density is 12 pounds per cubic foot. How much does the sculpture weigh?

Number of Sandwiches Sold 9 8 10 14 12 16 9 7 10 11 11 8 9 8 7 12 14 8 9 9

F mean G median

H mode J none


1

2

3

4

5

6

7

8

9

10

11

12

13

14

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6-1

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tud nt an Built-In Workbooks Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Mixed Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 Preparing for Standardized Tests . . . . . . . . . . . . . . . . . . . . . . 809

Reference English-Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R1 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R24 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R58 Photo Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R77 Mathematics Chart . . . . . . . . . . . . . . . . . . . . . Inside Back Cover

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738 Eclipse Studios

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A Student Handbook is the additional skill and reference material found at the end of books. The Student Handbook can help answer these questions.

What If I Forget What I Learned Last Year? Use the Prerequisite Skills section to refresh your memory about things you have learned in other math classes. Here’s a list of the topics covered in your book. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Factors Comparing and Ordering Decimals Rounding Decimals Estimating Sums and Differences of Decimals Adding and Subtracting Decimals Estimating Products and Quotients of Decimals Multiplying Decimals Dividing Decimals Estimating Sums and Differences of Fractions and Mixed Numbers Estimating Products and Quotients of Fractions and Mixed Numbers Converting Measurements within the Metric System Converting Measurements within the Customary System Measuring and Drawing Angles Displaying Data in Graphs

What If I Need More Practice? You, or your teacher, may decide that working through some additional problems would be helpful. The Extra Practice section provides these problems for each lesson so you have ample opportunity to practice new skills.

What If I Have Trouble with Word Problems? The Mixed Problem Solving pages provide additional word problems that use the concepts in each lesson. These problems give you real-world situations where the math can be applied.

What If I Need to Practice for a Standardized Test? You can review the types of problems commonly used for standardized tests in the Preparing for Standardized Test section. This section includes examples and practice with multiple-choice, griddable or grid-in, and extended response test items.

What If I Forget a Vocabulary Word? The English-Spanish Glossary provides a list of important words used in the textbook. It provides a definition in English and Spanish as well as the page number(s) where the word can be found.

What If I Need to Check a Homework Answer? The answers to odd-numbered problems are all included in Selected Answers. Check your answers to make sure you understand how to solve all of the assigned problems.

What If I Need to Find Something Quickly? The Index alphabetically lists the topics covered in the textbook and the page(s) on which each topic can be found.

What If I Forget a Formula? Inside the back cover of the book is a Mathematics Chart that lists formulas and symbols that are used in the book. Student Handbook

739

Prerequisite Skills

Prerequisite Skills 1 Factors Two or more numbers that are multiplied to form a product are called factors. 4 × 9 = 36

product

factors

So, 4 and 9 are factors of 36 because they each divide 36 with a remainder of 0. We can say that 36 is divisible by 4 and 9. However, 5 is not a factor of 36 because 36 ÷ 5 = 7 with a remainder of 1. Sometimes you can test for divisibility mentally. The following rules can help you determine whether a number is divisible by 2, 3, 5, 6, or 10. Divisibility Rules A number is divisible by . . .

Examples

Reasons

2 if the ones digit is divisible by 2.

54

4 is divisible by 2.

3 if the sum of its digits is divisible by 3.

72

7 + 2 = 9, and 9 is divisible by 3.

5 if the ones digit is 0 or 5.

65

The ones digit is 5.

6 if the number is divisible by 2 and 3.

48

48 is divisible by 2 and 3.

10 if the ones digit is 0.

120

The ones digit is 0.

EXAMPLE 1 Determine whether each number is divisible by 2, 3, 5, 6, or 10. a. 138 Number 2 3 5 6 10

Divisible? yes yes no yes no

Reason 8 is divisible by 2. 1 + 3 + 8 = 12, and 12 is divisible by 3. The ones digit is 8, not 0 or 5. 138 is divisible by 2 and 3. The ones digit is not 0.

So, 138 is divisible by 2, 3, and 6. b. 3050 Number 2 3 5 6 10

Divisible? yes no yes no yes

Reason 0 is divisible by 2. 3 + 0 + 5 + 0 = 8, and 8 is not divisible by 3. The ones digit is 0. 3050 is divisible by 2, but not 3. The ones digit is 0.

So, 3050 is divisible by 2, 5, and 10.

740 Prerequisite Skills

You can also use the rules for divisibility to find the factors of a number.

Prerequisite Skills

EXAMPLE 2 List all the factors of 72. Use the divisibility rules to determine whether 72 is divisible by 2, 3, 5, and so on. Then use division to find other factors of 72. Number 1 2 3 4 5 6 7 8 9

72 Divisible by Number? yes yes yes yes no yes no yes yes

Factor Pairs 1 · 72 2 · 36 3 · 24 4 · 18 —— 6 · 12 —— 8·9 9·8

Use division to find the other factor in each factor pair. 72 ÷ 2 = 36

⎫ ⎬ ⎭

You can stop finding factors when the numbers start repeating.

So, the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10. 1. 39

2. 135

3. 82

4. 120

5. 250

6. 118

7. 378

8. 955

9. 5010

10. 684

11. 10,523

12. 24,640

List all the factors of each number. 13. 75

14. 114

15. 57

16. 65

17. 90

18. 124

19. 102

20. 135

MUSIC For Exercises 21 and 22, use the following information. The band has 72 students who will march during halftime of the football game. For one drill, they need to march in rows with the same number of students in each row. 21. Can the whole band be arranged in rows of 7? Explain. 22. How many different ways could students be arranged? Describe the arrangements. 23. CALENDARS Years that are divisible by 4, called leap years, are 366 days long. Also, years ending in “00” that are divisible by 400 are leap years. Use the rule given below to determine whether 2000, 2004, 2015, 2018, 2022, and 2032 are leap years. If the last two digits form a number that is divisible by 4, then the number is divisible by 4. Prerequisite Skills

741

Prerequisite Skills

2 Comparing and Ordering Decimals To determine which of two decimals is greater, you can compare the digits in each place-value position, or you can use a number line.

METHOD 1 Use place value. Line up the decimal points of the two numbers. Starting at the left, compare the digits in each place-value position. In the first position where the digits are different, the decimal with the greater digit is the greater decimal.

METHOD 2 Use a number line. Graph each number on a number line. On a number line, numbers to the right are greater than numbers to the left.

EXAMPLE 1 Which is greater, 4.35 or 4.8? METHOD 1 Use place value. 4.35 Line up the decimal points. 4.8 The digits in the tenths place are not the same. 8 tenths > 3 tenths, so 4.8 > 4.35. METHOD 2 Use a number line. Compare the decimals on a number line. 4.35 4.0

4.1

4.2

4.3

4.4

4.8 4.5

4.6

4.7

4.8

4.9

5.0

4.8 is to the right of 4.35. So 4.8 > 4.35.

EXAMPLE 2 Order 0.8, 1.52, and 1.01 from least to greatest. 0.8 is less than both 1.52 and 1.01. 1.01 is less than 1.52 but greater than 0.8. Thus, the order from least to greatest is 0.8, 1.01, 1.52. Replace each 1. 4.05

with < or > to make a true sentence.

4.45

2. 2.26

2.28

3. 3.005

3.05

4. 8.7

82.1

5. 6.2

6.008

6. 15.601

7. 1.9

1.96

8. 8.9

7.99

9. 0.66

0.582

10. 7.14

7.2

11. 0.048

0.11

12. 10.1

1.01

13. 32.1

3.215

14. 1.098

2

15. 9.1

9.005

16. 16.8

16.791

17. 0.943

0.4991

18. 0.117

16.9

0.95

Order each set of decimals from least to greatest. 19. {0.2, 0.01, 0.6}

20. {1.2, 2.4, 0.04, 2.2}

21. {3.5, 0.6, 2.06, 0.28}

22. {0.8, 0.07, 1.001, 0.392}

23. {7.06, 7.026, 7.061, 7.009, 7.1}

24. {0.82, 0.98, 0.103, 0.625, 0.809}

742

Prerequisite Skills

3 Rounding Decimals

• If the digit to the right is less than or equal to 4, the digit being rounded stays the same. • If the digit to the right is greater than or equal to 5, the digit being rounded increases by one.

3 . 8

Thousandths

Hundredths

Tenths

Ones

Tens

The place-value chart below shows how to round 3.81 to the nearest one (or whole number).

1

• 3 is in the ones place • 8 is to the right of 3 • 8>5 So, 3.81 rounded to the nearest one (or whole number) is 4.

EXAMPLE 1 Round each number to the nearest one (or whole number). a. 8.3

b. 9.6

8.3 rounds to 8.

9.6 rounds to 10.

EXAMPLE 2 Round each number to the nearest tenth. a. 16.08

b. 29.54

16.08 rounds to 16.1.

29.54 rounds to 29.5.

EXAMPLE 3 Round each number to the nearest hundredth. a. 50.345

b. 19.998

50.345 rounds to 50.35.

19.998 rounds to 20.00.

Round each number to the nearest whole number. 1. 3.2

2. 64.8

3. 50.57

4. 16.08

5. 38.726

6. 86.299

Round each number to the nearest tenth. 7. 16.57 10. 80.349

8. 1.05 11. 49.5463

9. 53.865 12. 131.9884

Round each number to the nearest hundredth. 13. 62.624

14. 44.138

15. 85.5639

16. 99.9862

17. 458.7625

18. 153.2965

Round each number to the nearest dollar. 19. $40.29

20. $72.50

21. $36.82 Prerequisite Skills

743

Prerequisite Skills

Rewriting a number to a certain place value is called rounding. Look at the digit to the right of the place being rounded.

Prerequisite Skills

4 Estimating Sums and Differences of Decimals Estimation is often used to provide a quick and easy answer when an exact answer is not necessary. It is also an excellent way to quickly see if your answer is reasonable or not.

EXAMPLE 1 Estimate each sum or difference to the nearest whole number. a. 16.9 + 5.4 16.9 + 5.4 _____

b. 200.35 - 174.82 200.35 → 200 Round to the nearest -174.82 -175 ________ _____ whole number. 25

17 Round to the nearest + 5 whole number. ____ 22

→

You can also use rounding to estimate answers involving money.

EXAMPLE 2 Estimate each sum or difference to the nearest dollar. a. $67.07 + $52.64 + $0.85 $67.07 52.64 + 0.85 _______

b. $89.42 - $8.94

$67.00 Round to → 53.00 the nearest + 1.00 dollar. _______ $121.00

$89.42 → $89.00 Round to the - 8.94 9.00 nearest dollar. _______ _______ $80.00

Estimate each sum or difference to the nearest whole number. 1. 12.5 + 44.8

2. 8.6 + 11.9

3. 34.32 + 19.51

4. 15.9 + 20.32

5. 32 - 29.75

6. 125.8 - 22.4

7. 159.7 - 124.8

8. 8.890 + 15.98

9. 0.7 + 1.663

10. 52.4 - 21.01

11. 26.55 - 10

12. 2.79 + 5.9 + 0.02

13. 42.1 + 16.25 + 8.96

14. 209.5 - 110

15. 18 - 12.49

Estimate each sum or difference to the nearest dollar. 16. $6.89 + $1.20

17. $5.72 + $4.35

18. $1.68 - $0.99

19. $5.00 - $2.56

20. $20.00 - $15.34

21. $12.86 + $3.33

22. $4.99 + $3.29

23. $50.00 - $39.89

24. $92.30 - $40.00

25. $16.39 - $11.80

26. $84.99 + $5.52

27. $132.62 - $45.81

28. $20.19 + $3.60 + $5.08

29. $4.80 + $7.65 + $2.59

30. $325.44 + $125.10

31. Annual precipitation in Seattle, Washington, is about 37.19 inches. The city of Spokane receives only about 16.49 inches annually. About how much more precipitation does Seattle receive than Spokane? 32. The Adventure Club holds monthly aluminum can recycling drives. During the last three months, they collected $45.45, $45.19, and $44.95 from the drives. About how much did the club collect altogether? 744 Prerequisite Skills

5 Adding and Subtracting Decimals Prerequisite Skills

To add or subtract decimals, write the numbers in a column and line up the decimal points. Then add or subtract as with whole numbers, and bring down the decimal point.

EXAMPLE 1 Find each sum or difference. b. 36.98 - 15.22

a. 8.2 + 3.4

36.98 Line up the - 15.22 decimal points. _______ 21.76 Then subtract.

8.2 Line up the + 3.4 decimal points. _____ 11.6 Then add.

In some cases, you may want to annex, or place zeros at the end of the decimals, to help align the columns. Then add or subtract.

EXAMPLE 2 Find each sum or difference. a. 21.43 + 5.2 21.43 + 5.2 → ______

b. 7 - 1.75 21.43 + 5.20 ______ 26.63

Annex one zero to align the columns.

6 91

7 → - 1.75 ______

7.00 Annex two zeros to - 1.75 align the columns. ______ 5.25

Find each sum or difference. 1.

42.3 + 0.81 ______

2.

5.86 - 1.51 ______

3.

13 - 0.324 _______

4. 2.3 + 1.1

5. 11.5 + 4.2

6. 9.5 - 8.3

7. 24.8 - 3.6

8. 3.57 - 2.17

9. 7.43 - 5.34

10. 6.40 + 7.36

11. 15.20 + 0.16

12. 7.97 - 4.29

13. 8.70 + 0.64

14. 56.88 - 12.35

15. 4.192 + 1.255

16. 14.6 + 20.81

17. 5.2 - 3.01

18. 1.9 - 1.65

19. 6.38 - 1.1

20. 4.86 - 0.3

21. 9.43 + 1.8

22. 70.3 + 7.03

23. 0.5 + 1.674

24. 25 - 8.3

25. 18 - 12.31

26. 2.85 + 23.6

27. 0.8 + 9.612

28. 6.8 + 5.09 + 0.03

29. 0.5 + 2.41 + 6.7

30. 0.563 + 5.8 + 6.89

31. 41.30 + 0.28 + 6.15

32. 4.52 + 0.167 + 12.9

33. 23.4 + 9.865 + 18.26

34. Find the sum of 27.38 and 6.8. 35. Add $26.59, $1.80, and $13. 36. Find the difference of 42.05 and 11.621. 37. How much more than $102.90 is $115? 38. Karen plans to buy a softball for $6.50, a softball glove for $37.99, and sliders for $13.79. Find the cost of these items before tax is added. Prerequisite Skills

745

Prerequisite Skills

6 Estimating Products and Quotients of Decimals You can use rounding to estimate products and quotients of decimals.

EXAMPLE 1 Estimate each product or quotient to the nearest whole number. a. 3.8 × 2.1 3.8 × 2.1 → 4 × 2 = 8 Round 3.8 to 4 and round 2.1 to 2. 3.8 × 2.1 is about 8. b. 16.45 ÷ 3.92 16.45 ÷ 3.92 → 16 ÷ 4 = 4 Round 16.45 to 16 and round 3.92 to 4. 16.45 ÷ 3.92 is about 4. You can use mental math and compatible numbers to estimate products and quotients of decimals. Compatible numbers are rounded so it is easy to compute with them mentally.

EXAMPLE 2 Estimate each product or quotient to the nearest whole number. a. 7 × 98.24 7 × 98.24 → 7 × 100 = 700 Even though 98.24 rounds to 98, 100 is a compatible 7 × 98.24 is about 700.

number because it is easy to mentally compute 7 × 100.

b. 47.5 ÷ 5.23 47.5 ÷ 5.23 → 48 ÷ 6 = 8 47.5 ÷ 5.23 is about 8.

Even though 5.23 rounds to 5, 6 is a compatible number because 48 is divisible by 6.

Rewrite each expression using rounding and compatible numbers. Then estimate each product or quotient. 1. 9.2 × 4.89

2. 6.75 × 5.25

3. 12.19 ÷ 3.8

4. 39.79 ÷ 4.61

5. 11.2 × 6.25

6. 15.2 ÷ 2.7

7. 47.2 ÷ 5.1

8. 16.53 ÷ 8.36

9. 4.32(107.6)

10. 26 × 10.9

11. 73.2 ÷ 6.99

12. 19.1(21.60)

Estimate each product or quotient. 13. 4.6 × 8.3 14. 5.12 × 5.9

15. 7.5 ÷ 4.2

16. 9.27 ÷ 3.31

17. 19.8(2.6)

18. 41.75 ÷ 6

19. 36.24 ÷ 8.7

20. 5.85 × 7.55

21. 8.1 ÷ 2.2

22. 7.9(9.12)

23. 6.1 ÷ 2.1

24. 9 × 96.42

25. 13 × 9.1

26. 10.1 ÷ 4.7

27. 28.6(5)

28. 21 ÷ 7.6

29. 81 ÷ 10.5

30. 52.7÷ 5.3

31. 47.74 × 2

32. 204.5 × 3

33. 41.79 ÷ 7.23

34. The speed of the spine-tailed swift has been measured at 106.25 miles per hour. At that rate, about how far can it travel in 1.8 hours? 746

Prerequisite Skills

7 Multiplying Decimals Prerequisite Skills

You can use decimal models to multiply a decimal by a whole number. Recall that a 10-by-10 grid represents the whole number 1.

EXAMPLE 1 Find 0.3 × 0.6 using base-10 blocks or another decimal model.

ä°Î ä°È

Draw a decimal model, and shade 3 rows yellow to represent 0.3. Then shade 6 columns blue to represent 0.6 There are 18 hundredths in the overlapping green region. So, 0.3 × 0.6 = 0.18.

Sometimes you need to use more than one decimal model to find the product of two decimals.

EXAMPLE 2 Find 0.9 × 1.2 using base-10 blocks or another decimal model. Since 1.2 is greater than 1 but less than 2, you will need two decimal models.

ä°

£°Ó

Shade 9 rows of each model yellow to represent 0.9. Then shade all of one model and 2 columns of the other model blue to represent 1.2.

Cut off the squares that are not shaded green. Then rearrange the remaining squares to form 10-by-10 grids. There are 1 and 8 hundredths in the overlapping green region. So, 0.9 × 1.2 = 1.08.

Prerequisite Skills

747

Prerequisite Skills

To multiply decimals without models, multiply as with whole numbers. The product has the same number of decimal places as the sum of the decimal places of the factors.

EXAMPLE 3 Find each product. a. 6.3(2.1) 6.3 × 2.1 _____ 63 12.60 _____ 13.23

← 1 decimal place ← 1 decimal place ← Annex a zero. ← 2 decimal places

The product is 13.23. b. 9.47(0.5) 9.47 ← 2 decimal places × 0.5 ← 1 decimal place ______ 4.735 ← 3 decimal places The product is 4.735. c. 7.34(0.19) 7.34 × 0.19 ______ 6606 + 7340 _______ 1.3946

← 2 decimal places ← 2 decimal places ← Annex a zero. ← 4 decimal places

The product is 1.3946.

Use base-10 blocks or another decimal model to find each product. Explain how the model shows the product. 1. 0.4 × 0.4

2. 0.5 × 0.7

3. 0.4 × 0.5

4. 0.8 × 0.6

5. 0.9 × 0.9

6. 0.8 × 2.4

7. 0.5 × 1.7

8. 0.4 × 2.1

9. 0.6 × 3.5

Find each product without using models. 10. 1.2(3)

11. 8(3.4)

12. 0.2 × 7.2

13. 1.4(6.1)

14. 0.06 × 3

15. 3.9(8.2)

16. 0.2(3.1)

17. 15.6 × 38

18. 5.7(3.8)

19. 7.07(4)

20. 1.25 × 12

21. 6.5(0.13)

22. 14.9(0.56)

23. 0.47 × 3.01

24. 1.01(6.2)

25. 0.001(7.09)

26. 6.32 × 0.81

27. 0.0001(6.4)

28. Find the product of 13.6 and 9.15. 29. If one United States dollar can be exchanged for 128.46 Spanish pesetas, how many pesetas would you receive for $50? 748

Prerequisite Skills

8 Dividing Decimals Prerequisite Skills

You can use decimal models to divide decimals.

EXAMPLE 1 Find 2.6 ÷ 2 using base-10 blocks or another decimal model. Use two decimal models and 6 tenths to represent 2.6.

Ó°È Separate the models into two equal groups.

Each group contains one whole and 3 tenths. So, 2.6 ÷ 2 = 1.3.

EXAMPLE 2 Find 0.2 ÷ 0.05 using base-10 blocks or another decimal model. Use two tenths to represent 0.2.

ä°Ó Replace the tenths with hundredths since you are dividing by hundredths.

ä°Ó Separate the hundredths into groups of five hundredths to represent dividing by 0.05.

ä°äx ä°äx ä°äx ä°äx

There are four groups of five hundredths in 0.2. So, 0.2 ÷ 0.05 = 4.

Prerequisite Skills

749

Prerequisite Skills

To divide decimals without models, use the following steps. • If necessary, change the divisor to a whole number by moving the decimal point to the right. You are multiplying the divisor by a power of ten. • Move the decimal point in the dividend the same number of places to the right. You are multiplying the dividend by the same power of ten. • Divide as with whole numbers.

EXAMPLES 3 Find 1.20 ÷ 0.8. 1.5 0.8 1.2 0 Move each decimal point right 1 place. 8 ___ 40 4___ 0 0 The quotient is 1.5.

4 Find 32 ÷ 0.25. 128 0.25 32.00 Move each decimal point right 2 places. 25 ___ 70 50 ___ 200 200 ___ 0 The quotient is 128.

Use base-10 blocks or another decimal model to find each quotient. Explain how the model shows the quotient. 1. 4.8 ÷ 2

2. 3.9 ÷ 3

3. 1.2 ÷ 4

4. 0.54 ÷ 3

5. 0.4 ÷ 0.08

6. 0.3 ÷ 0.03

7. 0.8 ÷ 0.04

8. 0.6 ÷ 0.05

9. 0.9 ÷ 0.03

Find each quotient without using models. 10. 0.63 ÷ 0.9

11. 8.4 ÷ 0.4

12. 42 ÷ 0.8

13. 27 ÷ 0.3

14. 64 ÷ 0.4

15. 0.4 ÷ 2

16. 14.4 ÷ 0.16

17. 0.51 ÷ 0.03

18. 62.9 ÷ 100

19. 0.384 ÷ 1.2

20. 4.2 ÷ 1.05

21. 25.9 ÷ 2.8

22. 9 ÷ 0.375

23. 50 ÷ 0.25

24. 500 ÷ 3.2

25. What is the quotient of 72.05 and 0.11? 26. It takes Pluto 247.69 Earth years to revolve once around the Sun. It takes Jupiter 11.86 Earth years to revolve once around the Sun. About how many times longer does it take Pluto than Jupiter to revolve once around the Sun? 750 Prerequisite Skills

9 Estimating Sums and Differences of Fractions and Mixed Numbers Prerequisite Skills

You can use rounding to estimate sums and differences of fractions and mixed numbers. To estimate the sum or difference of proper fractions, round each 1 , or 1. fraction to 0, _ 2

EXAMPLE 1 Estimate each sum or difference. 5 9 +_ a. _

5 3 -_ b. _

8 10 9 1 1 _5 + _ →_ + 1 = 1_ 2 2 10 8 5 9 1 _ _ The sum of and is about 1_ . 8 2 10

6 8 _5 - _3 → 1 - _1 = _1 8 2 2 6 3 5 1 _ _ _ - is about . 8 2 6

To estimate the sum or difference of mixed numbers, round each mixed number 1 . to the nearest whole number or to the nearest _ 2

EXAMPLE 2 Estimate each sum or difference. 3 15 + 15_ a. 3_

8 16 15 3 1 1 _ _ 3 + 15 → 3_ + 16 = 19_ 2 2 16 8 3 15 1 The sum of 3_ and 15_ is about 19_ . 8 2 16

3 1 - 4_ b. 10_ 4

6

1 3 10_ - 4_ → 11 - 4 = 7

6 4 1 3 10_ - 4_ is about 7. 6 4

1 Round each fraction to 0, _ , or 1. 9 1. _

1 2. _ 8

10

2

13 3. _ 25

3 4. _ 14

9 5. _ 15

78 6. _ 81

Estimate each sum or difference. 8 1 7. _ +_

10. 13. 16. 19. 22. 25. 26. 27. 28. 29.

3 47 9. _ +_ 6 90 15 24 15 3 5 7 11. _ + 9_ 12. 1_ +_ 16 4 12 18 8 3 4 2 14. 21_ + 6_ 15. 32_ + 18_ 9 25 56 75 13 9 3 7 17. _ -_ 18. _ -_ 9 8 18 10 3 34 1 1 20. 8_ - 2_ 21. 16_ - 3_ 5 8 35 6 9 4 4 2 _ _ _ 23. 15 + 13 24. 140 - 120_ 9 5 11 15 5 1 minute is 4_ minutes? About how much longer than _ 6 2 3 4 1 + 2_ + 3_ . Estimate the sum 3_ 5 3 10 3 7 inches than 10_ inches? About how much more is 19_ 8 4 3 _ 15 1 _ , 6 4 , 6_ , 7 1 , and 6_ . Estimate the sum of 7_ 3 5 4 10 16 5 inches long is about how much longer than a board that is A board that is 63_ 8 1 _ 9 4 11 4 _ +_ 9 12 10 3 _ 5 +_ 5 11 1 _4 - _ 5 10 3 1 5_ - 2_ 5 4 7 1 _ 35 - 4_ 8 2

5 14 8. _ +_

62 inches long? 4

Prerequisite Skills

751

Prerequisite Skills

10 Estimating Products and Quotients of Fractions and Mixed Numbers You can estimate products and quotients of fractions and mixed numbers using rounding and compatible numbers. Compatible numbers are rounded to make it easy to compute with them mentally.

EXAMPLE 1 Estimate each product or quotient. 5 × 30 a. _ 16

5 1 _ × 30 → _ × 30 3

16

1 _ and 30 are compatible numbers.

5 5 5 1 _ is close to _ and _ = _. 16

15

15

3

3

1 × 30 = 10 Think: _

3 5 _ × 30 is about 10. 16 7 ÷5 b. 9_

8 7 9_ ÷ 5 → 10 ÷ 5 = 2 8 7 9_ ÷ 5 is about 2. 8

Round 9 _ to 10. 10 and 5 are compatible numbers. 7 8

Estimate each product or quotient. 1 · 11 1. _

4. 7. 10. 13. 16. 19.

4 _1 (15) 4 31 _ · 100 40 5 · 30 1_ 6 7 ÷2 5_ 8 7 50 ÷ 4_ 8 _ 79 ÷ 1 9 10

1 2. _ (20)

5. 8. 11. 14. 17. 20.

3 7 _ × 120 15 6 _ × 150 13 1 2_ · 22 4 1 8_ ÷4 4 4 61 ÷ 2_ 5 _ 75 ÷ 2 11 16

1 3. _ × 14 3

11 6. _ (62) 20 1 9. _ (44) 5

4 12. 4_ × 24 5

6 15. 14_ ÷3 7

1 18. 148 ÷ 3_ 4

1 21. 88 ÷ 2_ 8

1 1 batches of cookies. If one recipe calls for 2_ cups of flour, 22. Kim needs 3_ 2 4 about how many cups of flour are needed?

23. Mario wants to place photographs of people in one vertical row on a 3 1 inches long. If each photograph is 2_ inches long, poster board that is 17_ 2 4 about how many photographs can Mario place on the poster board? 1 inches. Estimate the circumference 24. A basketball hoop has a diameter of 18_ 2 of the hoop. (Hint: To estimate the circumference of a circle, multiply the diameter by 3.)

752

Prerequisite Skills

11 Converting Measurements within the Metric System

× 1000

× 100

× 10

kilometer

meter

centimeter

millimeter

km

m

cm

mm

÷ 1000

÷ 100

÷ 10

Comparing Metric and Customary Units of Length 1 mm ≈ 0.04 inch (height of a comma) 1 cm ≈ 0.4 inch (half the width of a penny) 1 m ≈ 1.1 yards (width of a doorway) 1 km ≈ 0.6 mile (length of a city block)

• To convert from larger units to smaller units, multiply. • To convert from smaller units to larger units, divide.

There will be a greater number of smaller units than larger units.

Converting From Larger Converting From Smaller Units to Smaller Units Units to Larger Units 1 km = 1 × 1000 = 1000 m 1 mm = 1 ÷ 10 = 0.1 cm 1 m = 1 × 100 = 100 cm 1 cm = 1 ÷ 100 = 0.01 m 1 cm = 1 × 10 = 10 mm 1 m = 1 ÷ 1000 = 0.001 km

There will be fewer larger units than smaller units.

EXAMPLE 1 Complete each sentence. ? m a. 3 km = _____ 3 × 1000 = 3000 To convert from kilometers to 3 km = 3000 m meters, multiply by 1000. b. 42 mm = _____ ? cm To convert from millimeters to 42 ÷ 10 = 4.2 42 mm = 4.2 cm centimeters, divide by 10.

The basic unit of capacity in the metric system is the liter (L). A liter and milliliter (mL) are related in a manner similar to meter and millimeter. × 1000

1 L = 1000 mL ÷ 1000

Comparing Metric and Customary Units of Capacity 1 mL ≈ 0.03 ounce (drop of water) 1 L ≈ 1 quart (bottle of ketchup)

EXAMPLE 2 Complete each sentence. ? mL a. 2.5 L = _____ 2.5 × 1000 = 2500 To convert from larger units 2.5 L = 2500 mL to smaller units, multiply. b. 860 mL = _____ ? L 860 ÷ 1000 = 0.86 To convert from smaller units 860 mL = 0.86 L to larger units, divide.

Prerequisite Skills

753

Prerequisite Skills

All units of length in the metric system are defined in terms of the meter (m). The diagram below shows the relationships between some common metric units.

Prerequisite Skills

The mass of an object is the amount of matter that it contains. The basic unit of mass in the metric system is the kilogram (kg). Kilogram, gram (g), and milligram (mg) are related in a manner similar to kilometer, meter, and millimeter. 1 kg = 1000 g

Comparing Metric and Customary Units of Mass 1 g ≈ 0.04 ounce (one raisin) 1 kg ≈ 2.2 pounds (textbook)

1g = 1000 mg

EXAMPLE 3 Complete each sentence. ? g a. 3400 mg = _____

? g b. 74.2 kg = _____ To convert from 74.2 × 1000 = 74,200 larger units to smaller units, 74.2 kg = 74,200 g

3400 ÷ 1000 = 3.4 To convert from smaller units to larger 3400 mg = 3.4 g

multiply.

units, divide.

State which metric unit you would probably use to measure each item. 1. amount of water in a pitcher

2. distance between two cities

3. thickness of a coin

4. amount of water in a medicine dropper

5. length of a textbook

6. mass of a pencil

7. length of a football field

8. width of a quarter

9. thickness of a pencil

10. gas in the tank of a car

11. vanilla used in a cookie recipe

12. mass of a table tennis ball

13. bag of sugar

14. mass of a horse

Complete each sentence. ? m 15. 5 km = _____

16. 3.5 cm = _____ ? mm

? mL 17. 6 L = _____

18. 370 mL = _____ ? L

? cm 19. 20 mm = _____

20. 4000 g = _____ ? kg

? mm 21. 18 cm = _____

22. 0.75 L = _____ ? mL

? m 23. 935 cm = _____

24. 210 mm = _____ ? cm

? kg 25. 65 g = _____

26. 2 m = _____ ? cm

? g 27. 52.9 kg = _____

28. 800 m = _____ ? km

? g 29. 9.05 kg = _____

30. 0.62 km = _____ ? m

? L 31. 1250 mL = _____

32. 20,000 mg = _____ ? g

? km 33. 3100 m = _____

34. 2.6 m = _____ ? cm

? g 35. 36 mg = _____

36. 7 mm = _____ ? cm

? mL 37. 0.085 L = _____

38. 125.9 g = _____ ? kg

39. The mass of a sample of rocks is 1.56 kilograms. How many grams are in 1.56 kilograms? 40. How many milliliters are in 0.09 liter? 41. Runners often participate in races that are 10 kilometers long. How many meters are in 10 kilometers? 42. How many centimeters are in 0.58 meter? 43. A can holds 355 milliliters of soft drink. How many liters is this? 754

Prerequisite Skills

12 Converting Measurements within the Customary System Customary Units of Length 1 foot (ft) = 12 inches (in.) 1 yard (yd) = 3 feet 1 mile (mi) = 5280 feet

• To convert from larger units to smaller units, multiply. • To convert from smaller units to larger units, divide. Larger Units

Smaller Units

→

5 ft = 5 × 12 4 yd = 4 × 3

= 60 in. = 12 ft

There will be a greater number of smaller units than larger units.

Smaller Units

→

Prerequisite Skills

The units of length in the customary system are inch, foot, yard, and mile. The table at the right shows the relationships among these units.

Larger Units

24 in. = 24 ÷ 12 = 2 ft 15 ft = 15 ÷ 3 = 5 yd There will be fewer larger units than smaller units.

EXAMPLE 1 Complete each sentence. ? ft a. 1.5 mi = _____ 1.5 × 5280 = 7920 To convert from miles to feet, multiply by 5280. 1.5 mi = 7920 ft b. 120 in. = _____ ? ft 120 ÷ 12 = 10 To convert from inches to feet, 120 in. = 10 ft divide by 12. The units of weight in the customary system are ounce, pound, and ton. The table at the right shows the relationships among these units.

Customary Units of Weight 1 pound (lb) = 16 ounces (oz) 1 ton (T) = 2000 pounds

• To convert from larger units to smaller units, multiply. • To convert from smaller units to larger units, divide. Larger Units

→

Smaller Units

3 T = 3 × 2000 = 6000 lb 2 lb = 2 × 16 = 32 oz

Smaller Units

→

Larger Units

84 oz = 48 ÷ 16 = 3 lb 4000 lb = 4000 ÷ 2000 = 2 T

EXAMPLE 2 Complete each sentence. ? lb a. 120 oz = _____ 120 ÷ 16 = 7.5 To convert from smaller units 120 oz = 7.5 lb to larger units, divide. b. 4 T = _____ ? lb 4 × 2000 = 8000 To convert from larger units to smaller units, multiply. 4 T = 8000 lb

Prerequisite Skills

755

Prerequisite Skills

Capacity is the amount of liquid or dry substance a container can hold. Customary units of capacity are fluid ounce, cup, pint, quart, and gallon. The relationships among these units are shown in the table. As with units of length and units of weight, to convert from larger units to smaller units, multiply. To convert from smaller units to larger units, divide.

Customary Units of Capacity 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 cups 1 quart (qt) = 2 pints 1 gallon (gal) = 4 quarts

EXAMPLE 3 Complete each sentence. ? qt a. 3 gal = _____ 3 × 4 = 12 larger unit → smaller unit 3 gal = 12 qt b. 2 c = _____ ? fl oz 2 × 8 = 16 larger unit → smaller unit 2 c = 16 fl oz c. 12 pt = _____ ? qt 12 ÷ 2 = 6 smaller unit → larger unit 12 pt = 6 qt d. 8 c = _____ ? qt 8 ÷ 2 = 4 First, convert cups to pints. 8 c = 4 pt 4 ÷ 2 = 2 Next, convert pints to quarts. 4 pt = 2 qt So, 8 c = 2 qt.

Complete each sentence. ? in. 1. 5 ft = _____

2. 2 gal = _____ ? qt

3. 96 oz = _____ ? lb

4. 2 T = _____ ? lb

5. 9 ft = _____ ? yd

6. 6 c = _____ ? pt

7. 2 mi = _____ ? ft

8. 72 in. = _____ ? ft

9. 3 lb = _____ ? oz

10. 7 yd = _____ ? ft

11. 32 fl oz = _____ ? c

12. 15,840 ft = _____ ? mi

? pt 13. 2 qt = _____

14. 5 pt = _____ ? c

15. 16 qt = _____ ? gal

? T 16. 3000 lb = _____

17. 6 pt = _____ ? qt

18. 8 pt = _____ ? c

19. 14 pt = _____ ? qt

20. 8 yd = _____ ? ft

21. 5 gal = _____ ? qt

? gal 22. 36 qt = _____

23. 5 c = _____ ? fl oz

24. 120 in. = _____ ? ft

25. 30 in. = _____ ? ft

26. 6.5 lb = _____ ? oz

27. 12 oz = _____ ? lb

Solve each problem by breaking it into simpler parts. 28. How many inches are in a yard? 29. How many ounces are in a ton? 30. How many cups are in a gallon? 756 Prerequisite Skills

13 Measuring and Drawing Angles Prerequisite Skills

A line is a never-ending straight path extending in two directions. A ray is part of a line. It has one endpoint and extends infinitely in one direction. A ray is named using the endpoint first, then another point on the ray. Rays AP, AQ, AM, and AN are shown.

P Q

The symbol for ray AM is AM.

The symbol for line NP is NP.

A N M

A

side

Two rays that have the same endpoint form an angle. The common endpoint is called the vertex, and the two rays that make up the angle are called the sides of the angle.

vertex

B

1 side

C

The symbol ∠ represents angle. There are several ways to name the angle shown above. • Use a vertex and a point from each side. ∠ABC or ∠CBA The vertex is always the middle letter. • Use the vertex only. ∠B • Use a number. ∠1 The most common unit of measure for angles is the degree(°). You can use a protractor to measure angles in degrees.

EXAMPLE

Measure Angles

1 Use a protractor to measure ∠CDE. C 70 60

110

0 12

50

80

90

100 80

100

110 70

12 0

0

15 0

30

20

160

0

180

$

E

180

0

170

10

#

160

20

10

30

0 15

14

40

D

Step 2 Use the scale that begins with 0° at DE . Read where the other side , crosses this scale. of the angle, DC

0

50

0

40

13

60

0 13

14

Place the center point of the protractor’s base on vertex D. Align the straight side with side DE so that the marker for 0° is on the ray.

170

Step 1

%

The measure of angle CDE is 120°. Using symbols, m∠CDE = 120°. Acute angles have measures less than 90°. Right angles have measures equal to 90°. Obtuse angles have measures between 90° and 180°. Straight angles have measures equal to 180°.

Prerequisite Skills

757

EXAMPLE

Draw Angles

2 Draw ∠X having a measure of 85°.

X

Step 1 Draw a ray with endpoint X. 85˚ 70 60 0

50

90

100

110

80

100

70

12 1

0 14

15 0

30

20

160 170

10 0

0

8. ∠WZU

180

7. ∠UZV

10

6. ∠XZT

X

170

5. ∠TZW

20

4. ∠UZX

160

3. ∠SZY

0

30

2. ∠SZT

13

0 15

Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or straight. 1. ∠XZY

0

40

Step 3 Use the scale that begins with 0. Locate the mark labeled 85. Then draw the other side of the angle.

12 60

50

30

0

40

110

80

14

Step 2 Place the center point of the protractor on X. Align the mark labeled 0 with the ray.

180

Prerequisite Skills

Protractors can also be used to draw an angle of a given measure.

V U

W X

T

Z

S

Y

Use a protractor to find the measure of each angle. 9.

10.

11.

Use a protractor to draw an angle having each measurement. Then classify each angle. 12. 40°

13. 70°

14. 65°

15. 85°

16. 95°

17. 110°

18. 155°

19. 140°

20. 38°

For Exercises 21 and 22, use the graphic. 21. Find the measure of each angle of the circle graph to the nearest degree. 22. Suppose 500 adults were surveyed. How many would you expect to spend between $250 and $349?

"ACK TO SCHOOL CLOTHES fÓ{ À iÃÃ {¯

fÓxä fÎ{ Ó£¯ £ä¯ fÎxä À Ài Óä¯

-ÕÀVi\ (ARRIS )NTERACTIVE FOR 3EARS 2OEBUCK

758

Prerequisite Skills

SPENDING PER CHILD

½Ì Ü

14 Displaying Data in Graphs

• Bar graphs are used to compare the frequency of data. The bar graph below compares the amounts of recycled materials.

• Double bar graphs compare two sets of data. The double bar graph below shows movie preferences for men and women.

0 H

er O

or

th

ro

r

a m

io

y ed m Co

Resource

ra

l ee

10

St

Pa pe A r lu m in um

tic as Pl

G

la

ss

0

n

3.8

0.5

Men Women

ct

3.0

20

50 40 30 20

D

63.6 35.1

A

80 60 40

Favorite Movies Number of People

Recycling (millions of tons)

Recycling Resources

Type of Movie

Source: Bureau of Mines

• Line graphs usually show how values change over a period of time. The line graph at the right shows the results of the women’s Olympic high jump event from 1972 to 2004.

7i½Ã "Þ«V } Õ«

i}Ì ®

Ó°äÈ Ó°ä{

Ó°äÓ Ó°ää

£°n £°È £°{

£°Ó ä ½ÇÓ

½nä

½nn 9i>À

½È

½ä{

3OURCE 4HE 7ORLD !LMANAC

• Double line graphs, like double bar graphs, show two sets of data. The double line graph below compares the number of boys and the number of girls participating in high school athletics.

ÕLiÀ v *>ÞiÀÃ Ã®

} -V ÌiÌVÃ {°x {°ä Î°x Î°ä

"OYS

'IRLS

Ó°x Ó°ä ä ½Îq½{ ½{q½x ½xq½È ½Èq½Ç ½Çq½n ½nq½ ½q½ää ½ääq½ä£ ½ä£q½äÓ ½äÓq½äÎ ½äÎq½ä{ ½ä{q½äx

-V 9i>À 3OURCE "ASED ON .ATIONAL &EDERATION OF 3TATE (IGH 3CHOOL STATISTICS

Prerequisite Skills

759

Prerequisite Skills

Statistics involves collecting, analyzing, and presenting information. The information that is collected is called data. Displaying data in graphs makes it easier to visualize the data.

Prerequisite Skills

• Circle graphs show how parts are related to the whole. The circle graph at the right shows how electricity is generated in the United States.

• Line plots organize data using a number line. The line plot below shows the number of Calories in a single serving of different brands of yogurt.

How America Powers Up Gas 9.0%

Nuclear 21.2%

Hydropower 9.3%

£ää

Coal 56.9%

Oil 3.5%

£xä

Óää

Óxä

Other 0.1% Source: Energy Information Administration

EXAMPLE A newspaper wants to display the high temperature of the past week. Should they use a line graph, circle graph, or double bar graph? Since the data would show how values change over a period of time, a line graph would give the reader a clear picture of what temperatures were and the changes in temperature.

Determine whether a bar graph, double bar graph, line graph, double line graph, circle graph, or line plot is the best way to display each of the following sets of data. Explain your reasoning. 1. the number of people who have different kinds of pets 2. the percent of students in class who have 0, 1, 2, 3, or more than 3 siblings 3. the number of teens who attended art museums, symphony concerts, rock concerts, and athletic events in 1990 compared to the number who attended the same events this year 4. the minimum wage every year from 1980 to the present 5. the number of boys and the number of girls participating in volunteer programs each year from 1995 to the present 6. The table below shows the number of events at recent Olympic games. Would the data be best displayed using a line graph, circle graph, or double bar graph? Explain your reasoning. Olympic Year

1968

1972

1976

1980

1984

1988

1992

1996

2000

2004

Number of Events

172

196

199

200

223

237

257

271

300

296

7. The prices of lawn seat tickets at events held at an amphitheater are shown in the chart at the right. Would the data be best displayed using a line graph, circle graph, or line plot? Explain your reasoning.

760

Prerequisite Skills

,AWN 3EAT 0RICES

Extra Practice Lesson 1-1

Pages 26–30

Solve. 1. POSTAL SERVICE The U.S. Postal Service offers air mail service to Canada. The rates for Canadian Air Mail letters and packages are shown in the table at the right. Determine the air mail rate for a package that weighs 6.5 ounces. a. Write the Explore step. What do you know and what do you need to find?

Weight not over (ounces) 1.0 2.0 3.0 4.0

Rate $0.60 $0.85 $1.10 $1.35

Extra Practice

b. Write the Plan step. What strategy will you use? What do you estimate the answer to be? c. Solve the problem using your plan. What is your answer? d. Check your solution. Is it reasonable? Does it answer the question? 2. POSTAL SERVICE In 1995, the state of Florida celebrated the 150th anniversary of its statehood. The U.S. Postal Service issued a stamp, the first to bear the 32-cent price, to honor the occasion. Ninety million of the commemorative stamps were issued. About how much postage did the stamps represent? a. Which method of computation do you think is most appropriate for this problem? Justify your choice. b. Solve the problem using the four-step plan. Be sure to examine your solution. Find the next term in each list. 3. 3, 8, 13, 18, 23, …

4. 32, 29, 26, 23, 20, …

5. 6, 7, 9, 12, 16, …

Lesson 1-2

Pages 32–36

Find the value of each expression. 1. 8 + 7 + 12 ÷ 4

2. 20 ÷ 4 - 5 + 12

3. (25 · 3) + (10 · 3)

4. 36 ÷ 6 + 7 - 6

5. 30 · (6 - 4)

6. (40 · 2) - (6 · 11)

86 - 11 7. __ 11 + 4

12 + 84 8. __ 11 + 13

5·5+5 9. __

10. (19 - 8)4

5 · 5 - 15

11. 75 - 5(2 · 6)

12. 81 ÷ 27 × 6 - 2

Write a numerical expression for each verbal phrase. 13. three increased by nine

14. fifteen divided by three 15. six less than ten

Lesson 1-3

Pages 37– 41

ALGEBRA Evaluate each expression if a = 2, b = 4, and c = 3. 1. ba - ac 2. 4b + a · a 3. 11 · c - ab 5. 7(a + b) - c 9(b + a) 9. __ c-1

6. 8a + 8b 10. abc - bc

8(a + b) 7. __ 4c

11. 28 - bc + a

4. 4b - (a + c) 8. 36 - 12c 12. a(b - c)

ALGEBRA Translate each phrase into an algebraic expression. 13. nine more than a

14. eleven less than k

15. three times p

16. the product of some number and five

17. twice Shelly’s score decreased by 18

18. the quotient of 16 and n Extra Practice

761

Lesson 1-4

Pages 43–47

Name the property shown by each statement. 1. 1 · 4 = 4

2. 6 + (b + 2) = (6 + b) + 2

3. 9(6n) = (9 · 6)n

4. 8t · 0 = 0 · 8t

5. 0(13n) = 0

6. 7 + t = t + 7

MENTAL MATH Find each sum or product. Explain your reasoning. 7. 6 + 8 + 14 10. 8 + 4 + 12 + 16

8. 5 · 18 · 2

9. 0(13 · 6)

11. 8 · 20 · 10

12. 4 · 14 · 5

ALGEBRA Simplify each expression.

Extra Practice

13. (12 + x) + 9

14. 2 · (6 · x)

15. (5 · m) · 3

Lesson 1-5

Pages 49–53

ALGEBRA Find the solution of each equation from the list given. 1. 16 - f = 11; 3, 5, 7

72 2. 9 = _ m ; 8, 9, 11

3. 4b + 1 = 17; 3, 4, 5

4. 17 + r = 25; 6, 7, 8

5. 9 = 7n - 12; 3, 5, 7

6. 67 = 98 - q; 21, 26, 31

7. 9z = 45; 5, 7, 9

8. 88 = 11d; 7, 8, 9

9. 5t = 0; 0, 5, 10

ALGEBRA Define a variable. Then write an equation and solve. 10. Six added to a number is 23. 11. Thirteen times a number is 39. 12. 84 divided by a number is 12. 13. Thirteen less a number is 7. 14. The sum of a number and 8 is 14. 15. Twelve less than a number is 50. 16. The product of a number and ten is seventy. 17. A number divided by three is nine.

Lesson 1-6

Pages 54–59

Use the grid at the right to name the point for each ordered pair.

D

1. (9, 7)

2. (5, 5)

3. (3, 1)

4. (2, 7)

5. (8, 4)

6. (4, 0)

Refer to the coordinate system shown at the right. Write the ordered pair that names each point. 7. R

8. P

9. W

10. C

11. D

12. F

y

762 Extra Practice

14. {(2, 1), (4, 4), (6, 7), (4, 3)}

P

C N S

F Q O

Express each relation as a table and as a graph. Then determine the domain and range. 13. {(3, 6), (4, 9), (5, 1)}

R B

W T

x

Lesson 1-7

Pages 61–66

Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. 1. speed of airplane and miles traveled in three hours 2. weight and shoe size 3. outside temperature and heating bill GAMES For Exercises 4–6, use the following information. The number of pieces in a jigsaw puzzle and the number of minutes required for a person to complete it is shown below. 60 20

500 175

750 1000 800 315 395 270

75 25

Extra Practice

Number of Pieces 100 35 Time (min)

4. Make a scatter plot of the data. 5. Does the scatter plot show any relationship? If so, is it positive or negative? Explain your reasoning. 6. Suppose Dave purchases a puzzle having 650 pieces. Predict the length of time it will take him to complete the puzzle.

Lesson 2-1 Replace each

Pages 78–83


1. -4

-8

2. -6

5. 12

-25

6. 3

9. 5

-7

10. ⎪6⎥

3

3. 0

-5

4. -12

-9

-7

7. 0

-2

8. -15

12

11. -2

-2

12. ⎪-7⎥

⎪-3⎥

⎪-4⎥

Order the integers in each set from least to greatest. 13. {-1, 2, -5}

14. {0, -2, 8, 5, -9}

15. {100, -34, -86, 21, 0}

16. {-1, 16, -43, 8, 27, -40}

17. {0, -23, 75, -15, 24}

18. {-6, 6, -5, 18}

Evaluate each expression. 19. ⎪-3⎥ + ⎪9⎥

20. ⎪-18⎥ - ⎪5⎥

21. ⎪12 + 7⎥

22. -⎪6⎥

23. ⎪-8⎥ + ⎪4⎥

24. -⎪-20⎥

25. ⎪15 - 12⎥

26. ⎪8 + 9⎥

27. -⎪4⎥ · ⎪-5⎥

28. ⎪-6⎥ · ⎪8⎥

29. -⎪12⎥ · ⎪9⎥

30. -⎪⎪-16⎥ + ⎪-22⎥⎥

Lesson 2-2

Pages 86–90

Find each sum. 1. 5 + (-6)

2. -17 + 24

3. 15 + (-29)

4. -6 + 13

5. 50 + (-14)

6. -21 + (-4)

7. 30 + (-7)

8. (-3) + (-10)

9. -15 + 26

10. -17 + 4 + (-2)

11. 50 + (-16) + (-11)

12. -17 + 8 + (-14)

13. -11 + 15 + (-6)

14. 23 + (-64)

15. -1 + 14 + (-13)

16. 33 + (-18) + 7

17. -75 + (-13)

18. 26 + 14 + (-71)

19. 8 + (-9) + (-1)

20. -16 + (-12) + 13

21. 35 + (-60)

22. 12 + (-20) + 16

23. 100 + (-54) + (-17)

24. 11 + (-22) + (-33) Extra Practice

763

Lesson 2-3

Pages 93–97

Extra Practice

Find each difference. 1. 8 - 17

2. -15 - 3

3. 10 - 21

4. 20 - (-5)

5. 5 - (-9)

6. -12 - (-7)

7. -19 - (-6)

8. -16 - (-23)

9. -56 - 32

10. -49 - (-52)

11. -6 - 9 - (-7)

12. -6 - (-10) -7

13. 17 - 33

14. -21 - 19

15. 12 - (-24)

16. -35 - (-18)

17. -54 - 27

18. 32 - (-18)

19. -26 - (-41)

20. 99 - (-1)

21. -12 - (-25)

ALGEBRA Evaluate each expression if x = 6, y = -8, z = -3, and w = 4. 22. y - z

23. 3 - z

24. y - 5

25. x - y

26. 14 - y - x

27. 6 + x - z

28. y + z + w

29. w - z + 11

Lesson 2-4

Pages 100–104

Find each product. 1. -4(2)

2. -8(-5)

3. 13(-4)

4. -5 · 6 · 10

5. -6(-2)(-14)

6. 18(-3)(6)

7. 4(-10)(-3)

8. -9(3)(2)

9. 12(-8)


11. 7(-8m)

12. -10(-3k)

13. -4y(-8z)

14. (-2r)(-3s)

15. 6(-2m)(3n)

ALGEBRA Evaluate each expression. 16. -6t, if t = 15

17. 7p, if p = -9

18. -4k, if k = -16

19. aw, if a = 0 and w = -72 20. dk, if d = -12 and k = 11 21. st, if s = -8 and t = -10 22. 3hp, if h = 9 and p = -3 23. -5bc, if b = -6 and c = 2 24. -4wx, if w = -1 and x = -8

Lesson 2-5

Pages 106–110

Find each quotient. 1. -36 ÷ 9

2. 112 ÷ (-8)

3. -72 ÷ 2

4. -26 ÷ (-13)

5. -144 ÷ 6

6. -180 ÷ (-10)

7. 304 ÷ (-8)

8. -216 ÷ (-9)

9. 80 ÷ (-5)

10. -105 ÷ 15

11. 120 ÷ (-30)

12. -200 ÷ (-8)

13. 42 ÷ (-6)

14. 144 ÷ (-12)

15. -360 ÷ 9

16. -84 ÷ -6

17. 125 ÷ (-5)

18. 180 ÷ (-15)

19. -400 ÷ 20

20. 72 ÷ (-9)

21. -156 ÷ (-2)

ALGEBRA Evaluate each expression if x = -5, y = -3, z = 2, and w = 7. 22. 25 ÷ x

23. -42 ÷ w

24. 3 ÷ y

25. 2x ÷ z

26. -3x ÷ y

27. x ÷ (-1)

28. xyz ÷ 10

29. yz ÷ 2

3y 30. _ -3

6-y 31. _ y

w 32. _ -7

w-x 33. _ y

764

Extra Practice

Lesson 2-6

Pages 111–115

Name the point for each ordered pair graphed at the right. 1. (-6, 8)

2. (1, -2)

3. (9, 2)

4. (1, 4)

5. (-3, -4)

6. (2, 5)

7. (3, 0)

8. (5, -1)

Graph and label each point on a coordinate plane. Name the quadrant in which each point is located.

E

N L G K x

O

J

M

F B

10. P(1, 5)

11. R(-3, 1)

12. M(4, -2)

13. K(-4, 5)

14. G(3, -5)

Lesson 3-1

C

Extra Practice

9. H(-2, -5)

y

D

A

Pages 124–128

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. 1. 2(4 + 5)

2. 4(5 + 3)

3. 3(7 - 6)

4. (2 + 5)9

5. (10 - 4)3

6. -6(1 + 3)

ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 7. 3(m + 4)

8. (y + 7)5

9. -6(x + 3)

10. (p - 4)5

11. -3(s - 9)

12. 5(x + y)

13. b(c + 3d)

14. (a - b)(-5)

15. -6(v - 3w)

16. 5(x + 12)

17. (m - 6)(4)

18. -2(a - b)

19. (8 - m)(-3)

20. 8(p - 3q)

21. (2x + 3y)(4)

Lesson 3-2

Pages 129–133

Identify the terms, like terms, coefficients, and constants in each expression. 1. 3 + 4x + x

2. 5n + 2 - 3n

3. 6 + 1 + 7y

4. 2c + c + 8d

5. 3a - 9 + b

6. 2 + 6k + 7 - 5k

ALGEBRA Simplify each expression. 7. 8k + 2k + 7

8. 3 + 2b + b

9. t + 2t

10. 9(3 + 2x)

11. 4(y + 2) - 2

12. (6 + 3e)4

13. 4 + 9c + 3(c + 2)

14. 5(7 + 2s) + 3(s + 4)

15. 9(f + 2) + 14f

16. 5a - 9a

17. -6 + 4x + 9 - 2x

18. 6a + 11 + (-15) + 9a

19. 2(8w - 7)

20. 3(2d + 5) + 4d

21. 2 + 4p - 6(p - 2)

22. -3(b + 4)

23. -6 + 3s + 11 - 5s

24. 3(x - 5) + 7(x + 2)

25. 3q - r + q + 6r

26. 8(r + 1) + 7

27. 3p - 2(p + 6q)

28. a + 2b + 4a

29. 9x - 12 + 12

30. 1 + g + 5g - 2 Extra Practice

765

Lesson 3-3

Pages 136–140

Extra Practice

ALGEBRA Solve each equation. Check your solution. 1. y + 49 = 26

2. d + 31 = -24

3. q - 8 = 16

4. x - 16 = 32

5. 40 = a + 12

6. b + 12 = -1

7. 21 = u + 6

8. -52 = p + 5

9. -14 = 5 - g

10. 121 = k + (-12)

11. -234 = m - 94

12. 110 = x + 25

13. f - 7 = 84

14. y - 864 = 652

15. 475 + z = -18

16. x + 12 = -9

17. 15 - h = 11

18. 16 = p + 21

19. -13 + t = -2

20. 86 = x + 43

21. y - 11 = -14

ALGEBRA Write and solve an equation to find each number. 22. The sum of -6 and a number is 8. 23. When 3 is subtracted from a number, the result is -5. 24. When 7 is added to a number, the result is -9. 25. When a number is decreased by 8, the result is 5.

Lesson 3-4

Pages 141–145

ALGEBRA Solve each equation. Check your solution. 1. -y = -32

2. 7r = -56

t = 12 3. _

s 4. 4 = _

b 5. _ = -2

6. 64 = -4n

7. -144 = 12q

r 8. _ = -12

9. -5g = -385

10. -16x = -176

y 11. -21 = _ -4

-14

k 13. 84 = _ 5

-3

47 11

12. -372 = 31k

14. -b = 19

v 15. _ = -9

16. -3x = -27

p 17. _ = 4

18. 5q = -100

d 19. _ = -8

20. -9n = -45

21. 125 = -25z

11

112

-12

ALGEBRA Write and solve an equation for each sentence. 22. The product of 8 and a number is -40. 23. The quotient of a number and -3 is 27. 24. When 6 is multiplied by a number, the result is -24.

Lesson 3-5

Pages 147–151

ALGEBRA Solve each equation. Check your solution. 1. 3t - 13 = 2

2. -8j - 7 = 57

3. 9d - 5 = 4

4. 6 - 3w = -27

k + 8 = 12 5. _

6. -4 = _ - 19

n 7. 15 - _ = 13

8. 44 = -4 + 8p

9. 21 - h = -32

7

6

q 8

10. -19 = 11b - (-3)

x 11. 6 = 20 + _

12. 9 + 3a = -3

13. 2x - 8 = 10

m 14. _ - 6 = 10

15. -12 + 3p = 3

16. -18 = 6a - 6 k 19. 16 = _ - 11 3

766

Extra Practice

3

4 _ 17. t + 11 = 23 -3

20. -6g - 12 = -60

18. 3 + 2v = 11 21. 15 - 4c = -21

Lesson 3-6

Pages 153–157

Translate each sentence into an equation. Then find the number. 1. Five less than three times a number is 13. 2. The product of 2 and a number is increased by 9. The result is 17. 3. Ten more than four times a number is 46. 4. The quotient of a number and -8, less 5 is -2. 5. Three more than two times a number is 11. 6. The quotient of a number and six, increased by 2 is -5. 7. The product of -3 and a number, decreased by 9 is 27. 8. A number is divided by 2. The sum of the result and 6 is -2.

Extra Practice

9. The sum of 6 and a number divided by 3 is 7.

Lesson 3-7

Pages 158–161

Describe each sequence using words and symbols. 1. 25, 50, 75, …

2. 39, 40, 41, …

3. 4, 8, 12, …

4. 1.5, 2.5, 3.5, …

1 1 , 1, 1_ ,… 5. _ 2 2

6. 4, 7, 10, …

Write an equation that describes each sequence. Then find the indicated term. 7. 5, 10, 15, …; 23rd term

8. 18, 17, 16, …; 13th term

9. 2, 1, 0, …; 7th term

10. 3, 5, 7, …; 99th term

11. 6, 13, 20, …; 38th term

12. -6, -4, -2, …; 67th term

Lesson 3-8

Pages 162–167

Find the perimeter and area of each rectangle. 1. a rectangle 23 centimeters long and 9 centimeters wide 2. a 16-foot by 14-foot rectangle 3. a rectangle with a length of 31 meters and a width of 3 meters 4. a square with sides 7 meters long Find the missing dimension of each rectangle. Length

5.

9 ft

6. 7. 8. 9.

Width 18 in.

13 yd 12 cm 3m

Area

Perimeter

126

ft2

46 ft

108

in2

48 in.

273

yd2

68 yd

168

cm2

52 cm

162

m2

114 m

10. The perimeter of a rectangle is 50 meters. Its width is 10 meters. Find the length. 11. The area of a rectangle is 96 square inches. Its length is 12 inches. Find the width. Extra Practice

767

Lesson 4-1

Pages 180–184

ALGEBRA Write each expression using exponents. 2. 9

3. (-6)(-6)(-6)(-6)(-6)

4. (y · y · y) · (y · y · y · y)

5. a · b · b

6. 4 · 4 · 4 · 4 · x · x · x · y

7. 3q · 3q · 3q · 3q · 3q · 3q

8. n · n · n · … · n

9. (x + y)(x + y)

⎧ ⎨ ⎩

1. 8 · 8 · 8 · 8

17 factors

Extra Practice

ALGEBRA Evaluate each expression if m = 3, n = 2, and p = -4. 10. 3m2

11. n0 + m

12. 74

13. -53

14. p3

15. 2(m - p)2

16. -2n3 + m

17. m - p2

18. (m + n + p)3

19. 5p - m2

20. (n + p)4

21. (m - n)8

Lesson 4-2

Pages 186–190


2. 369

3. 116

4. 125

5. 83

6. 99

7. 91

8. 79


10. 44

11. 51

12. 65

13. 30

14. 28

15. 117

16. 88

17. 54

18. 32

19. 300

20. 210

ALGEBRA Factor each monomial. 21. 40y

22. 630a

23. 187c2

24. 310p2

25. 510xy

26. 1589cd

27.

-18ab2

28.

-117x3

29. 105j2k5

Lesson 4-3

Pages 191–195

ALGEBRA Find the GCF of each set of numbers or monomials. 1. 27, 45

2. 30, 12

3. 16, 40, 28

4. 18, 17, 15

5. 112, 216

6. 120, 245

8. 135ab,171b

9. 185fg, 74f 2g

7. 84k,

108k2

10. 44m, 60n 13. 16w,

28w3

11. 90gh, 225k 14. 24a, 30ab,

66a2

12. 8, 28h 15. 13z, 39yz, 52y

ALGEBRA Factor each expression. 16. 3m + 12

17. 5x + 15

18. 4 + 8b

19. 7x + 21

20. 2a + 100

21. 42 - 14b

22. 5f - 25

23. 11p - 66

24. 7y - 21

25. 48 + 12s

26. 18 - 2w

27. 24k + 96

28. 2y + 14

29. 42 - 7b

30. 13w + 39

768

Extra Practice

Lesson 4-4

Pages 196–200


4. 7. 10. 13.

19.

3 2. _

6 3. _

5.

20 6. _ 49

8. 11. 14. 17.

58

16 10 _ 90 99 _ 9 40 _ 76 42 _ 49 16p _ 24p

18 9. _

54 49 12. _ 56 110 15. _ 200 21x2y 18. _ 81y

Extra Practice

16.

54 15 _ 55 8 _ 20 21 _ 64 22 _ 66 b _ b4 32d2 _ 6d

120z3x 21. __

72ab 20. _

18zx

8b

22. Fourteen inches is what part of 1 yard? 23. Nine hours is what part of one day?

Lesson 4-5

Pages 203–207

ALGEBRA Find each product or quotient. Express using exponents. 1. r4 · r2

29 2. _ 3

b18 3. _ 5

4. 123 · 128

5. x · x9

6. (2s6)(4s2)

7. w3 · w4 · w2

8. (-2)2(-2)5(-2)

47 9. _ 6

2

b

10. 3(f 17)(f 2)

11. (5k)2 · k7

13. (3x4)(-6x)

14. (4k 4)(-3k)3

4 6m8 12. _ 3m2 42 15. _ -6

( )( g

Lesson 4-6

10

)

g _ 3

Pages 209–213

ALGEBRA Write each expression using a positive exponent. 1. y-9

2. m-4

3. 5-3

4. 2-7

5. 6-3

6. a-11


p _ 11. 1 2 15

1 8. _ 9

1 9. _ 3

b _ 12. 1 25

1 10. _ 4

5 _ 13. 17 c

7

1 14. _ 64

Write each decimal using a negative exponent. 15. 0.01

16. 0.00001

17. 0.0001

18. 0.001

19. 0.1

20. 0.000001

Evaluate each expression if x = 3 and y = -2. y

21. x-2

22. 9

23. y-3

24. x-3

25. y-4

26. (xy)-2 Extra Practice

769

Lesson 4-7

Pages 214–218

Express each number in scientific notation. 1. 9040

2. 0.015

3. 6,180,000

4. 27,210,000

5. 0.00004637

6. 0.00546

7. 500,300,100

8. -0.0000032

9. 0.00047

10. 10,471,300

Extra Practice

Express each number in standard form. 11. -9.5 × 10-3

12. 8.245 × 10-4

13. 8.2 × 104

14. -9.102040 × 102

15. 4.02 × 103

16. 1.6 × 10-2

17. 2.41023 × 106

18. 4.21 × 10-5

19. 1.0012 × 10-3

20. 8.604 × 102

Lesson 5-1

Pages 228–233

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. 6 1. _

4 2. _

1 3. -_

3 4. 1_

3 10. -_

7 7. -4_ 12 36 11. 8_

8 8. _ 11 6 12. _

25 _ 6. 9 20

10 _ 5. 5 6 4 9. 3_ 18

8

44

16

Replace each

14. 0.04

1 17. _

0.75

18. 0.3

_1

_5

− 22. 2.1

1 2_

− 21. 0.5

9

15


5 7 _ 13. _ 8 6 2

4

_5 9

3

10

1 15. _

_2 3 7 2 19. _ 0.64 3 7 23. 3_ 3.78 8

Lesson 5-2

3 16. _

12 _ 5 20 2 20. _ 0.10 20 6 5 24. -_ -_ 7 6

Pages 234–238

Write each number as a fraction. 4 1. 3_

2 2. -1_

3. 15

3 4. 2_

5. -13

6 6. 2_ 7

7. 36

3 8. -1_

5

9

8

5

Write each decimal as a fraction or mixed number in simplest form. 9. 0.6 13. 0.375

10. 0.05

11. 0.38

14. -3.24

15. 0.222…

12. 4.12 − 16. -0.4

Identify all sets to which each number belongs. 2 17. -4_

18. 6

21. 5.9

3 22. -_

5

770 Extra Practice

1 19. 3_ 1

3 16 _ 23. 8

20. -10 24. 7.02002000…

Lesson 5-3

Pages 239–244

Find each product. Write in simplest form. 2 _ · 3 1. _

1 _ 2. 3_ · 2

3 _ 3. _ -5

4.

9 _ 5. -_ · 5

6.

7. 10. 13.

5 16 _5 · _2 8 3 _4 · _1 5 8 3 4 -9_ 1_ 7 5 p3 _ 12 _ · p 4

(

4

( )

11

10 24 2 2 _ 8. 2 · 6_ 6 7 6 _ _ 11. - - 6 7 7 3 ab _ _ 14. · 9 b2

9.

( )

)

12. 15.

5 12 21 _1 · _ 7 22 7 2 -_ 12 6c _ _ ·2 10 c 12y4 4x _ _ · 2 3y x

(

)

MEASUREMENT Complete. 5 yard ? inches = _

3 18. _ pound = ? ounces

5

7 19. _ day = ? hours 8

4

20.

1 ? minutes = _ hour

17.

12

Extra Practice

16.

1 ? inches = 1_ feet

21.

2

? minutes = 180 seconds

Lesson 5-4

Pages 245–249

Find the multiplicative inverse of each number. 4 1. _ 7

5 2. -_

1 3. _

5. 6

6. -18

8. 2.35

9. -1.4

4

9

3 4. 5_ 8 7 7. _ 10

Find each quotient. Write in simplest form. 4 2 ÷_ 10. _

3

9 5 3 1 15. _ ÷ _ 12 4 1 1 _ 18. 2 ÷ -1_ 6 5

7

8 4 14. _ ÷ -_ 5 15 7 17. 16 ÷ 1_ 8 8 10 _ _ 20. ÷ 27 45 ab _ _ 23. ÷ b 12 16

( )

1 19. -11 ÷ 3_ 7

w w 22. _ ÷_ 5

4 1 12. _ ÷_

6 1 11. -_ ÷_

5 5 2 1 13. _ ÷ _ 3 9 3 _ _ 16. ÷ 15 4 16

35

(

)

1 21. -22 ÷ -5_

(

24.

21y _

Lesson 5-5

8x2

÷

2

7y _

)

16x

Pages 250–254

Find each sum or difference. Write in simplest form. 3 2 +_ 1. _ 7

8 4 2. _ -_

7

8 1 +_ 4. -_ 9 9 5 11 _ _ 7. + 12 12 3 1 _ 10. - -1_ 8 8

(

)

3 4 3. _ +_

7 7 5 7 6. _ - _ 12 12 3 1 9. 3_ + -_ 4 4 3 1 _ 12. -5 + -2_ 5 5

15 15 5 1 5. _ - _ 6 6 3 5 _ 8. - - _ 14 14 9 1 11. 4_ - 1_ 10 10

( ) ( )

ALGEBRA Find each sum or difference. Write in simplest form. 3n n +_ 13. _ 5

5

4 2 16. -6_ t - 3_ t 9 9

15 8 14. _ -_ ,k≠0

3 7 15. 12_ s - 7_ s 8

k k 3 1 _ 17. 6 g + -6_ g 4 4

(

)

8

2 2 18. 7_ n - -4_ n 5 5

(

)

Extra Practice

771

Lesson 5-6

Pages 257–261

Find the least common multiple (LCM) of each set of numbers or monomials. 1. 30, 18 4. 6a,

17a5

2. 4, 16

3. 3m, 12

5. 2, 5, 7

6. 9x2y, 12xy3

Find the least common denominator (LCD) of each pair of fractions. 2 _ , 6 7. _

3 _ 8. _ ,4

5 25 5 _ 11. 1 , _ 4 6

Replace each

5 _ 10. _ , 7

6 9 7 _ 13. 3 , _ 10p 5p3

9 12

1 _ 14. _ , 24 2 a

3a


2 15. _

Extra Practice

4 _ 9. _ ,7

12 5 3 _ 12. 11 , _ 20 8

_3 3 4 33 11 _ 18. _ 18 54

5 16. -_ 8

4 19. _ 19

4 17. _

7 _ 6 12 9 _1 20. _ 15 2

3 -_ 5

8 _ 38

Lesson 5-7

Pages 263–267

Find each sum or difference. Write in simplest form. 1 2 +_ 1. _

4 7 2. _ +_

5 7 8 4 _ -_ 4. 5 11 5 3 7. -_ + -1_ 12 8 2 4 + 2_ 10. 3_ 5 7

(

1 7 3. _ -_

5 9 7 4 _ 5. - -_ 12 11 1 2 8. -_ - 3_ 5 15 1 _ _ 11. -4 + 2 5 8 9

9

( )

)

12

15 9 6. -_ +_ 16 14 1 1 9. -5_ + -_

( ) ( )

3 6 3 5 _ 12. 11 - -6_ 5 8

Lesson 5-8

Pages 268–272

ALGEBRA Solve each equation. Check your solution. 1. a - 4.86 = 7.2

2. n + 6.98 = 10.3

3. 87.64 = f -(-8.5)

8 2 = -_ 4. x - _ 5 15

3 5 5. 3_ + m = 6_ 8 4

1 1 6. 4_ = r + 6_

1 4 =c-_ 7. 7_ 3 5 2 1 10. _w = _ 3 6

6

4 1 2 9. w - 1_ = _ 5 9 1 12. -_t = 7 9

8. -4.62 = h + (-9.4) 11. -0.5m = -10

Lesson 5-9

Pages 274–279

Find the mean, median, and mode for each set of data. Round to the nearest tenth, if necessary. 1. 82, 79, 93, 91, 95

2. 88, 85, 76, 94, 85, 97

3. 23, 32, 19, 27, 41, 21, 26, 32, 23

4. 7.4, 8.3, 6.1, 5.4, 6.8, 7.1, 8.0, 9.2

5.

6.

⫻ 8

9

10

⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻

⫻

⫻ ⫻

11

12

13

14

⫻ ⫻ 15

⫻ ⫻ ⫻ ⫻

⫻ ⫻ ⫻

⫻ ⫻ ⫻

⫻

⫻

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

7. POPULATION The population of the Canadian provinces and territories in 2000 is shown in the table. Find the mean, median, and mode of the data. Round to the nearest tenth, if necessary. Populations (thousands)

538.8 138.9 1023.6

772 Extra Practice

941.0 756.6 7372.4 11,669.3 1147.9

42.1 2997.2

30.7 4063.8

27.7

Lesson 6-1

Pages 292–296

Express each ratio as a fraction in simplest form. 1. 15 vans out of 40 vehicles

2. 6 pens to 14 pencils

3. 12 dolls out of 18 toys

4. 8 red crayons out of 36 crayons

5. 18 boys out of 45 students

6. 30 birds to 6 birds

7. 98 ants to 14 ladybugs

8. 140 dogs to 12 cats

9. 321 pennies to 96 dimes

10. 3 cups to 3 quarts

Express each ratio as a unit rate. Round to the nearest tenth, if necessary. 12. $7.95 for 5 pounds

13. $52 for 8 tickets

14. $43.92 for 4 CDs

15. 450 miles in 8 hours

16. $3.96 for 12 cans of soda

17. $3.84 for 64 ounces

18. 200 yards in 32.3 seconds

Extra Practice

11. 343.8 miles on 9 gallons

19. MONEY Which costs more per notebook, a 4-pack of notebooks for $3.98 or a 5-pack of notebooks for $4.99? Explain. 20. ANIMALS A cheetah can run 70 miles in 1 hour. How many feet is this per second? Round to the nearest whole number.

Lesson 6-2

Pages 297–300

Determine whether the set of numbers in each pattern forms a proportion. 1.

3.

5.

7.

Baskets

2

4

6

8

Apples

10

20

30

40

2.

Cups of Flour

2

4

6

8

Cups of Sugar

1.5

3

4.5

6

Age (years)

5

10

15

20

Height (inches)

46

59

72

74

Visitors

1

2

3

4

Shoes Left by the Door

2

4

6

8

4.

6.

8.

Days

5

10

15

20

Grass Height (inches)

4

6

8

10

Men’s Shoe Size

6

6.5

7

7.5

Women’s Shoe Size

7.5

8

8.5

9

Temperature (°C)

0

25

60

100

Temperature (°F)

32

77

140

212

Age (months)

8

10

14

20

Number of Teeth

2

6

12

16

Extra Practice

773

Lesson 6-3

Pages 302–306

ALGEBRA Solve each proportion. 49 7 =_ 1. _

30.6 s 2. _ =_

63 k 8 b _ =_ 4. 65 13 6.5 w =_ 7. _ 8 20

28.8 4.8 26 x _ _ 5. = 12 24 10 7 _ 8. =_ y 4.21

6 19.2 3. _ =_ g 11 21 _3 6. _ p = 9

12 1 9. _ =_ x 2.54

Extra Practice

Write a proportion that could be used to solve for each variable. Then solve. 10. 6 plums at $1 10 plums at d

11. 8 gallons at $9.36 f gallons at $17.55

12. 3 packages at $53.67 7 packages at m

13. 10 cards at $7.50 p cards at $18

14. 12 cookies at $3.00 16 cookies at s

15. 6 toy cars at $4.50 c toy cars at $6.75

Lesson 6-4

Pages 308–312

1 On a set of architectural drawings for a school, the scale is _ inch = 4 feet. 2

Find the actual length of each room. Room

Drawing Distance (in.)

1. 2.

Classroom

5

Principal’s Office

1.75

3.

Library

7_

4.

Cafeteria

5. 6.

Gymnasium Nurse’s Office

1 2 1 9_ 4

12.2 1.3

Lesson 6-5

Pages 313–318

Express each decimal or fraction as a percent. Round to the nearest tenth percent, if necessary. 1. 0.42

2. 0.06

3. 1.35

4. 0.001

5. 0.99

6. 3.6

7. 0.8

8. 0.0052

9. 0.00009

17 10. _ 50

9 11. _ 25

12 12. _

7 13. _ 40

11 14. _ 33

36 15. _ 27

8


17. 15%

1 % 18. 88 _

19. 250%

20. 21%

21. 64%

22. 25%

23. 131%

24. 72.5%

2 25. 66_ %

26. 0.06%

27. 315%

3

774

Extra Practice

2

Lesson 6-6

Pages 322–326

Use the percent proportion to solve each problem. Round to the nearest tenth. 1. What is 81% of 134?

2. 52.08 is 21% of what number?

3. 11.18 is what percent of 86?

4. What is 120% of 312?

5. 140 is what percent of 400?

6. 430.2 is 60% of what number?


8. What is 15% of 125?

9. 22 is what percent of 110? 11. What is 41.5% of 95?

10. 9.4 is 40% of what number? 12. 17.92 is what percent of 112?

Extra Practice

13. FOOD If 28 of the 50 soup cans on a shelf are chicken noodle soup, what percent of the cans are chicken noodle soup? 14. SCHOOL Of the students in a classroom, 60% are boys. If there are 20 students, how many are boys?

Lesson 6-7

Pages 327–331

Find the percent of each number mentally. 1. 40% of 60

2. 25% of 72

3. 50% of 96

1 % of 24 4. 33_

5. 150% of 42

1 6. 37_ % of 80 2

7. 200% of 125

2 8. 66_ % of 45

3

3

Estimate. Explain which method you used to estimate. 9. 60% of 49

10. 19% of 41

11. 82% of 60

12. 125% of 81

1 % of 502 13. _ 2

14. 31% of 19

Lesson 6-8

Pages 332–336

Solve each problem using an equation. 1. 9.28 is what percent of 58?

2. What number is 43% of 110?

3. 80% of what number is 90?

4. What number is 61% of 524?


6. 52% of what number is 109.2?

7. 62% of what number is 29.76?


9. Find 78% of 125. 11. 66% of what number is 49.5?

10. What is 0.2% of 12? 12. 36.45 is what percent of 81?

Find the discount to the nearest dollar. 13. $35 skirt, 20% off

14. $108 lamp, 25% off

Find the interest to the nearest cent. 15. $1585 at 6% for 5 years

1 years 16. $2934 at 5.75% for 3_ 2

17. BOOKS A dictionary is on sale at a 15% discount. Find the sale price of the dictionary if it normally sells for $29.99. Extra Practice

775

Lesson 6-9

Pages 338–342

Extra Practice

Find the percent of change. Round to the nearest tenth, if necessary. Then state whether each change is a percent of increase or a percent of decrease. 1. from $56 to $42

2. from $26 to $29.64

3. from $22 to $37.18

4. from $137.50 to $85.25

5. from $455 to $955.50

6. from $3 to $15

7. from $750.75 to $765.51

8. from $953 to $476.50

9. from $101.25 to $379.69

10. from $836 to $842.27

11. BASEBALL CARDS A baseball card collection contains 340 baseball cards. What is the percent of change if 25 cards are removed from the collection?

Lesson 6-10

Pages 343–347

Identify each sample as biased or unbiased and describe its type. Explain your reasoning. 1. To determine the most popular kind of fish to eat, every third person coming out of a grocery store is interviewed. 2. To determine where to hold the senior prom, a survey is taken of the entire school. 3. To determine the popularity of blogging, members of the Blog Society are polled. 4. To determine whether a city’s outerbelt should be widened, crews filmed traffic every six weeks at different times of day.

Lesson 7-1

Pages 359–363

Determine whether each relation is a function. Explain. 1. {(3, 6), (35, 64), (1, 1), (21, 7)} 5. {(1, 0), (1, 9), (1, 18)}

2. {(32, 24), (27, 24), (36, 24), (45, 24)} ⎧ ⎫ 1 1 1 1 1 ,3 , _ ,5 , _ ,7 , _ ,9 , _ , 11 ⎬ 4. ⎨ _ 2 6 8 4 10 ⎩ ⎭ 6. {(5, 5), (6, 6), (7, 7), (8, 7)}

7.

8.

3. {(2, 9), (3, 18), (4, 27), (2, 36)}

x

y

8

x

y

8

-2

4

15

8

-1

5

22

51

0

6

29

22

-1

7

-2

8

9.

10.

y

O

776

( )( ) ( ) ( ) (

Extra Practice

x

y

O

x

)

Lesson 7-2

Pages 365–369

Find four solutions of each equation. Show each solution in a table of ordered pairs. 1. x = 4

2. y = 0

3. x + y = 2

4. y = 2x - 6

5. x - y = 5

6. 3x - y = 8

1 x-3 7. y = _

1 8. y = _ x+1

9. 2x + y = -2

10. 2x + 3y = 12

11. x + 2y = -4

2

3

12. 2x - 4y = 8

Graph each equation by plotting ordered pairs. 14. y = 4x

15. x + y = 3

16. y = x - 3

17. y = -2x + 5

18. 2x + y = 6

Lesson 7-3

Extra Practice

13. y = x + 4

Pages 371–375

Find the rate of change for each linear function. 1.

5.

y Distance (ft)

Height (in.)

40 30 20 10

x

0

3.

2.

y 70 60 50

20 15 10 5

x

0

0.5 1 1.5 2 2.5 3 3.5 Time (yr)

Cookies Purchased Balance ($)

35 30 25

x

0

1

2

3

y

6

5.6

5.2

4.8

4.

Sales ($)

1240

1580

2250

2885

Commission ($)

49.60

63.20

90.00

115.40

6.

5 10 15 20 25 30 35 Time (s)

Miles Walked

0

1

2

3

Calories Burned

0

74

148

222

Time (min)

10

30

40

90

Candle Height (in.)

5.9

5.7

5.6

5.1

Lesson 7-4

Pages 376–381


2.

ÕÀ ,i>} V®

>} iÃ £È £{ £Ó £ä n È { Ó ä

Y

{ n £Ó £È Óä Ó{ Ón ÎÓ

iÃ

X

Time (min)

Volume (gal)

x

y

5

60

10

120

15

180

20

240

JOBS Lenora works at a job where her pay varies directly as the number of hours she works. Her pay for 6.5 hours is $49.40. 3. Write a direct variation equation relating Lenora’s pay x to the hours worked y. 4. Find Lenora’s pay if she works 25 hours in a week? Extra Practice

777

Lesson 7-5

Pages 384–389


2.

y

(⫺1, 4)

y

(⫺2, 1) x

O

x

O (⫺2, ⫺2)

(3, ⫺2)

Extra Practice

Find the slope of the line that passes through each pair of points. 3. P(3, 8), Q(4, -3)

4. D(4, 5), E(-3, -9)

5. L(-1, 2), M(0, 5)

6. J(6, 2), K(6, -4)

7. B(8, -3), C(-4, 1)

8. D(1, 5), E(3, 10)

9. H(7, 2), I(-2, -2)

10. K(2, -4), L(5, -19)

11. G(5, 6), H(7, 6)

12. A(-6, -3), B(-9, 4)

13. P(-1, -6), Q(-5, -10)

14. B(5, 9), C(-4, -5)

Lesson 7-6

Pages 391–394

State the slope and the y-intercept of the graph of each equation. 1. y = x + 9

2. y = 2x - 5

3. y = -6x

3 x 4. y = _ 2

1 5. y = _ x+8 3

6. x + 2y = 12

Graph each equation using the slope and y-intercept. 7. y = 3x - 2

8. x - 3y = 9

1 x+4 9. y = _ 2

2 10. y = -_ x-1

11. x - y = -4

12. 2x + 4y = -4

13. y = x + 5

14. 3x + y = 9

3

Lesson 7-7

Pages 397–402

Write an equation in slope-intercept form for each line. 1. slope = 3, y-intercept = -4

3 2. slope = _ , y-intercept = 1

3. slope = -7, y-intercept = -2

5 , y-intercept = 9 4. slope = _ 8

1 , y-intercept = 0 5. slope = -_


2

4

Write an equation in slope-intercept form for the line passing through each pair of points. 7. (4, 7) and (0, 3) 9. (8, 7) and (0, 0) 11. (-2, 5) and (3, 9) 778 Extra Practice

8. (3, -6) and (-1, 2) 10. (1, 4) and (3, -6) 12. (3, -1) and (5, -1)

Lesson 7-8

Pages 403–407

TECHNOLOGY For Exercises 1–3, use the table that shows the percent of U.S. households owning more than one television set. Year Percent

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

4

12

22

35

43

50

57

65

71

76

1. Make a scatter plot and draw a line of fit. 2. Write an equation in slope-intercept form for the line of fit. 3. Use the equation to predict what percent of U.S. households will own more than one television set in 2010.

Extra Practice

Lesson 8-1

Pages 420–423

ALGEBRA Solve each equation. Check your solution. 1. -7h - 5 = 4 - 4h

2. 5t - 8 = 3t + 12

3. m + 2m + 1 = 7

4. 2y + 5 = 6y + 25

5. 3z - 1 = 23 - 3z

6. 5a - 5 = 7a - 19

7. 5x + 12 = 3x - 6

8. 3x - 5 = 7x + 7

9. 5c + 9 = 8c

10. 3p = 4 - 9p

11. 6z + 5 = 4z - 7

12. 2a + 4.2 = 3a - 1.6

13. 3.21 - 7y = 10y - 1.89

14. 1.9s + 6 = 3.1 - s

15. 12b - 5 = 3b

16. 9 + 11a = -5a + 21

17. 6 - x = -5

18. 2.8 - 3w = 4.6 - w

19. 2.9y + 1.7 = 3.5 + 2.3y

20. 2.85a - 7 = 12.85a - 2

Lesson 8-2

Pages 424–428

ALGEBRA Solve each equation. Check your solution. 1. 6(m - 2) = 12

2. 4(x - 3) = 4

3. 5(2d + 4) = 35

4. w + 6 = 2(w - 6)

5. 3(b + 1) = 4b - 1

6. 7w - 6 = 3(w + 6)

7. 4(k - 6) = 6(k + 2)

8. 3x - 0.8 = 3x + 4

5 1 g+8=_ g+1 9. _ 9

6

s+5 s-3 10. _ =_ 7

9

11. ALGEBRA Find the solution of 3(3x + 4) - 2 = 9x + 10. 12. NUMBER THEORY Four times the sum of three consecutive integers is 48. a. Write an equation that could be used to find the integers. b. What are the integers? Extra Practice

779

Lesson 8-3

Pages 430–434

ALGEBRA For the given value, state whether each inequality is true or false. 1. 5 ≥ 2t - 12; t = 11

2. 7 + n < 25; n = 4

3. 6r - 18 > 0; r = 3

4. 3n + 2 < 26; n = 3

5. h - 19 < 13; h = 28

6. 20m ≥ 10; m = 0

Extra Practice

ALGEBRA Graph each inequality on a number line. 7. b ≥ 4

8. x < -2

9. y > 2

10. m ≤ 0

11. p > -1

12. q ≥ -3

ALGEBRA Write an inequality for each sentence. 13. At least 295 students attend Greenville Elementary School. 14. An electric bill increased by $15 is now more than $80. 15. If 8 times a number is decreased by 2, the result is less than 15. 16. Citizens who are 18 years of age or older can vote. 17. One dozen jumbo eggs must weigh at least 30 ounces. 18. A healthful breakfast cereal should contain no more than 5 grams of sugar.

Lesson 8-4

Pages 435–439

ALGEBRA Solve each inequality. Check your solution. 1. m + 9 < 14

2. k + (-5) < -12

3. -15 < v - 1

4. -7 + f ≥ 47

5. r > -15 - 8

6. 18 ≥ s - (-4)

7. 38 < r - (-6)

8. z - 9 ≤ -11

9. -16 + c ≥ 1

10. d + 1.4 < 6.8

11. -3 + x > 11.9

12. -0.2 ≥ 0.3 + y

13. h + 5.7 > 21.3

14. t - 8.5 > -4.2

15. -13.2 > w - 4.87

5 7 ≥_ 16. a + _ 12 18 5 2 ≤a-_ 18. _ 3 6

1 7 17. 7_ < n - -_ 2 8

( )

19. -7.42 ≤ d - 5.9

Lesson 8-5

Pages 441–445

ALGEBRA Solve each inequality. Check your solution. 1. 6p < 78

m > 24 2. _

3. -18 < 3b

4. -5k ≥ 125

a 5. -75 > _

w 6. _ < -5

2 c 7. 8 < _ 3

m 8. _ ≥ 0.5 1.3

9. 0.4y > -2

1 1 d ≤ -5_ 10. -_

2 2 y 13. _ > -20 -13

-3

5

2 11. _ t 10 - 2x

4. -6a + 2 ≥ 14

5. 3y + 2 < -7

d + 3 ≥ -11 6. _

x -5 12

3

3(n + 1) n+4 11. __ ≥ _

10. _r - 6 ≤ 3 5

7

5

4

n + 10 12. __ ≤ 6 -3

13. Five plus three times a number is less than the difference of two times the same number and 4. What is the number?

Extra Practice

Lesson 9-1

Pages 464–468

Find each square root. 36 1. √ 4. - √ 144 100 7. √

√_14

2. - √ 81

3.

5. √ 25 8. - √ 0.49

6. √ 1.96 9. √ 400

Use a calculator to find each square root to the nearest tenth. 11. √99 12. - √ 60 10. √21 14. - √350 15. √18.6 13. √124 42 16. - √

18. √ 182

17. - √ 84.2

Estimate each square root to the nearest whole number. Do not use a calculator. 20. - √85 21. √7.3 19. √21 1.99 22. √ 25. √810

24. √ 74.1 27. √1000

23. - √ 62 26. - √88.8

Lesson 9-2

Pages 469–474

Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. 1. 15

2. 0

4. 0.666…

5. 1.75 8. - √36

7. 5.14726… Replace each 3 √ 15 10. 3_ 4

12. 5.2

3 3. _ 8 6. √2

9. 0.3535…

with , or = to make a true statement. 11. - √ 41 -6.8

√ 27.04

13. - √ 110

-10.5

ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 14. x2 = 14

15. y2 = 25

16. 34 = p2

17. 55 = h2

18. 225 = k2

19. 324 = m2

20. d2 = 441

21. r2 = 25,000

22. 10,000 = x2 Extra Practice

781

Lesson 9-3

Pages 476–481


2.

x˚

3.

x˚

18˚

42˚

16˚

56˚ 63˚

4.

x˚

5.

6. 65˚

40˚

31˚ 65˚

Extra Practice

x˚ 95˚

x˚

x˚

7. ALGEBRA The measure of the angles of a triangle are in the ratio 1:2:3. What is the measure of each angle? 8. ALGEBRA Determine the measures of the angles of ABC if the measures of the angles of a triangle are in the ratio 1:1:2. 9. ALGEBRA Suppose the measures of the angles of a triangle are in the ratio 1:9:26. What is the measure of each angle?

Lesson 9-4

Pages 485–490


1. c ft

4 ft

8 in.

6 in.

3. cm

c in.

3 ft

24 m

10 m

If c is the measurement of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 4. a = 7 m, b = 24 m

5. a = 18 in., c = 30 in.

6. b = 10 ft, c = 20 ft

7. a = 3 cm, c = 9 cm

8. b = 8 m, c = 32 m

9. a = 32 yd, c = 65 yd

Lesson 9-5

Pages 492–496

Find the distance between each pair of points. Round to the nearest tenth, if necessary. 1. A(2, 6), B(-4, 2)

2. C(-3, 9), D(2, 4)

3. E(6, -4), F(1, -6)

4. G(0, -1), H(9, -1)

5. I(-8, -3), J(2, 2)

6. K(3, 0), L(-7, -2)

7. M(3, 5), N(7, 1)

8. O(-6, 2), P(0, 8)

9. Q(4, -9), R(-2, 7) 782

Extra Practice

10. S(13, -1), T(-5, -3)

Lesson 9-6

Pages 497–502

The triangles are similar. Find each missing measure. 2.

1.

20 m 8 in.

12 in.

xm 8m

6 in.

4m

x in.

3.

4.

C m 6c

H

m 9c B

x in.

12 in.

L

4 cm x cm

18 in.

G

Extra Practice

A

J

24 in.

10 in.

K

E

D

Lesson 10-1

Pages 512–517

In the figure at the right, m and p is a transversal. If the m∠2 is 38°, find the measure of each angle. 1. ∠1

2. ∠4

3. ∠3

4. ∠6

5. ∠5

6. ∠8

ᐉ

p

m 2

4 5 8 6 7 1 3

Find the value of x in each figure. 8.

7.

9. x˚

x˚ 32˚

x˚ 52˚

18˚

Lesson 10-2

Pages 518–523


A

2. G

D

C

K

H

E

I

J

B F

ABC _?_

GHI _?_

Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. 3. ! "

4.

$ KM KM

KM

#

%

KM

MI

&

+

*

(

MI

)

MI

,

MI

Extra Practice

783

Lesson 10-3

Pages 524–530

Find the coordinates of the vertices of each figure after the given translation. Then graph the translation image. 1. (2, -1)

2. (-3, -2)

y

K y

A B

O

L

N

x

x

O

C

Extra Practice

M

Find the coordinates of the vertices of each figure after a reflection over the given axis. Then graph the reflection image. 3. x-axis

4. y-axis

y

D y

I E G x

O

K

x

O

J F

Lesson 10-4

Pages 532–536

ALGEBRA Find the value of x. Then find the missing angle measure. 1.

115˚

2.

x˚

65˚

x˚ 125˚

65˚

4.

3.

85˚ 40˚

93˚

86˚

2x ˚

96˚

(x ⫺ 5 )˚

134˚ 82 ˚

(x ⫹10 )˚

Lesson 10-5

Pages 539–543

Find the area of each figure. 2.

1.

10.6 cm

9m 14.2 cm 6m

3.

8.5 ft

4.

3 in.

2.5 ft 10 in. 6 ft

5. What is the height of a parallelogram with a base of 3.4 inches and an area of 32.3 inches? 6. The bases of a trapezoid measure 8 meters and 12 meters. Find the measure of the height if the trapezoid has an area of 70 square meters. 784

Extra Practice

Lesson 10-6

Pages 545–550


1.

Find the sum of the measures of the interior angles of each polygon. 3. decagon

4. pentagon

5. nonagon

6. hexagon

7. octagon

8. 15-gon

Extra Practice

Lesson 10-7

Pages 551–556


1.

9 cm

5 in.

4.

3. 18 ft

7.3 m

5. The radius is 8.2 feet.

6. The diameter is 1.3 yd.

7. The diameter is 5.2 yd.

8. The radius is 4.8 cm.

9. Find the diameter of a circle if its circumference is 18.5 feet. Round to the nearest tenth. 10. A circle has an area of 62.9 square inches. What is the radius of the circle? Round to the nearest tenth.

Lesson 10-8

Pages 558–562

Find the area of each figure. Round to the nearest tenth. 1.

2.

8 ft

6 cm 5.5 ft

8 cm

3 ft

3.

4.

2.1 yd 4.8 yd

10 in.

6.4 yd

3.2 yd

12 in.

Extra Practice

785

Lesson 11-1

Pages 575–581

Identify each solid. Name the bases, faces, edges, and vertices. 1.

B A

2.

C

D

N K

F E

L

G H

H

M

J

ARCHITECTURE The sketch shows plans for steps to the front door of a house. Each unit on the diagram is 1.5 feet.

Extra Practice

3. Draw a top view and find the area of the ground that the entrance takes up. 4. Draw a side view and find the height of the steps.

Lesson 11-2

Pages 583–588


2.

6 in.

6m

15 in. 10 m

3.

3m

4.

4 cm 6 cm

5 ft

8 ft

25 cm

4 ft

1 yd, width 7 yd, height 12 yd 5. rectangular prism: length 2_ 2

6. cylinder: diameter 9.2 mm, height 16 mm 7. triangular prism: base of triangle 3.1 cm, altitude of triangle 1.7 cm, height of prism 5.0 cm

Lesson 11-3

Pages 589–594


1. 10 ft

12 cm

4 ft 8 cm

8 cm

3. cone: diameter 10 yd, height 7 yd 4. rectangular pyramid: length 6 in., width 6 in., height 9 in. 1 5. square pyramid: length 3_ ft, height 12 ft 4

786 Extra Practice

Lesson 11-4

Pages 597–601

Find the lateral area and surface area of each solid. Round to the nearest tenth, if necessary. 1.

2.

6 in.

3 cm

20 in.

9 cm

5 cm

3. cube: side length 6 ft

4. cylinder: diameter 8 m, height 12 m

5. cylinder: radius 2.5 cm, height 5 cm

6. cube: side length 4.9 m

7. rectangular prism: length 7.6 mm, width 8.4 mm, height 7.0 mm

Lesson 11-5

Extra Practice

8. triangular prism: right triangle 3 in. by 4 in. by 5 in., height of prism 10 in. Pages 602–606


2.

5.2 in. 6 in.

6 in.

15 cm 8 cm

6 in.

6 in.

4.

3.

4.2 m

9 ft 9.3 m 4 ft

4 ft

5. square pyramid: base side length 1.8 mm, slant height 3.0 mm 6. cone: radius 4 in., slant height 7 in. 7. cone: diameter 15.2 cm, slant height 12.3 cm

Lesson 11-6

Pages 608–613


4 in.

2.

20 in.

8 cm 12 cm 3 in.

10 in. 4 cm

4 cm 6 cm

6 cm

Find the missing measure of each pair of similar solids. 3.

4.

21 ft

10 yd

x 12 ft

x

4 ft 4 yd

4 yd 10 yd

10 yd

Extra Practice

787

Lesson 12-1

Pages 626–631

Display each set of data in a stem-and-leaf plot. 1. 37, 44, 32, 53, 61, 59, 49, 69

2. 3, 26, 35, 8, 21, 24, 30, 39, 35, 5, 38

3. 157, 74, 6, 5, 153, 79, 73

4. 172, 198, 181, 182, 193, 171, 179, 186, 181

5. 55, 62, 81, 75, 71, 69, 74, 80, 67

6. 121, 142, 98, 106, 111, 125, 132, 109, 117, 126

7. 17, 54, 37, 86, 24, 69, 77, 92, 21

8. 73, 61, 89, 67, 82, 54, 93, 102, 59, 75, 83

For Exercises 9–11, use the stem-and-leaf plot shown at the right. 9. What is the greatest value?

Extra Practice

10. In which interval do most of the values occur?

Stem 7 8 9

11. What is the median value?

Lesson 12-2

Leaf 22359 01146689 348 9 ⎢4 = 94

Pages 633–637

Find the range, interquartile range, and any outliers for each set of data. 1. {44, 37, 23, 35, 61, 95, 49, 96}

2. {30, 62, 35, 80, 12, 24, 30, 39, 53, 38}

3. {7.15, 4.7, 6, 5.3, 30.1, 9.19, 3.2}

4. {271, 891, 181, 193, 711, 791, 861, 818}

5.

6.

Stem 2 3 4 5

Leaf 0112479 336888 245799 29 3 ⎢6 = 36

Stem 4 5 6 7 8 9

Leaf 0223456678 125559 4788 0014999 179 00135 8 ⎢7 = 87

Lesson 12-3

Pages 638–642

Draw a box-and-whisker plot for each set of data. 1. 32, 54, 88, 17, 29, 73, 65, 52, 99, 103, 43, 13, 8, 59, 40, 37, 23 2. 42, 23, 31, 27, 32, 48, 37, 25, 19, 26, 30, 41, 32, 29 3. 124, 327, 215, 278, 109, 225, 186, 134, 251, 308, 179 4. 126, 432, 578, 312, 367, 400, 275, 315, 437, 299, 480, 365, 278

VOLLEYBALL For Exercises 5–7, use the box-and-whisker plot shown. Heights (in.) of Players on Volleyball Team

50

55

60

65

70

75

5. What is the height of the tallest player? 6. What percent of the players are between 56 and 68 inches tall? 7. Explain what the length of the box-and-whisker plot tells us about the data. 788 Extra Practice

Lesson 12-4

Pages 644–649


2.

Weekly Exercise Time

Weekly Grocery Bill

Time (h)

Tally

Frequency

Amount ($)

0–2

|||| |||

8

0–49

3–5

||||

4

50–99

6–8

||

2

100–149

|||| |||

8

9–11

|||

3

150–199

||||

4

200–249

||

2

3.

4.

Temperature (°F) 60–69

Tally ||

Frequency 6 12

|||| |||| ||

Score on Math Test

Frequency

Score

Tally

2

50–59

||

2

Frequency

70–79

|||| ||||

10

60–69

|

1

80–89

|||| |

6

70–79

|||| |||

8

90–99

|||

3

80–89

|||| |||| ||||

14

90–99

Lesson 12-5

Extra Practice

Daily High Temperatures in August

Tally |||| |

0

Pages 651–656

Choose an appropriate type of display for each data set. 1. the relationship among the people who have at least one dog, another animal, have both, or have neither 2. number of people entering a museum by age groups 3. number of U.S. Congressional Representatives from each state 4. number of MP3 players sold in the last year by model 5. number of runs for the home team in each inning 6. the populations of the middle 50% of cities in a state

Lesson 12-6

Pages 659–663

MONEY For Exercises 1–2, refer to the graphs below. À>«

À>«

Vi

Vi

Óäää £äää

Óäää £Çxä £xää £Óxä £äää

ä

£

Ó Î { x È 7ii

£

Ó Î { x È 7ii

1. Explain why the graphs look different. 2. Which graph appears to show that the income has been fairly consistent? Explain your reasoning. Extra Practice

789

Lesson 12-7

Pages 665–669

There are 4 blue marbles, 6 red marbles, 3 green marbles, and 2 yellow marbles in a bag. Suppose you select one marble at random. Find the probability of each outcome. Express each probability as a fraction and as a percent. Round to the nearest percent. 1. P(green)

2. P(blue)

3. P(red)

4. P(yellow)

5. P(not green)

6. P(white)

7. P(blue or red)

8. P(not yellow)

Extra Practice

9. P(neither red nor green)

10. P(red or yellow)

11. P(not orange)

12. P(neither blue nor yellow)

13. P(not red)

14. P(not green or yellow)

15. Suppose two number cubes are rolled. What is the probability of rolling a sum greater than 8? 16. COOKIES A sample from a package of assorted cookies revealed that 20% of the cookies were sugar cookies. Suppose there are 45 cookies in the package. How many can be expected to be sugar cookies?

Lesson 12-8

Pages 670–674

Find the number of possible outcomes for each situation. 1. Engagement rings come in silver, gold, and white gold. The diamond can 1 1 1 karat, _ karat, or _ karat. The diamond can have 4 possible shapes. weigh _ 2

3

4

2. A dress can be long, tea-length, knee-length, or mini. It comes in 2 colors and the dress can be worn on or off the shoulders. 3. The first digit of a 7-digit phone number is a 2. The last digit is a 3. 4. A chair can be a rocker, recliner, swivel, or straight back. It is available in fabric, vinyl, or leather. Find the probability of each event. 5. Three coins are tossed. What is the probability of three tails? 6. Two six-sided number cubes are rolled. What is the probability of getting an odd sum? 7. A ten-sided die is rolled and a coin is tossed. Find the probability of the coin landing on tails and the die landing on a number greater than 3.

Lesson 12-9

Pages 676–680

Tell whether each situation is a permutation or a combination. Then solve. 1. Seven people are running for four seats on student council. How many ways can the students be elected? 2. How many ways can the letters of the word ISLAND be arranged? 3. How many ways can five candles be arranged in three candlesticks? 4. How many ways can six students line up for a race? 5. How many ways can you select three books from a shelf containing 12 books? 6. GEOMETRY Determine the number of line segments that can be drawn between any two vertices of a pentagon. 790

Extra Practice

Lesson 12-10

Pages 682–687

A deck of Euchre cards consists of 4 nines, 4 tens, 4 jacks, 4 queens, 4 kings, and 4 aces. Suppose one card is selected and not replaced. Find the probability of each outcome. 1. 3 nines in a row 2. a black jack and a red queen 3. a nine of clubs, a black king, and a red ace 4. 4 face cards in a row A number from 6 to 19 is drawn. Find the probability of each outcome. 6. P(13 or less than 7)

7. P(even or odd)

8. P(14 or greater than 20)

9. P(even or less than 10)

Extra Practice

5. P(13 or even)

10. P(odd or greater than 10)

Lesson 13-1

Pages 701–704

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. 1. 3x2 + 5

6 + 9x 2. _ x

2 4 3. _ p

4. -6x2 + 3x - 5

5.

6. 16 - 3m + m3

d 7. _

8. t2 - 2

15

√w

3

-6

x 9. _ y +z

Find the degree of each polynomial. 10. 38

11. 4b + 9

12. cd

13. 4x

14. a2 - 6

15. 11r + 5s

16.

x2y

17.

n2

18. 6a2b2

-n

19. 3y2 - 2

20. 9cd3 - 5

21. -5p3 + 8q2

22. w2 + 2x - 3y3 - 7z

23. _ - x

24. -17n2p - 11np3

x3 6

Lesson 13-2

Pages 706–709

Find each sum. 1.

-6m + 7 (+) 9m - 2

2.

12y - 4 (+) - 8y + 9

3.

5x + y (+) 9x - 2y

4.

7c2 - 10c + 5 (+) 4c2 - 4c - 8

5.

2a2 + 5ab + 6b2 (+) 3a2 - b2

6.

3d3 + 2d2 + 6d - 4 (+) - 4d2 -3

7. (3a + 4) + (a + 2) 9. (5x - 3y) + (2x - y) 11. (-11r2 + 3s) + (5r2 - s)

8. (8m - 3) + (4m + 1) 10. (8p2 - 2p + 3) + (-3p2 - 2) 12. (3a2 + 5a + 1) + (2a2 - 3a - 6)

Find each sum. Then evaluate if m = -2, n = 4, and p = 3. 13. (3m - 5n) + (-6m + 8n) 14. (m2 + 2p2) + (-4m2 - 6p2) 15. (-2m + 3n + 4p) + (5m - 6n - 8p) Extra Practice

791

Lesson 13-3

Pages 710–713

Find each difference. 1.

2a + 7 (-) a + 3

2.

-3k2 + 6k (-) 4k2 + k

3.

6x2 - 4x + 11 (-) 5x2 + 5x - 4

4.

+1 9r2 (-) 2r2 + 3r - 7

5.

8n2 + 3mn - 9 (-) 4n2 + 2mn

6.

-5b2 - 2ab (-) -10ab + 6a2

7. (3n + 2) - (n + 1)

Extra Practice

9.

(4x2

+ 1) -

(3x2

8. (-3c + 2d) - (7c - 6d) 10. (5a - 4b) - (-a + b)

- 4)

12. (3w3 + 5w - 6) - (5w3 - 2w + 5)

11. (-12a + 9b) - (3a - 7b)

Lesson 13-4

Pages 716–719

Find each product. 1. 2(3a - 7)

2. (8c + 1)4

3. n(5n + 6)

4. t(2 - t)

5. (3k - 5)k

6. (a + b)a

7. 4n(5n - 3)

8. -3x(4 - x)

9. 6m(-m2 + 3)

10. 5(3x - 2)

11. (2p + 9)8

12. m(3m - 4)

13. -2w(6 - w)

14. ab(a + b)

15. 7t(-3t + 4w)

-5x(2x2

7r(r2

16.

17.

- 3x + 1)

18. -3az(2z2 + 4az + a2)

- 3r + 7)

Lesson 13-5

Pages 720–725


2.

y

O

4. y = -3x 8. y =

(-2)x

x

5. y = 2x3 - 5 6 9. y = _ x

3.

y

y

x

O

6. -2x + 5y = 10 10.

x

O

7. x = 7y 11.

x

y

x

y

2

5

5

7

4

7

10

13

6

9

15

19

8 11

Lesson 13-6

20 25 Pages 726–730

Graph each function.

792

1. y = 3x2

2. y = -2x2

1 2 3. y = _ x

4. y = x3

5. y = 0.3x3

6. y = x3 - 2

7. y = x2 + 4

8. y = -0.5x2 + 1

Extra Practice

2

Mixed Problem Solving Chapter 1 The Tools of Algebra

figure 1 figure 2

figure 3


2. SPACE EXPLORATION On one flight, the space shuttle Endeavour traveled 6.9 million miles and circled Earth 262 times. About how many miles did the shuttle travel on each trip around Earth? (Lesson 1-1) 3. TREES A conservation group collects seeds from trees, grows them into saplings, and sells them to the public. Each sapling costs $35, plus $7 for shipping and handling per order. Write and then evaluate an expression for the total cost of one order of six saplings. (Lesson 1-2)

4. SALES Mia sold 15 white chocolate hearts at $4.25 each, 36 milk chocolate hearts at $3.75 each, and 22 milk chocolate assortments at $7.45 each. How much money did Sophia raise? (Lesson 1-2) SPACE For Exercises 5 and 6, use the following information. Objects weigh six times more on Earth than they do on the moon because the force of gravity is greater. (Lesson 1-3) 5. Write an expression for the weight of an object on Earth if its weight on the moon is x. 6. A scientific instrument weighs 34 pounds on the moon. How much does the instrument weigh on Earth? VOLLEYBALL For Exercises 7 and 8, use the following information. A volleyball net is 3 feet 3 inches tall. The bottom of the net is to be set 4 feet 8 inches from the floor. (Lesson 1-4)

9. NEWSPAPERS Nick sold 86 newspapers on Monday, 79 on Tuesday, 68 on Wednesday, and 83 on Friday. How many newspapers did Nick sell on Thursday if he sold a total of 391 in the five days? (Lesson 1-5) 10. FOOD Kristen buys twice as many blueberry bagels as plain bagels. Write a relation to show the different possibilities. (Lesson 1-6)

GEOLOGY For Exercises 11 and 12, use the following information. The underground temperature of rocks in degrees Celsius is estimated by the expression 35x + 20, where x is the depth in kilometers. (Lesson 1-6) 11. Make a list of ordered pairs in which the x-coordinate represents the depth and the y-coordinate represents the temperature for depths of 0, 2, and 4 kilometers. 12. Graph the ordered pairs. 13. EMPLOYMENT The scatter plot shows the years of experience and salaries of twenty people. Do the data show a positive, negative, or no relationship? Explain. (Lesson 1-7)

Salary ($ thousands)

1. PATTERNS How many cubes are in the tenth figure in the pattern below? (Lesson 1-1)

(pages 24–75)

y 55 50 45 40 35 30 25 0

1 2 3 4 5 6 7 x Years of Experience

14. BIRDS The table shows the average lengths and widths of five bird eggs. Bird

Length (cm)

Width (cm)

Canadian goose robin turtledove hummingbird raven

8.6 1.9 3.1 1.0 5.0

5.8 1.5 2.3 1.0 3.3

7. Write an expression for the distance from the floor to the top of the net.

Source: Animals as Our Companions

8. Find the distance from the floor to the top of the net.

Make a scatter plot and predict the width of an egg 6 centimeters long. (Lesson 1-7)

794 Mixed Problem Solving

Chapter 2 Integers 1. ASTRONOMY Mars is about 228 million kilometers from the Sun. Earth is about 150 million kilometers from the Sun. Write two inequalities that compare the two distances. (Lesson 2-1) 2. GAMES One contestant finished the regular round with a score of -200, and another contestant finished with a score of -500. Write two inequalities that compare their scores. (Lesson 2-1) 3. MONEY Tino had $250 in his checking account at the beginning of April. During the month he wrote checks in the amounts of $72, $37, and $119. He also made one deposit of $45. Find Tino’s account balance at the end of April. (Lesson 2-2)

(Lesson 2-2)

5. SUBMARINES The research submarine Alvin is located 1500 meters below sea level. It descends another 1250 meters to the ocean floor. How far below sea level is the ocean floor? (Lesson 2-3) 6. METEOROLOGY Windchill factor is an estimate of the cooling effect the wind has on a person in cold weather. If the outside temperature is 10°F and the wind makes it feel like -25°F, what is the difference between the actual temperature and how cold it feels? (Lesson 2-3) 7. GEOGRAPHY The highest point in Africa is Mount Kilimanjaro. Its altitude is 5895 meters. The lowest point in Africa is Lake Assal. Its altitude is -155 meters. Find the difference between these altitudes. (Lesson 2-3) 8. GEOLOGY In December, 1994, geologists found that the Bering Glacier had come to a stop. The glacier had been retreating at a rate of about 2 feet per day. If the retreat resumes at the old rate, what integer represents how far the glacier will have retreated after 28 days? (Lesson 2-4)

9. SPORTS The Wildcat football team was penalized the same amount of yardage four times during the third quarter. The total of the four penalties was 60 yards. If -60 represents a loss of 60 yards, write a division sentence to represent this situation. Then express the number of yards of each penalty as an integer. (Lesson 2-5)

10. AEROSPACE To simulate space travel, NASA’s Lewis Research Center in Cleveland, Ohio, uses a 430-foot shaft. If the free fall of an object in the shaft takes 5 seconds to travel the -430 feet, on average how far does the object travel in each second? (Lesson 2-5)


4. ASTRONOMY At noon, the average temperature on the Moon is 112°C. During the night, the average temperature drops 252°C. What is the average temperature of the Moon’s surface during the night?

(pages 76–121)

11. MAPS A map of a city can be created by placing the following buildings at the given coordinates: City Hall (1, 2), High School (-3, 6), Fire Department (4, -2), Recreation Center (0, 3). Draw and label the map. (Lesson 2-6)

GEOMETRY For Exercises 12 and 13, use the following information. A vertex of a polygon is a point where two sides of the polygon intersect. (Lesson 2-6) 12. Identify the coordinates of the vertices in the triangle below. y

A

x

O

B C

13. Add 2 to each x-coordinate. Graph the new ordered pairs. Describe how the position of the new triangle relates to the original triangle. Mixed Problem Solving

795

Chapter 3 Equations

(pages 122–175)

1. BUSINESS A local newspaper can be ordered for delivery on weekdays or Sundays. A weekday paper is 35¢, and the Sunday edition is $1.50. The Stadlers ordered delivery of the weekday papers. The month of March had 23 weekdays and April had 20. How much should the carrier charge the Stadlers for those two months? (Lesson 3-1) SHOPPING For Exercises 2 and 3, use the following information. One pair of jeans costs $23, and one T-shirt costs $15. (Lesson 3-1)

FENCING For Exercises 10 and 11, use the following information. Wanda uses 130 feet of fence to enclose a rectangular flower garden. She also used the 50-foot wall of her house as one side of the garden. What is the width of the garden? (Lesson 3-6)

10. Write an equation that represents this situation.

2. Write two equivalent expressions for the total cost of 3 pairs of jeans and 3 T-shirts.

11. Solve the equation to find the width of the garden.

3. Find the total cost.


9. PURCHASING Mr. Rockwell bought a television set. The price was $362. He paid $75 down and will pay the balance in 7 equal payments. How much is each payment? (Lesson 3-5)

4. ENTERTAINMENT Kyung bought 3 CDs that each cost x dollars, 2 tapes that each cost $10, and a video that cost $14. Write an expression in simplest form that represents the total amount that Kyung spent. (Lesson 3-2)

WORKING For Exercises 12 and 13, use the following information. Halley worked a 40-hour week and was paid $410. This amount included a $50 bonus. (Lesson 3-6)

12. Write an equation that represents this situation.

TRANSPORTATION For Exercises 5 and 6, use the following information. A minivan is rated for maximum carrying capacity of 1500 pounds. (Lesson 3-3)

13. What was Halley paid per hour? 14. EXERCISE The table shows the amount of time Gary spends jogging every week. He increases the time he jogs every week. How many minutes will he jog during week 12?

5. If the luggage weighs 150 pounds, what is the maximum weight allowable for passengers? 6. About what is the maximum average weight allowable for each of 7 passengers?

Week

7. GEOMETRY The perimeter of any square is 4 times the length of one of its sides. If the perimeter of a square is 72 centimeters, what is the length of each side of the square? (Lesson 3-4)

x

x

x

x

8. SPORTS Marcie paid $75 to join a tennis club for the summer. She will also pay $10 for each hour that she plays. If Marcie has budgeted $225 to play tennis this summer, how many hours can she play tennis? (Lesson 3-5)

796


1 2 3 4 5

Time Jogging (min) 7 15 23 31 39

15. GEOMETRY The perimeter of the triangle below is 27 yards. Find the lengths of the sides of the triangle. (Lesson 3-8) x⫹2

x⫺2 x

Chapter 4 Factors and Fractions

(pages 178–225)

1. PATTERNS In a pattern, the number of colored tiles used in row x is 3x. Find the number of tiles used in rows 4, 5, and 6 of the pattern. (Lesson 4-1)

7. BASKETBALL Ciera made 8 out of 14 free throws in her last basketball game. Write her success as a fraction in simplest form.

2. ELECTRICITY The amount of power lost in watts P can be found by using the formula P = I2R, where I is current in amps, and R is resistance in ohms. The resistance of the wire leading from the source of power to a home is 2 ohms. If an electric stove causes a current of 41 amps to flow through the wire, find the power lost from the wire powering the stove. (Lesson 4-1)

8. TRANSPORTATION Cameron spends 18 minutes traveling to work. What fraction of the day is this? (Lesson 4-4)

3. ELEPHANTS At 660 days, African elephants have the longest gestation period. Write the prime factorization of this number. (Lesson 4-2)

5. INTERIOR DESIGN Mrs. Garcia has two different fabrics to make square pillows for her living room. One fabric is 48 inches wide, and the other fabric is 60 inches wide. How long should each side of the pillows be if they are all the same size and no fabric is wasted? (Lesson 4-3) 6. ECONOMICS The graph below shows how each dollar spent by the Federal Government is used. Government Spending Education 3¢

Income Security 32¢

Medical Care 13¢

Housing 6¢

Debt 14¢

Earthquake A B

Richter Scale 8 4

Intensity 107 103

How much more intense was Earthquake A than Earthquake B? (Lesson 4-5)

ASTRONOMY For Exercises 10 and 11, use the following information. Any two objects in space have an attraction that can be calculated by using a formula that includes the universal gravitational constant, 6.67 × 10-11 Nm2/kg2. (N is newtons.) (Lesson 4-6)

10. Write the universal gravitational constant using positive exponents. 11. Write the constant as a decimal. BIOLOGY For Exercises 12 and 13, use the following information. Deoxyribonucleic acid, or DNA, contains the genetic code of an organism. The length of a DNA strand is about 10-7 meter. (Lesson 4-6) 12. Write the length of a DNA strand using positive exponents. 13. Write the length of a DNA strand as a decimal.

Defense 20¢

Veterans 2¢

9. EARTHQUAKES The table below describes different earthquake intensities.


4. CODES Prime numbers are used to code and decode information. Suppose two prime numbers p and q are chosen so that n = pq. Then the key to the code is n. Find p and q if n = 1073. (Lesson 4-2)

(Lesson 4-4)

Other 10¢

14. SCIENCE Atoms are extremely small particles about two millionths of an inch in diameter. Write this measure in standard form and in scientific notation. (Lesson 4-7)

Source: The Tax Foundation

Write a fraction in simplest form comparing the amount spent on housing assistance and the total amount spent.

15. BUSINESS A large corporation estimates its yearly revenue at $4.72 × 108. Write this number in standard form. (Lesson 4-7)

(Lesson 4-4) Mixed Problem Solving

797

Chapter 5 Rational Numbers

(pages 226–287)

5 1. FURNITURE A shelf 16_ inches wide is to 8 3 inches be placed in a space that is 16_ 4 wide. Will the shelf fit in the space? Explain. (Lesson 5-1)

2. MEASUREMENT A piece of metal is 0.025 inch thick. What fraction of an inch is this? (Lesson 5-2) 3. HEALTH You can stay in the Sun 15 times longer than usual without burning by applying SPF number 15. If you usually 1 hour in the Sun, how long burn after _ 4 could you stay in the Sun using SPF 15 lotion? (Lesson 5-3)


4. MONEY A dollar bill remains in circulation 1 1 years. A coin lasts about 22_ about 1_ 2 4 times longer. How long is a coin in circulation? (Lesson 5-3) 5. FOOD If each guest at a party eats twothirds of a small pizza, how many guests would finish 12 small pizzas? (Lesson 5-4) 6. PUBLISHING A magazine page is 8 inches wide. The articles are printed in three 1 inch of space in between columns with _ 4 3 _ and -inch margins on each side. 8

1" 4

120

3" 8

Gsdg as jsdbf ashfuhau f s ath fgsbd g dOaoiwrn a addf sad sa a s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf gjsd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d f sfih sdfk awe i dzfh a sdjf f da fasd.Gsdg as jsdbf as fgsbd g dOaoiwrn a addf sad sa a s sjnfksa asjf. Yas d skdfo asdjoa awi e sjhf sddhf g sd. ash asdfh wfp j g aug asjf su sfih sdfk a e i dzfh a sdjf f da sdg as jsdbf a fgsbd g dOao ddf sad sa a s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf g sd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d f sfih sdfk awe i dzfh a sdjf f da fasd.Gsdg as jsdbf as jfgsbd g dOaoiwrn a addf sad sa a s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf g sd. Odjbfh ash asdfh wfp j gsaug asjf sue g f sfih sdfk awe i dg

?

1" 4

Gsdg as jsdbf ashfuhau f sfath fgsbd g dOaoiwrn a addf sad sa a s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf gjsd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d f sfih sdfk awe i dzfh a sdjf f da fasd.Gsdg as jsdbf as fgsbd g dOaoiwrn a addf sad sa a s sjnfksa asjf. Yas d skdfo asdjoa awi e sjhf sddhf g sd. ash asdfh wfp j g aug asjf su sfih sdfk a e i dzfh a sdjf f da sdg as jsdbf a fgsbd g dOao ddf sad sa a s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf g sd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d f sfih sdfk awe i dzfh a sdjf f da fasd.Gsdg as jsdbf as jfgsbd g dOaoiwrn a addf sad sa a s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf g sd. Odjbfh ash asdfh wfp gsaug asjf s f sfih sdfk awe i f as

?

(Lesson 5-7)

10. OIL PRODUCTION Texas and Alaska produced a total of 1372.2 million barrels of oil. Alaska produced 684.0 million barrels. How many barrels of oil were produced in Texas? (Lesson 5-8) 11. EMPLOYMENT The table below shows the earnings per woman for every $100 earned by a man in the same occupation for two years. Earnings ($) Occupation Year 1

Year 2

Nurse

99.50

104.70

Teacher

88.60

90.30

Police Officer

91.20

94.20

Food Service

102.50

105.60

Postal Clerk

93.40

94.60

November

Gsdg as jsdbf ashfuhau f sfath fgsbd g dOaoiwrn a addf sad saia s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf gjsd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d f sfih sdfk awe dzfh a sdjf f da fasd.Gsdg as jsdbf as fgsbd g dOaoiwrn a addf sad saia s sjnfksa asjf. Yas d skdfo asdjoa awi e sjhf sddhf g sd. ash asdfh wfp j g aug as f su sfih sdfk a e i dzfh a sdjf f da sdg as jsdbf a fgsbd g dOaoi ddf sad saia s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf g sd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d f sfih sdfk awe i dzfh a sdjf f da fasd.Gsdg as jsdbf as jfgsbd g dOaoiwrn a addf sad saia s sjnfksa asjf. Yasjd skdfo asdjoa awi e sjhf sddhf g sd. Odjbfh ash asdfh wfp jhgsaug asjf sue gi d sdfk awe i djf

3" 8

Find the mean, median, and mode of the earnings for each year. (Lesson 5-9)

?

How wide should an author set the columns on her computer so that they are the same width as in the magazine? (Lesson 5-4)

7. REMODELING In their basement, the 3 -inch thick paneling Jacksons installed _ 8

5 -inch over a layer of dry wall that is _ 8 thick. How thick are the wall coverings? (Lesson 5-5)

3 8. COLLEGE In a college dormitory, _ of the

WEATHER For Exercises 12–14, use the following information. The table shows the average wind speeds for sixteen windy U.S. cities. (Lesson 5-9) 8.9

7.1

9.1

9.0

10.2

12.5

11.9

11.0

12.8

10.4

10.5

8.6

7.7

9.6

9.1

8.1


12. What is the difference between the least and greatest values?

2 of the residents are from Rockford, and _

13. Find the mean, median, and mode of the data.

residents are from Chicago. Which city has a greater representation? (Lesson 5-6)

14. Does the mode represent the data well? Explain.

8

5

798

1 9. NUTRITION A survey found that _ of 6 American households bought bottled 1 of American water in 2000. Only _ 17 households bought bottled water in 1993. What fraction of the population bought bottled water in 2000 that did not in 1993?


Chapter 6


1. SHOPPING Best buys in grocery stores are generally found by comparing unit rates such as cents per ounce. Which bag of nachos shown in the table below is the better buy? (Lesson 6-1) Size

Price

16-oz

$2.49

32-oz

$3.69

(pages 290–355)

8. HEALTH Doctors estimate that 3 babies out of every 1000 are likely to get a cold during their first month. What percent is this? (Lesson 6-5) 9. NUTRITION Refer to the nutritional label from a bag of pretzels shown below. Nutrition Facts Serving Size 1 package (46.8g) Servings per container 1

2. LENGTH There are 3.28 feet in a meter. Write an equation comparing the number of feet in a meter. How many meters are there in 42.64 feet? (Lesson 6-2)

$1 US is equal to: Rate Unit

0.835 0.565 1.178 115.549 45.125

European euros British pounds Canadian dollars Japanese yen Indian rupees

% Daily Value* Total Fat 1.5g 3% Saturated Fat 0g 0% Cholesterol 0mg 0% Sodium 760mg 32% Total Carbohydrate 37g 12%

The 760 milligrams of sodium (salt) in one serving is 32% of the recommended daily value. What is the total recommended daily value of sodium? (Lesson 6-6) 10. FAST FOOD A certain hamburger has 560 Calories, and 288 of these are from fat. About what percent of the Calories are from fat? (Lesson 6-7)

Source: xe.com

3. Write an equation for each exchange rate. 4. How many Japanese yen can you receive for $25? 5. COOKING A recipe that makes 72 cookies 1 cups of flour. How many calls for 4_ 2 cups of flour would be needed to make 48 cookies? (Lesson 6-3) 6. FERRIS WHEEL In a scale model of a Ferris wheel, the diameter of the wheel is 5 inches. If the actual height of the wheel is 55 feet, what is the scale of the model? (Lesson 6-4)

7. BUSINESS An executive of a marshmallow company said that marshmallows are 80% air. What fraction of a marshmallow is air? (Lesson 6-5)

11. MONEY If Simone wants to leave a tip of about 15% on a dinner check of $23.85, how much should she leave? (Lesson 6-7) 12. BUSINESS Many car dealers offer special interest rates as incentives to attract buyers. How much interest would a person pay for the first month of a $5500 car loan if the monthly interest rate is 0.24%? (Lesson 6-8) 13. PETS Hedgehogs are becoming so popular as pets that some breeders have reported a 250% increase in sales in recent years. If a breeder sold 50 hedgehogs one year before the increase, how many should he or she expect to sell a year from now? (Lesson 6-9) 14. SKATEBOARDING A sporting goods store holds a survey to find how many people in the United States have skateboarded. Is this sampling method valid? (Lesson 6-10) Mixed Problem Solving

799


CURRENCY For Exercises 3–4, use the following information. The table shows the exchange rates for selected countries compared to the U.S. dollar on a certain day. (Lesson 6-2)

Amount per serving Calories 190 Calories from Fat 15

Chapter 7 Functions and Graphing SHIPPING RATES For Exercises 1–3, use the following information. The shipping costs for mail-order merchandise are given in the table below. (Lesson 7-1) Total Price of Merchandise $0-$30.00

Shipping Cost $4.25

$30.01-$70.00

$5.75

$70.01 and over

$6.95

1. What is the shipping cost of merchandise totaling $75? 2. For what price of merchandise is the shipping cost $5.75?


3. Does the table represent a function? Explain. PHYSICS For Exercises 4 and 5, use the following information. As a thunderstorm approaches, you see lightning as it occurs, but you hear the accompanying thunder a short time afterward. The distance y in miles that sound travels in x seconds is given by y = 0.21x. (Lesson 7-2) 4. Find three ordered pairs that relate x and y. 5. How far away is lightning when thunder is heard 2.5 seconds after the lightning is seen? 6. HOUSING The median price of existing single-family homes in the United States can be represented by y = 12,600x + 132,667 when x represents the number of years since 2000. Use the equation to find the median prices in 2008, 2009, and 2010. Round to the nearest thousand dollars. (Lesson 7-2)

7. The table shows the growth of the Pacific giant kelp plant. Find the rate of change for the linear function. (Lesson 7-3)

800

Time (days) x

Growth (cm) y

3

135

4

180

7

315


(pages 356–415)

AVIATION For Exercises 8 and 9, use the following information. The table shows the altitude of a jet liner. (Lesson 7-4)

8. Find the constant rate of change for the linear function and interpret its meaning. 9. Determine whether a proportional linear relationship exists between time and altitude. Explain your reasoning.

Time (min) x

Altitude (ft) y

1

22,500

2

21,000

5

16,500

10

9000

10. KITES Arthur is flying a kite in the park. The kite is a horizontal distance of 20 feet from Arthur’s position and a vertical distance of 70 feet. Find the slope of the kite string. (Lesson 7-5) 11. BUSINESS A company’s monthly cost y is given by y = 1500 + 12x, where x represents the number of items produced. State the slope and y-intercept of the graph of the equation and describe what they represent. (Lesson 7-6) CAR RENTAL For Exercises 12 and 13, use the following information. It costs $59 per day plus $0.12 per mile driven to rent a minivan. (Lesson 7-7) 12. Write an equation in slope-intercept form that shows the cost y for renting a minivan for one day and driving x miles. 13. Find the daily rental cost if 30 miles are driven. OLYMPICS For Exercises 14 and 15, use the table that shows winning Olympic women’s high jump heights.

Year

Height (m)

1928

1.59

1952

1.67

(Lesson 7-8)

1968

1.82

14. Write an equation in slope-intercept form for a line of fit.

1988

2.03

2004

2.06


15. Estimate the height of the winning high jump in the 2028 Olympics.

Chapter 8 Equations and Inequalities CENSUS For Exercises 1–3, use the following information. The table below shows the 2000 populations and the average rates of change in population in the 1990s for Buffalo, New York, and Corpus Christi, Texas. Suppose the population of each city continued to increase or decrease at these rates. (Lesson 8-1) City Buffalo, NY Corpus Christi, TX

Population in 2000 293,000

Yearly Rate of Change -3500

277,000

+2000

1. Write an expression for the population of Buffalo after x years. 2. Write an expression for the population of Corpus Christi after x years.

4. INTERNET One Internet provider charges $19.95 a month plus $0.21 per minute, and a second provider charges $24.95 a month plus $0.16 per minute. For how many minutes is the cost of the plans the same?

8. SAVINGS Randall is saving money to buy a new mountain bike. The bikes that he likes start at $375, and he has already saved $285. Write and solve an inequality to find the amount he must still save. (Lesson 8-4) 9. STATISTICS The Boston Marathon had more than 2,600,000 spectators along its 26-mile route. Write and solve an inequality to find the average number of spectators per mile. (Lesson 8-5) 10. GROCERY SHOPPING Mrs. Hiroshi spends at least twice as much on her weekly grocery shopping as she did one year ago. Last year, she spent $54 each week. How much is Mrs. Hiroshi now spending each week on groceries? (Lesson 8-5) 11. GEOMETRY An acute angle has a measure less than 90°. If the measure of an acute angle is 2x, write and solve an inequality to find the possible values of x. (Lesson 8-5)

2x ˚

(Lesson 8-1)

5. GEOMETRY The length of a rectangle is three times the difference between its width and two. Find the width if the length is 15 inches. (Lesson 8-2)

6. SPORTS More than 100,000 fans attended the opening football game of the season. Write an inequality for the number of people who attended. (Lesson 8-3)

7. SCHOOL Liliana has math and English homework tonight. She has no more than 90 minutes to spend on her homework. Suppose Liliana spends 35 minutes completing her math homework. Write and solve an inequality to find how much time she can spend on her English homework. (Lesson 8-4)

12. SHOPPING Armando plans to spend at most $85 on jeans and shirts. He bought 2 shirts for $15.30 each. How much can he spend on jeans? (Lesson 8-6) 13. CAR SALES A car salesperson receives a monthly salary of $1000 plus a 3% commission on every car sold. For what amount of monthly sales will the salesperson earn more than $2500? (Lesson 8-6)

14. SCHOOL Dave has earned scores of 73, 85, 91, and 82 on the first four of five math tests for the grading period. He would like to finish the grading period with a test average of at least 82. What is the minimum score Dave needs to earn on the fifth test in order to achieve his goal? (Lesson 8-6) Mixed Problem Solving

801


3. In how many years would the population of the two cities be the same?

(pages 416–457)

Chapter 9


1. CONSTRUCTION A banquet facility must allow at least 4 square feet for each person on the dance floor. Reston’s Hotel is adding a square dance floor that will be large enough for 100 people. How long should it be on each side? (Lesson 9-1) 2. PHYSICS The time t in seconds that it takes an object to fall d feet can be estimated by using d = 0.5gt2. In this formula, g is acceleration due to gravity, 32 ft/s2. If a ball is dropped from the top of a 55-foot building, how long does it take to hit the ground? (Lesson 9-2)


3. GEOMETRY Use the formula A = ␲r2 to find the radius of a circle if the area is 380.13 square millimeters. Round to the nearest millimeter. (Lesson 9-2) 4. DISTANCE The table shows the distance between Cincinnati, Ohio, and two other cities. Suppose a triangle was formed by drawing a line between each pair of cities. Classify the triangle by its sides. (Lesson 9-3) Distance from Cincinnati City

Distance (mi)

Minneapolis

692

Omaha

692

5. PETS Jason’s elderly dog needs a ramp to get into Jason’s truck. A pet store sells a ramp that is 60 inches long and will be attached to the back of Jason’s truck 14 inches off the ground. What kind of angle does the ramp make with the ground? (Lesson 9-3)

(Lesson 9-3)


7. BASEBALL A baseball diamond is actually a square with 90 feet between the bases. What is the distance between home plate and second base?

second base 90 ft

90 ft

90 ft

90 ft home plate

(Lesson 9-4)

8. SAILING A rope from the top of a sailboat mast is attached to a point 6 feet from the base of the mast. If the rope is 24 feet long, how high is the mast? (Lesson 9-4) 9. TRAVEL Trey’s home is at (4, 9) on the map. His friend Nicolas’ home is at (6, 3) on the same map. If each unit on the map is 1 mile, how far do the two friends live from each other? Round to the nearest tenth. (Lesson 9-5) 10. HISTORY The largest known pyramid is Khufu’s pyramid. At a certain time of day, a yardstick casts a shadow 1.5 feet long, and the pyramid casts a shadow 241 feet long. Use shadow reckoning to find the height of the pyramid. (Lesson 9-6) 11. SURVEYING A surveyor needs to find the distance across a river and draws the sketch shown below.


6. UTILITIES A support cable is sometimes attached to give a utility pole stability. If the cable makes an angle of 65° with the ground, what is the measure of the angle formed by the cable and the pole?

(pages 460–509)

Q

xm

V

16 m

10 m

S

20 m

T

Find the distance across the river. (Lesson 9-6) x˚

65˚

12. DOLLHOUSES Ella has a dollhouse with furniture that matches the furniture in her room. The dollhouse furniture was 1 1 -inch scale. That is, _ inch produced on a _ 2 2 on the dollhouse furniture corresponds to 1 foot on Ella’s furniture. If the dollhouse 1 inches wide, how wide is Ella’s bed is 2_ 2 bed? (Lesson 9-6)

Chapter 10


1. TRANSPORTATION The angle at the corner where two streets intersect is 125°. If a bus cannot make a turn at an angle of less than 70°, can bus service be provided on a route that includes turning that corner in both directions? Explain. (Lesson 10-1)

(pages 510–571)

SIGNS For Exercises 7 and 8, use the following information. Part of a driver’s license exam includes identifying road signs by color and by shape. Identify the shape of each road sign pictured below. (Lesson 10-6) 7.

125˚

8.

STOP MANUFACTURING For Exercises 9 and 10, use the following information.

BRIDGES For Exercises 2 and 3, use the following information. The figure below shows part of the support structure of a bridge. Name a triangle that seems to be congruent to each triangle below. (Lesson 10-2) B

C

H

D

E

2. AFB

9. What is the shape of the tray?

G

F

10. Find the measure of each angle of the tray so that the trays will fit side-to-side around the table.

3. CHG

4. MOVING A historic house in the shape of a rectangle has coordinates A(-3, 5), B(4, 5), C(4, -3), and D(-3, -3) on a map. The house is going to be moved to a new site 3 units east and two units north. Find the coordinates of the house once it reaches the new site. (Lesson 10-3) 5. SHAPES Name three items in your room that are quadrilaterals. Classify the shapes. (Lesson 10-4) 6. GEOGRAPHY The state of Indiana is shaped almost like a trapezoid. Estimate the area of the state. (Lesson 10-5)

140 mi

200 mi 280 mi

11. PUBLIC SAFETY A tornado warning system can be heard for a 2-mile radius. Find the area that will benefit from the warning. Round to the nearest square mile. (Lesson 10-7)

12. CITY PLANNING The circular region inside the streets at DuPont Circle in Washington, D.C., is 250 feet across. What is the area of the region? Round to the nearest tenth. (Lesson 10-7) 13. MONUMENTS The Stonehenge monument in England is enclosed within a circular ditch that has a diameter of 300 feet. Find the area within the ditch to the nearest tenth. (Lesson 10-7) 14. GEOMETRY Find the area of a figure that is formed using a rectangle having width equal to 8 feet and length equal to 5 feet and a half circle with a diameter of 6 feet. (Lesson 10-8) Mixed Problem Solving

803


A

Some cafeteria trays are designed so that four people can place their trays around a square table without bumping corners, as shown below. The top and bottom of the tray are parallel. (Lesson 10-6)

Chapter 11 Three -Dimensional Figures 1. PRESENTS Zacarias received a gift wrapped in the shape of a rectangular pyramid. How many faces, edges, and vertices are on the gift box? (Lesson 11-1)

(pages 572–621)

9. CAMPING What is the minimum amount of canvas that was used to make the A-frame tent shown below? (Hint: Be sure to include the floor of the tent.) (Lesson 11-4)

2. PET CARE Tina has an old fish tank in the shape of a circular cylinder. The tank is 2 feet in diameter and 6 feet high. How many cubic feet of water does it hold? Round to the nearest cubic foot. (Lesson 11-2)


3. STORAGE A portable storage container takes up 90 square feet of floor space. If the walls of the container are 8 feet tall, what is the volume of the storage container? (Lesson 11-2)

4.5 ft 4 ft 6 ft 4 ft

10. HISTORY The Pyramid of Cestius is a monument in Rome. It is a square pyramid with the dimensions shown below.

4. CHEMISTRY A quartz crystal is a hexagonal prism. It has a base area of 1.41 square centimeters and a volume of 4.64 cubic centimeters. What is its height? Round to the nearest hundredth, if necessary.

39.9 m

30 m 30 m

(Lesson 11-2)

5. BAKING A rectangular cake pan is 30 centimeters by 21 centimeters by 5 centimeters. A round cake pan has a diameter of 21 centimeters and a height of 4 centimeters. Which holds more batter, the rectangular pan or two round pans? (Lesson 11-2)

6. MONUMENTS The top of the Washington Monument is a square pyramid 54 feet high and 34 feet long on each side. What is the volume of this top part of the monument? (Lesson 11-3) 7. PACKAGING Olinda plans to ship a minibasketball with a circumference of 24 inches in a box that measures 8 inches by 8 inches by 8 inches. Will the basketball fit in the box? Explain. 8. MANUFACTURING A carton of canned fruit holds 24 cans. Each can has a diameter of 7.6 centimeters and a height of 10.8 centimeters. Approximately how much paper is needed to make the labels for the 24 cans? Round to the nearest tenth, if necessary. (Lesson 11-4) 804 Mixed Problem Solving

What is its lateral area? If necessary, round to the nearest tenth. (Lesson 11-5) 11. TEPEES The largest tepee in the United States is in the shape of a cone with a diameter of 42 feet and a slant height of about 47.9 feet. How much canvas was used for the cover of the tepee? Round to the nearest tenth, if necessary. (Lesson 11-5) 12. SHIPPING Are the two packing tubes shown below similar solids? (Lesson 11-6) 1 2

22 in.

10 in. 4 in.

9 in.

13. MODELS A miniature greenhouse is a rectangular prism with a volume of 16 cubic feet. The scale factor of this greenhouse to a larger greenhouse of the 1 . What is the volume of same shape is _ 4 the larger greenhouse? (Lesson 11-6)

Chapter 12 More Statistics and Probability 1. ARCHITECTURE The World Almanac lists fifteen tall buildings in New Orleans, Louisiana. The number of floors in each of these buildings is listed below. 51 53 45 39 36 47 42 33 32 31 33 28 28 25 23

Make a stem-and-leaf plot of the data. (Lesson 12-1)

2. WORLD CULTURES Many North American Indians hold conferences called powwows, to celebrate their culture and heritage through various ceremonies and dances. The ages of participants and observers in a Menominee Indian powwow are shown in the chart below. 20, 18, 12, 13, 14, 72, 65, 23, 25, 43, 67, 35, 68, 13, 56

Observers

43, 55, 70, 63, 15, 41, 9, 42, 75, 25, 16, 18, 51, 80, 75, 39, 23, 55, 50, 54, 60, 43

Find the range and interquartile range for each group. (Lesson 12-2) 3. CONSUMERISM The average retail prices for one gallon of unleaded gasoline at a certain station are shown in the table below. 1 2.21 6 2.02

Year Price ($) Year Price ($)

2 2.20 7 2.16

3 1.92 8 2.14

4 1.95 9 2.13

5 1.95 10 2.11

Make a box-and-whisker plot of the data. (Lesson 12-3)

4. HOMEWORK The frequency table below shows the amount of time students spend doing homework each week. Weekly Homework Time Number of Hours

Tally

Frequency

0–3

||||

4–7

|||| |||| ||||

14

8–11

|||| |||| |||| |||

18

|||| |||| |

11

12–15

5

Display the data in a histogram. (Lesson 12-4)

5. ENTERTAINMENT The graph below displays data about movie attendance. 1.3 1.28 1.26 1.24 1.22 1.2 1.18 1.16 1.14 1.12 1.1

’01

’02

’03

’04

’05

’06

Tell why the graph appears to be misleading. (Lesson 12-6) 6. CANDY In a small bag of colored chocolate candies, there are 15 green, 23 red, and 18 yellow candies. What is the probability of selecting a red candy if one is taken from the bag at random? (Lesson 12-7) 7. BUSINESS The Yogurt Oasis advertises that there are 1512 ways to enjoy a one-topping sundae. They offer six flavors of frozen yogurt, six different serving sizes, and several different toppings. How many toppings do they offer? (Lesson 12-8) 8. VOLLEYBALL How many different 6-player starting squads can be formed from a volleyball team of 15 players? (Lesson 12-9) 9. BUSINESS An auto dealer finds that of the cars coming in for service, 70% need a tune up and 50% need a new air filter. What is the probability that a car brought in for service needs both a tune up and a new air filter? (Lesson 12-10) 10. ECONOMICS Thirty-one percent of minimum-wage workers are between 16 and 19 years old. Twenty-two percent of the minimum-wage workers are between 20 and 24 years old. If a person who makes minimum wage is selected at random, what is the probability that he or she will be between 16 and 24 years old? (Lesson 12-10) Mixed Problem Solving

805


Participants

(pages 624–697)

Chapter 13 Polynomials and Nonlinear Functions ARCHITECTURE For Exercises 1 and 2, use the following information. The polynomial 2xy + 2y2 + 2yz represents the total area of the first floor shown in the plan below. (Lesson 13-1)

(pages 698–737)

MANUFACTURING For Exercises 6 and 7, use the following information. The figure below shows a pattern for a cardboard box before it has been cut and folded. (Lesson 13-4) 2x ⫹ 1

2y 2y

x

x Kitchen

HIS SIDE UP

y

z

6. Find the area of each rectangular region and add to find a formula for the number of square inches of cardboard needed.


1. Find the degree of the polynomial. 2. Find an expression to represent the area of the living room. Then classify the expression as a monomial, binomial, or trinomial.

3. CONSTRUCTION A standard unit of measurement for a window is the united inch. You can find the united inches of a window by adding the length and width of the window. If the length of a window is 3x - 5 inches and the width is x + 7 inches, what is the size of the window in united inches? (Lesson 13-2)

4. GEOMETRY The perimeter of the triangle below is 4x + 4 centimeters. Find the length of the hypotenuse of the triangle. (Lesson 13-3) ?

7. Find the surface area if x is 2.5 inches. 8. PRODUCTION The XYZ Production Company states that the cost y of producing x items is given by the equation y = 2500 + 3.2x. Does this equation represent a linear or nonlinear function? (Lesson 13-5)

9. INTERNET The graph below shows the increase in electronic mailboxes in the United States.

iVÌÀV >LÝiÃ Ã®

4y

2x ⫺ 4 cm

2x ⫹ 1 x

x

Living Room

x

HIS SIDE UP

Dining Room

Y Îxä Îää Óxä Óää £xä £ää xä ä

¼n{ ¼nÇ ¼ä ¼Î ¼È ¼ ¼äÓ X 9i>À

3OURCE -ESSAGING /NLINE

x ⫹ 3 cm

Does this graph represent a linear or nonlinear function? Explain. (Lesson 13-5)

5. GEOMETRY Find the area of the shaded region. Write in simplest form. (Lesson 13-4) 2s s


s 3

10. POPULATION The population growth of a particular species of insect is given by the equation y = 2x3, where x represents time elapsed in days and y represents the population size. Graph this equation. (Lesson 13-6)

EXTEND

10-6

Main Idea Generate formulas involving area.

Graphing Geometric Relationships In this lab, you will generate formulas involving area.

ACTIVITY 1 Draw five parallelograms that have a height of 4 centimeters. Create a table to record the length of the base, the height, and the area of each parallelogram. Then graph the data to show the relationship between the area and base of this set of parallelograms.

CM BASE

Step 1 Draw each parallelogram on centimeter grid paper. Step 2 Copy and complete the table shown for each parallelogram. Base (cm)

Height (cm)

Area (cm2)

4 4 4 4 4

Step 3 Graph the ordered pairs (base, area) on a coordinate plane.

ANALYZE THE RESULTS 1. What does an ordered pair on your graph represent? 2. Sketch and describe the shape of the graph. 3. Write an equation in which x represents the length of the base and y represents the area. What does your equation mean? 4. Repeat Steps 1 – 3 and Exercises 1-3 using five parallelograms that have a height of 3 centimeters and 5 triangles that have a height of 6 centimeters. 5. As the length of the base of the parallelogram or triangle increases,what happens to its area? Does this happen at a constant rate? How can you tell this from the table? from the graph?

Extend 10-6 Graphing Geometric Relationships

807

ACTIVITY 2 Draw five rectangles that have an area of 36 square centimeters and whose length is longer than or equal to its width. Create a table to record the length and width of each rectangle. Then graph the data to show the relationship between the length and width of this set of triangles.

! CM

LENGTH

Step 1 Draw each rectangle on centimeter grid paper. Step 2 Copy and complete the table shown for each rectangle. Length (cm)

Width (cm)

Area (cm2) 36 36 36 36 36

Step 3 Graph the ordered pairs (length, width) on a coordinate plane.

ANALYZE THE RESULTS 6. What does an ordered pair on your graph represent? 7. Sketch and describe the shape of the graph. 8. Write an equation in which x represents the length of the base and y represents the area. What does your equation mean? 9. As the length of the rectangle increases, what happens to its width? Does this happen at a constant rate? How can you tell this from the table? from the graph? 10. MAKE A PREDICTION Draw five cubes with different edge lengths. Predict the shape of the graph of the relationship between the edge length and the volume of the cube. 11. Create a table to record the edge length and volume of each cube. Then graph the data to show the relationship betwen the edge length and the volume of the cube. Sketch and describe the shape of of the graph. 12. Write an equation for your graph. What does your equation mean? 13. As the length of the cube’s edge increases, what happens to the volume? 808 Extend 10–6 Graphing Geometric Relationships

WIDTH

Preparing for Standardized Tests

Becoming a Better Test-Taker At some time in your life, you will probably have to take a standardized test. Sometimes this test may determine if you go on to the next grade level or course, or even if you will graduate from high school. This section of your textbook is dedicated to making you a better test-taker.

TYPES OF TEST QUESTIONS In the following pages, you will see examples of four types of questions commonly seen on standardized tests. A description of each type is shown in the table below. Type of Question multiple choice gridded response short response extended response

Description

See Pages

Four or five possible answer choices are given from which you choose the best answer. You solve the problem. Then you enter the answer in a special grid and shade in the corresponding circles. You solve the problem, showing your work and/or explaining your reasoning. You solve a multi-part problem, showing your work and/or explaining your reasoning.

810–813 814–817 818–821 822–826

PRACTICE After being introduced to each type of question, you can practice that type of question. Each set of practice questions is divided into five sections that represent the concepts most commonly assessed on standardized tests. • • • • •

Number and Operations Algebra Geometry Measurement Data Analysis and Probability

USING A CALCULATOR On some tests, you are permitted to use a calculator. You should check with your teacher to determine if calculator use is permitted on the test you will be taking, and if so, what type of calculator can be used.

TEST-TAKING TIPS In addition to the Test-Taking Tips like the one shown on the right, here are some additional thoughts that might help you. • Get a good night’s rest before the test. Cramming the night before does not improve your results. • Budget your time when taking a test. Don’t dwell on problems that you cannot solve. Just make sure to leave that question blank on your answer sheet. • Watch for key words like NOT and EXCEPT. Also look for order words like LEAST, GREATEST, FIRST, and LAST.

If you are allowed to use a calculator, make sure you are familiar with how it works so that you won’t waste time trying to figure out the calculator when taking the test.


809


Multiple-Choice Questions Multiple-choice questions are the most common type of question on standardized tests. These questions are sometimes called selected-response questions. You are asked to choose the best answer from four or five possible answers. To record a multiple-choice answer, you may be asked to shade in a bubble that is a circle or an oval or to just write the letter of your choice. Always make sure that your shading is dark enough and completely covers the bubble.

Incomplete Shading A

A

A

Nathan wants to save $500 for a season ski pass. He has $200 and can save $25 per week. About how many months will he need to save in order to have enough money? A 20 months B 15 months C 12 months D 8 months E 3 months Notice that the problem gives the amount that Nathan can save each week, but asks for the time it will take to save the money in months. Since Nathan can save $25 per week and there are about 4 weeks in a month, he can save about 4 · $25 or $100 each month. Let x represent the number of months. So, the amount he will have after x months is 200 + 100x. Since Nathan wants to save $500, set the expression equal to 500. Write the equation 500 = 200 + 100x. You can use backsolving to find the correct answer.

500 = 200 + 100x 500 200 + 100(20) 500 ≠ 2200

Replace x with 20.

500 = 200 + 100x 500 200 + 100(15) 500 ≠ 1700

Replace x with 15.

500 = 200 + 100x 500 200 + 100(12) 500 ≠ 1400

Replace x with 12.

500 = 200 + 100x 500 200 + 100(8) 500 ≠ 1000

Replace x with 8.

500 = 200 + 100x 500 = 200 + 100(3) 500 = 500

Replace x with 3.

The answer is E.

810


D

B

C

D

Correct shading

EXAMPLE

Next, test each value given in the answer choices.

C

Too light shading

To make sure you have the correct solution, you must check to make sure that your answer satisfies the conditions of the original problem.

STRATEGY

B

B

C

D


Many multiple-choice questions do not include a diagram. Drawing a diagram for the situation can help you to answer the question.

EXAMPLE Isabelle and Belinda take a hiking trip. They want to get to Otter’s pond but cannot walk directly to it. They start from the beginning of the trail and follow the trail for 4 miles to the west. Then they turn south and walk 6 miles. How far is Otter’s Pond from the start of the trail? Round to the nearest tenth of a mile. F 1.4 mi STRATEGY Whenever possible, use a drawing to help you solve a problem.

G 4.5 mi

H 6.3 mi

J 7.2 mi

Draw a diagram of the hiking trip. Isabelle and Belinda have walked in a path that creates a right triangle. Looking at the diagram, you can eliminate 1.4 miles since it is too small.

4 miles

6 miles

Start

?

The legs of the right triangle are 4 and 6. Use the Pythagorean Theorem to find the hypotenuse. c2 = a2 b2 c2 = 62 42 c2 = 36 16 c2 = 52 √ c2 √ 52 c ≈ 7.2

Otter’s Pond

Pythagorean Theorem Replace a with 6 and b with 4. Evaluate 62 + 42. Add 36 and 16. Take the square root of each side. Round to the nearest tenth.

The answer is J, 7.2 miles.

Often multiple-choice questions require you to convert measurements to solve. Pay careful attention to each unit of measure in the question and the answer choices.

EXAMPLE Malik is planning to draw a large map of his neighborhood for a school project. He wants the scale for the map to be 1 inch = 8800 feet. His house is 2.5 miles from school. How far on the map will his house be from the school? A 0.5 in.

B 0.8 in.

C 1.0 in.

D 1.5 in.

E 1.8 in.

The actual distance from Malik’s house to the school is given in miles, and the scale is given in feet. You need to convert from miles to feet to solve the problem. Since 1 mile is equal to 5280 feet, 2.5 miles = 2.5(5280) or 13,200 feet. Now use the scale factor to find the distance. STRATEGY Have you really answered the question?

1 in. or 1.5 inches 13,200 ft × _ 8800 ft

On the map, Malik’s house will be 1.5 inches from the school. Choice D is the answer.


811

Choose the best answer.

Number and Operations 1. In 2001, the population of China was approximately 1,273,000,000. Write the population in scientific notation. A 12.73 × 101

C 1.27 × 109

B 1273.0 × 106

D 1.273 × 109

2. Tyler uses 4 gallons of stain to cover 120 square feet of fence. He still has 520 square feet left to cover. Which proportion could he use to calculate how many more gallons of stain he will need? 120 x F _ =_ 520

120 520 H _ =_ x

4

4 120 4 _ _ J = x 520

x 4 G _ =_ 520

120

3. If 1.5 cups of nuts are in a bag of trail mix that serves 6 people, how many cups of nuts will be needed for a trail mix that will serve 9 people? A 1c

C 2.25 c

B 1.5 c

D 2.5 c

6. Six friends go to a movie and each buys a large container of popcorn. If a movie ticket costs $8.75 and a large popcorn costs $2.25, which expression can be used to find the total cost for all six people? F 2.25(8.75 + 5)

H 6(8.75 + 2.25)

G 2.25 + 8.75(6)

J 6(2.25) + 8.75

7. Midtown Printing Company charges $50 to design a flyer and $0.25 per flyer for printing. If y is the total cost in dollars and x is the number of fliers, which equation describes the relationship between x and y? A y = 50 - 0.25x

C y = 50 + 0.25x

B y = 50x + 0.25

D y = 0.25x 50

8. The simple interest formula, I = Prt, gives the interest I earned for an amount of money invested P at a given rate r for t years. If Nicholas invests $2100 at an annual interest rate of 7.5%, how long will it take him to earn $3000? Round to the nearest year. F 3 yr

4. Hannah has a roll of ribbon for wrapping 2 presents that is 10 yards long. If it takes _ yard of 3 ribbon to wrap each present, how many presents can she wrap with the 10 yards? F 8

H 15

G 10

J 20

J 52 yr

Geometry

A reflection

5. Ms. Blackwell needs to rent a car for her family vacation. She has found a company that offers the following two options. Plan

Flat Rate

Cost per Mile

Option A Option B

$40 $30

$0.25 $0.35

How many miles must the Blackwells drive for the plans to cost the same?

A 0 mi

C 25 mi

B 10 mi

D 100 mi


H 20 yr

9. Alyssa wants to redecorate her room. She makes a scale diagram of her room on paper. She then cuts out scale pictures of her bed, dresser, and desk. If she slides her dresser along the wall, what type of transformation is this?

Algebra

812

G 19 yr

B rotation

Dresser


Multiple-Choice Practice

Bed

C dilation D translation

Desk

Question 8 Some multiple-choice questions have you use a formula to solve a problem. You can check your solution by replacing the variables with the given values and your answer. The answer choice that results in a true statement is the correct answer.

6 blocks

F 9 blocks

Work

G 12 blocks H 13 blocks

15. Shopper’s Mart sells two sizes of Corn Crunch cereal. The 16-ounce box costs $4.95. The 12-ounce box costs $3.55. What is true about these two cereals?


10. To get to work from her house, Amanda walks 6 blocks west then turns and walks 11 blocks south. If she could walk directly home from work, how many blocks would she need to walk? Round to the nearest block.

A The 16-ounce box is a better buy. B The 12-ounce box is a better buy. C They are the same cost per ounce. D None of these statements are true.

11 blocks

J 15 blocks Home

11. Andrés plans to lay sod in his yard. How many square feet of sod will he need? A 5100 ft2

Question 15 Always read every answer choice, particularly in questions that ask what is true about a given situation.

120 ft

B 8925 ft2 C 9000 ft2 D 9650

85 ft

Data Analysis and Probability

ft2 90 ft

Measurement

16. The graph below shows the number of hours that students in Mr. Cardona’s math class watch television and the number of hours that they exercise in the same week. Based on the trend in the scatter plot, what number of hours of exercise would you expect a student to get that watches 15 hours of television each week?

12. The world’s largest ball of Sisal Twine is located in Cawker City, Kansas. It consists of about 7,009,942 feet of twine. How many miles of twine is this? F 586,080 mi

H 1402 mi

G 1328 mi

J 37,012,493,760 mi

13. The world’s largest pizza was made on October 11, 1987 by Lorenzo Amato and Louis Piancone. The pizza measured 140 feet across. If a regular size pizza at a local restaurant measures 12 inches across, how many times more area does the largest pizza cover than the regular pizza? A about 12 times

C 140 times

B about 136 times

D 19,600 times

14. A cookie recipe requires 6 cups of chocolate chips. The recipe serves 20 people. How many cups of chocolate chips would be needed to serve 35 people?

F 2h

H 10 h

G 6h

J 15 h

17. Mika received the following scores on 4 of his five social studies exams: 89, 75, 82, and 77. What score does he need on his fifth exam to ensure that he receives at least an average of 85? Note that a score cannot be over 100. A 75 B 85

F 10.5 c

H 12 c

C 95

G 11 c

J 15 c

D not possible Preparing for Standardized Tests

813


Gridded-Response Questions Gridded-response questions are another type of question on standardized tests. These questions are sometimes called student-produced response or grid in. For gridded response, you must mark your answer on a grid printed on an answer sheet. The grid contains a row of four or five boxes at the top, two rows of ovals or circles with decimal and fraction symbols, and four or five columns of ovals, numbered 0–9. At the right is an example of a grid from an answer sheet.

EXAMPLE

.

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

.

/ .

/ . 0

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

Solve the equation 10 - 4a = -38 for a. What value do you need to find? You need to find the value of a. 10 - 4a = -38 10 - 4a - 10 = -38 - 10

Original equation Subtract 10 from each side.

-4a = -48

Simplify.

-48 -4a _ =_


-4

-4

a = 12

Simplify.

How do you fill in the answer grid? • Print your answer in the answer boxes. • Print only one digit or symbol in each answer box. • Do not write any digits or symbols outside the answer boxes. • You may print your answer with the first digit in the left answer box, or with the last digit in the right answer box. You may leave blank any boxes you do not need on the right or the left side of your answer.

1 2 .

2 3 4 5 6 7 8 9

1 2

/ .

/ .

.

0 1

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

3 4 5 6 7 8 9

2 3 4 5 6 7 8 9

. 0 1 3 4 5 6 7 8 9

• Fill in only one bubble for every answer box that you have written in. Be sure not to fill in a bubble under a blank answer box.

Many gridded-response questions result in an answer that is a fraction or a decimal. These values can also be filled in on the grid.

EXAMPLE A recipe for orange chicken calls for 1 cup of orange juice and serves 6 people. If Emily needs to serve 15 people, how many cups of orange juice will she need? What value do you need to find? You need to find the number of cups of orange juice Emily will need for 15 servings.

814



Write and solve a proportion for the problem. Let s represent the number of cups. 1 cup s cups _ =_

6 servings 15 servings s 1 _=_ Write the proportion. 6 15

15 = 6s 15 6s _ =_ 6 6 5 _=s 2

Find the cross products. Divide each side by 6. Simplify.

How do you fill in the answer grid? 5 You can either grid the fraction _ , or rewrite it as 2.5 and grid the decimal. 2 Be sure to write the decimal point or fraction bar in the answer box. The 5 and 2.5. following are acceptable answer responses that that represent _ 2

5 / 2 . 1 2 3 4 6 7 8 9

. 0 1 2 3 4 5 6 7 8 9

/ . 0 1 3 4 5 6 7 8 9

5 / 2 . 0 1 2 3 4 5 6 7 8 9

/ .

.

0 1 2 3 4

1 2 3 4 5 6 7 8 9

6 7 8 9

2 . 5 /

.

.

.

0 1 2 3 4 5 6 7 8 9

0 1

1

3 4 5 6 7 8 9

3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

/ . 0 1 2 3 4 6 7 8 9

Do not leave a blank answer box in the middle of an answer.

2 . 5 . 0 1 2 3 4 5 6 7 8 9

. 1 2 3 4 5 6 7 8 9

/ .

/

0 1

0 1 2 3 4 5 6 7 8 9

3 4 5 6 7 8 9

. 0 1 2 3 4 6 7 8 9

Some problems may result in an answer that is a mixed number. Before filling in the grid, change the mixed number to an equivalent improper fraction or decimal. For 1 11 example, if the answer is 1_ , do not enter 11/2, as this will be interpreted as _ . Instead, 2 2 enter 3/2 or 1.5.

EXAMPLE The Corner Candy Store sells 16 chocolates in a gift box. For Mother’s Day the store offers the chocolates in boxes of 20. What is the percent of change? Write the ratio for percent of change. new amount - original amount percent of change = ___ original amount

Remember percent of change has the original amount as the denominator.

20 - 16 =_ 16

.

4 =_

Subtraction

= 0.25

Rewrite as a decimal.

16

2 5

Substitution

1

= 25%

Since the question asks for the percent, be sure to grid 25, not 0.25.

3 4 5 6 7 8 9

/ .

/ .

.

0 1 2 3 4

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

6 7 8 9


815


Gridded-Response Practice Solve each problem. Then copy and complete a grid like the one shown on page 815.

Number and Operations 1. The Downtown Department Store has a 75% markup on all clothing items. A certain sweater cost the store $20. What will be the selling price of the sweater in dollars? 2. The following table shows the number of billionaires in the following countries in the year 2000. Country

Number of Billionaires

USA Germany Japan China France/Mexico/ Saudi Arabia

70 18 12 8 7

What percent of the billionaires were in the USA? Round to the nearest percent.

8. Solve 4x - 5 = 2x + 3 for x. 9. Ayana has $125 dollars to spend on CDs and DVDs. CDs cost $20. The equation y = 125 - 20c represents the amount she has left to buy DVDs. If she buys 3 CDs, how much money in dollars will she have to buy DVDs? 10. For her birthday, Allison and her five friends went out for pizza. They ate an entire pizza that was cut into 16 pieces. If two of her friends ate 4 pieces, one of her friends ate 3 pieces and two of her friends ate 1 piece, how many pieces did Allison eat? 11. A parking garage has two different pay parking options. You can pay $11 for the day or $2 for the first hour and $0.75 for each additional half hour. How many hours would you need to stay for both rates to be the same?

Geometry 3. Pepperoni is the most popular pizza topping. Each year approximately 251,770,000 pounds of pepperoni are eaten. If this number were written in scientific notation, what would be the power of 10? 4. Dylan wants to buy an ice cream cone with three different flavors of ice cream. He has 18 flavors from which to choose. How many different ways can he have his cone? 5. Ignacio has $40 to spend at the mall. He spends half of it on a CD. He then spends $6 for lunch. Later he decides to go to a movie that costs $5.75. How much money does he have left in dollars? 6. There are 264 tennis players and 31 coaches at a sports camp. What is the ratio of tennis players to coaches as a decimal rounded to the nearest tenth?

Algebra 7. Find the x-intercept of the graph of the equation 2x + y = 5.

816


12. The bridge over the Hoover Dam in Lake Mead, Nevada, stretches 1324 feet. A model is built that is 60 feet long. What is the scale factor of the model to the actual bridge? Round to the nearest hundredth. 13. Use the figure to find the value of x. 65˚

(x ⫹ 5)˚

x˚

14. Quadrilateral ABCD is translated 2 units to the right and 3 units down to get ABCD. What is the y-coordinate of A? y

A B x

O

D

C

40 ft

20 ft

16. A building casts a shadow that is 210 feet long. If the angle of elevation from the end of the shadow to the top of the building is 40°, how tall is the building? Round to the nearest foot.

20. Darnell’s commute takes 1.5 hours. He drove at 20 miles per hour for 0.75 hour, and the rest of the time he drove at 50 miles per hour. In miles, how far is his work from his house? 21. A juice company wants to make cylindershaped juice cans that hold approximately 196 cubic inches of juice. The base has to have a diameter of 5 inches. What will be the height of the can in inches? Use π = 3.14 and round to the nearest whole number.

Data Analysis and Probability 22. The following table shows the average number of vacation days per year for people in selected countries. What is the mean number of vacation days per year for these countries? Round to the nearest whole day. Average Number of Vacation Days per Year for 2000 Country

Number of Days

Korea Japan United States

40˚ 210 ft

Measurement 17. The Student Council held a car wash for a fund-raiser. On Saturday, approximately 8 containers of car wash were used to wash 52 cars. Assuming the rate stayed the same, how many cars did they wash if they used 11 containers on Sunday? 18. Ethan wants to know how much water his sister’s swimming pool holds. If the pool is 1 foot high and 6 feet across, what is the volume in cubic feet? Use π = 3.14 and round to the nearest tenth of a cubic foot.

25 25 13

Brazil

34

Italy

42

France

37


23. Kaya and her family went on a vacation for spring break. The following table shows the distance traveled for each hour. What is the rate of change between hours 3 and 4? Time (hours)

Distance (miles)

1 2 3 4 5

60 132 208 273 328

1 ft 6 ft

19. Marisa needs to make tablecloths for a wedding party. Each table is a rectangle measuring 8 feet by 3 feet. She wants the tablecloth to extend 1 foot on each of the four sides. In square feet, what will be the area of each tablecloth?

24. If a standard six-sided die is rolled, what is the probability of rolling a multiple of 2?

Question 24 Fractions do not have to be written in simplest form. Any equivalent fraction that fits the grid is correct.


817


15. Toshiro swam across a river that was 20 feet wide. When he got across the river, the current had pushed him 40 feet farther down stream than when he had started. How far did he travel? Round to the nearest tenth of a foot.


Short-Response Questions Short-response questions require you to provide a solution to the problem, as well as any method, explanation, and/or justification you used to arrive at the solution. These are sometimes called constructed-response, open-response, open-ended, free-response, or student-produced questions. The following is a sample rubric, or scoring guide, for scoring short-response questions. Credit

Score

Full

2

Partial

1

None

0

Criteria Full credit: The answer is correct and a full explanation is provided that shows each step in arriving at the final answer. Partial credit: There are two different ways to receive partial credit. • The answer is correct, but the explanation provided is incomplete or incorrect. • The answer is incorrect, but the explanation and method of solving the problem is correct. No credit: Either an answer is not provided or the answer does not make sense.

On some standardized tests, no credit is given for a correct answer if your work is not shown.

EXAMPLE In the first four events of a gymnastics competition, Nicole scored 7.6, 7.9, 8.5, and 8. Sandra scored 9.3, 7.4, 8.5, 7.9, and 8.1 after five events. If the entire competition consists of five events, what does Nicole need to get on her fifth event to score higher than Sandra?

Full Credit Solution First find the total number of points that Nicole has scored so far. 7.6 + 7.9 + 8.5 + 8 = 32 Let x represent the score that Nicole will receive on her final event. Then Nicole’s total score will be 32 + x. Next, find the total number of points that Sandra has scored. 9.3 + 7.4 + 8.5 + 7.9 + 8.1 = 41.2 Since Nicole needs to get a higher score than Sandra, we can write and solve an inequality.

The solution of the problem is clearly stated.

818

Nicole’s score > Sandra’s score 32 + x > 41.2 Substitution x > 9.2 Subtract 32 from each side. In order to score higher than Sandra in the competition, Nicole must score higher than a 9.2 on her fifth event.


The steps, calculations, and reasoning are clearly stated.

Partial Credit Solution


In this sample solution, the answer is correct; however there is no justification for any of the calculations.

9.3 + 7.4 + 8.5 + 7.9 + 8.1 = 41.2 There is not an explanation of how 32 was obtained.

32 + x > 41.2 x > 9.2 Nicole will need to score higher than 9.2.

Partial Credit Solution In this sample solution, the answer is incorrect because the student added Nicole’s points incorrectly. However, after this error, the calculations and reasoning are correct. An equation is used in this solution and the student reasons that the score must be greater than 9.7 in the final answer.

The answer is incorrect, but the reasoning used to find the answer is correct.

First find the total number of points that Nicole has scored so far. 7.6 + 7.9 + 8.5 + 8 = 31.5 Let x represent the score that Nicole will receive on her final event. Then Nicole’s total score will be 31.5 + x. Next, find the total number of points that Sandra has scored. 9.3 + 7.4 + 8.5 + 7.9 + 8.1 = 41.2 Now we can write an equation to solve. 31.5 + x = 41.2 x = 9.7 In order to score higher than Sandra in the competition, Nicole must score higher than a 9.7.

No Credit Solution

7.6 + 7.9 + 8.5 + 8 = 32 The student averaged all of the scores demonstrating no understanding of the problem.

9.3 + 7.4 + 8.5 + 7.9 + 8.1 = 41.2 41.2 + 32 _ = 7.32 10

Nicole needs a 7.32.


819


Short-Response Practice Solve each problem. Show all your work.

Number and Operations 1. Crispy Crunch cereal has a new box that says it is 25% larger than the original size. If the original box had 16 ounces, how many ounces does the new box have? 1 2. Austin has _ of a gallon of paint to paint his go 3 1 cart. He knows that he will need 3_ gallons to 2 paint his entire cart. How many more gallons does he need?

10. Roller Way Amusement park closes the park for a day and allows only schools to visit. The schools must bring students in buses that carry 30 students. If 32 buses are at the park, write an equation to represent the total number of students S that are at the park that day.

Geometry 11. Triangle RST is translated 3 units up and, 1 unit to the left. Find the coordinates of translated RST.

3. In 1999, the United States took in $6.206 × 109 from Canadian tourists. Express this value in standard notation.

y

R

5. Jacob works at a computer factory making $8.50 per hour. The company has not been selling as many computers lately so they have decreased each employee’s wage by 6%. What will Jacob’s new hourly wage be?

x

O

4. Prairie High School assigns every student a student identification code. The codes consist of one letter and five digits. What is the greatest number of students that can attend Prairie High School before codes will need to be reused?

T S

12. Chi-Yo wants to make a quilt pattern using similar triangles as shown. What is the length of the third side of the larger triangle?

3 in.

6 in. 8 in.

Algebra 6. Francisca ran for 1.25 hours at an average rate of 5 miles per hour. What distance did Francisca run? (Use the formula d = rt, where d represents distance, r represents rate, and t represents time.) 7. Simplify the expression -2(y + 4) - 3. 8. Tariq plans to go to Raging Waters Water Park. The park charges a $12 admission and rents inner tubes for $1.50 per hour. If he has $15, for how many hours can he rent an inner tube? 9. Solve and graph the inequality 6 - 2m < 13.

820 Preparing for Standardized Tests

15 in.

x

20 in.

13. Lawana has a window planter in the shape of a cylinder cut in half. She needs to fill it with dirt before she can plant flowers. What is the volume of the planter? Use π = 3.14 and round to the nearest tenth of a cubic foot. 3 ft

6 ft

Question 13 After finding the solution, always go back and read the problem again to make sure your solution answers what the problem is asking.

K

21. Mateo wants to fill the cylindrical container shown with water. How many pints of water will he need? (Hint: One pint of liquid is equivalent to 28.875 cubic inches.) Use π = 3.14 and round to the nearest pint.

X (3x ⫺ 1)˚

L

(x ⫹ 7)˚

M

Y

20. In a recent bicycle race, Kevin rode his bike at a pace of 16 miles per hour. How many feet per minute is this?


14. ∠KLM and ∠XYZ are complementary. If m∠KLM = 3x - 1 and m∠XYZ = x + 7, find the measure of each angle.

Z

15. A cylindrical grain bin holds approximately 28,260 cubic feet of grain. If the bin has a diameter of 50 feet, what is the bin’s height? Use π = 3.14 and round to the nearest tenth of a foot.

10 in.

3 in.

V ⫽ 28,260 ft3

Data Analysis and Probability 50 ft

22. Theo needs a four digit Personal Identification Number (PIN) for his checking account. If he can choose any digit from 0 to 9 for each of the digits, how many different PIN numbers are possible?

17. Jordan wants to make cone-shaped candles as shown in the diagram. What is the volume of one candle? Use π = 3.14 and round to the nearest tenth of a cubic inch.

8 in.

6 in.

RED LOW YEL

E BLU

YEL LOW RED

19. At Oakwood Lawncare Service, a lawn crew can do 5 jobs in 3 days. At this rate, how many days would it take the crew to do 20 jobs?

24. Nikki is playing a game in which you spin the spinner below and then roll a six-sided die labeled 1 through 6. Each section of the spinner is equal in size. What is the probability that the spinner lands on blue and the die lands on an even number? E BLU

18. Abigail is planning a cookout at her house. She wants to put trim around the edges of round tables. If the diameter of each table is 8 feet, how much trim will she need for each table? Use π = 3.14 and round to the nearest tenth of a cubic foot.

23. Matthew is choosing some CDs to take on a long car drive. He has 10 jazz CDs, 3 classical CDs, and 9 soundtracks. If he chooses two CDs without replacement, what is the probability that the first CD is a soundtrack and the second CD is a jazz CD?

YEL LOW

16. One kilometer is equal to about 0.62 mile. Tito is running a 10-kilometer race. How many miles is this?

BLU E

Measurement

25. Two cards are drawn from a hand of eight cards numbered 1 to 8. The first card is not replaced after it is drawn. What is the probability that a 2 and a 5 are drawn? Preparing for Standardized Tests

821

Extended-response questions are often called open-ended or constructed-response questions. Most extended-response questions have multiple parts. You must answer all parts to receive full credit. Extended-response questions are similar to short-response questions in that you must show all of your work in solving the problem and a rubric is used to determine whether you receive full, partial, or no credit. The following is a sample rubric for scoring extended-response questions. Credit

Score

Full

4

Partial

3, 2, 1

None

0

Criteria On some standardized tests, no credit is given for a correct answer if your work is not shown.

Full credit: A correct solution is given that is supported by well-developed, accurate explanations. Partial credit: A generally correct solution is given that may contain minor flaws in reasoning or computation or an incomplete solution. The more correct the solution, the greater the score. No credit: An incorrect solution is given indicating no mathematical understanding of the concept, or no solution is given.

Make sure that when the problem says to Show your work, you show every aspect of your solution including figures, sketches of graphing calculator screens, or reasoning behind computations.

EXAMPLE Northern Sofa Store’s delivery charges depend upon the distance furniture is delivered. The graph shows the charge for deliveries according to the distance. a. Write an equation to show the relationship between delivery cost y and distance from the store x. b. Name the y-intercept and slope of the line that models the data. Explain what each means in this situation.

Delivery Rate 62 Delivery Cost ($)


Extended-Response Questions

60 58 56 54 52 50 0

2

4 6 8 10 12 14 16 18 20 22 Distance from Store (mi)

c. Suppose Mr. Hawkins wants a sofa delivered. His house is 21 miles from the store. What will be the cost of delivery?

Full Credit Solution Part a A complete solution for writing the equation shows all the computations needed and the reasoning behind those computations.

To write the equation, first I need to find the slope. I will use the two points marked on the graph, (0, 50) and (10, 55). 822


=

_ 55 - 50 10 - 0


y2 - y1 _

m= x -x 2 1

slope formula (x1, y1) = (0, 50) and (x2, y2) = (10, 55)

5 =_ or 0.5 10

The calculations and reasoning are clearly stated. The solution of the problem is also clearly stated.

The slope of the line is 0.5 and the y-intercept is 50. y = mx + b y = 0.5x + 50

Slope-intercept form m = 0.5 and b = 50.

So an equation that fits the data is y = 0.5x + 50.

Part b In this sample answer, the student demonstrates clear understanding of the y-intercept and slope of the graph.

I can see from the graph that the y-intercept is 50 which means $50. This is the initial charge for any delivery and then additional money is charged for each mile. The slope of the line that I found in Part a is 0.5 which means $0.50. This is the additional charge per mile to deliver the furniture. Part c In this sample answer, the student knows how to use the equation to find the delivery cost for a given distance.

I will substitute 21 for x into the equation since Mr. Hawkins lives 21 miles from the store. y = 0.5x + 50 y = 0.5(21) + 50 y = 60.50 The cost of delivery will be $60.50.

Partial Credit Solution Part a This sample answer includes no explanations for the calculations performed. However, partial credit is given for correct calculations and a correct answer.

y2 - y1 _

The equation is 1 correct because _ 2 is the same as 0.5.

_ 55 - 50

m= x -x = = _1 10 0 2 2 1

_1

y = x + 50 2

More credit would have been given if an explanation had been given.


823


Part b This part receives full credit because the student demonstrates understanding of the y intercept and slope.

The y-intercept 50 is the charge for just making any _1 delivery. It means $50. The slope is the charge per mile 2 which is 50 cents. Part c Partial credit is given for Part c because the student makes a calculation error.

To get the delivery cost, substitute 21 miles into the equation.

_1

y = x + 50 2

y = _1 (21) + 50 2

y = 42 + 50 = 92 The cost is $92. This sample answer might have received a score of 2, depending on the judgment of the scorer. Had the student gotten Part c correct, the score would probably have been a 3.

No Credit Solution Part a The student demonstrates no understanding of how to write an equation for a line.

If I use the points (0, 50) and (10, 55), an equation is 10y = 10x + 55. Part b The student does not understand the meaning of the y-intercept or the slope.

The y-intercept is 0 because that is when the truck leaves the store. The slope is 10 because that is the distance from the store in miles when the truck makes its first stop. Part c The student does not understand how to read the graph to find the cost or how to use an equation to find the cost.

$21, because it is 21 miles from the store.

824


Solve each problem. Show all your work.

Number and Operations 1. In a recent survey, Funtime Amusement Park found that 6 out of 8 of their customers had been to the park before. In one week, 4500 people attended the park.

4. Jasmine wants to get her portrait taken for her senior pictures. She finds that three different portrait studios charge a sitting fee and charge a separate fee for each ordered picture. The table below shows their prices. Studio

a. What percent of customers have been to the park before?

Famous Photos Picture Perfect Timeless Portraits

b. How many of the park goers in that week had been there before? c. If 3000 people were at the park the next week, how many would you expect had not been there before?

2. The following table shows the number of people of each age living in the U.S. in the year 2000. Age

Number of People

Under 15 15 to 24 25 to 34 35 to 44 45 to 64 65 and over

60,253,375 39,183,891 39,891,724 44,148,527 61,952,636 34,991,753


a. To the nearest million, how many people were under age 25?

Sitting Fee $50 $80 $45

Cost per Portrait $10 $8 $15

a. For each studio, write an equation that represents the total cost. In each of the three equations, use C to represent the total cost and p to represent the number of pictures. b. If Jasmine wants to order 30 portraits, which studio would be the least expensive? c. How many portraits will she need to order for Famous Photos and Picture Perfect to cost the same?

Geometry 5. Rodrigo stands 50 yards away from the base of a building. From his eye level of 5 feet, Rodrigo sees the top of the building at an angle of elevation of 25°.

b. What percent of people were under 15? Round to the nearest percent. c. What is the total population of the United States? Write in scientific notation.

Algebra 3. Victoria needs to study for a math exam. The exam is in 18 days. She has decided to begin right away by studying 15 minutes the first night and increasing her study time by 5 minutes each day. a. Write an expression for the total number of minutes that Victoria will study T for a given time d days from today.

5 ft

25˚ 50 yards

a. What is the height of the building? Round to the nearest tenth of a foot. b. If Rodrigo moved back 15 more yards, how would the angle of elevation to the top of the building change? c. If the building were taller, would the angle of elevation be greater than or less than 25°?

b. If she has 18 days to study, how many minutes will she study on the last day before the exam? c. Victoria begins to study on a Monday, 18 days before the exam. On what day will she study exactly one hour?

Question 5 Be sure to use a trigonometric ratio that includes the measurement you are asked to find in order to solve the problem correctly.


825


Extended-Response Practice


6. The diagram below shows polygon GHIJ. y

Data Analysis and Probability 9. The table shows the life expectancy in years of people in different countries. The figures are for the year 2000.

H G J

Country x

O

I

a. Find the coordinates of the vertices of quadrilateral GHIJ after a reflection of quadrilateral GHIJ in the x-axis. b. Graph GHIJ after a reflection of quadrilateral GHIJ in the x-axis. c. If ABCD is the image of GHIJ dilated by a 1 scale factor of _ with respect to the origin, 4 what are the coordinates of C?

Life Expectancy (yr)

Afghanistan Australia Brazil Canada France Haiti Japan Madagascar Mexico United States

45.9 79.8 62.9 79.4 78.8 49.2 80.7 55.0 71.5 77.1


a. What is the median life expectancy?

Measurement 7. The diagram shows a pattern for a garden that Marie plans to plant. Use π = 3.14.

b. What is the mean life expectancy? c. In 2000, Cambodians had a life expectancy of 56.5. If Cambodia was added to the table, how would the mean be affected?

8 ft

10. The table shows the average cost for a year of higher education for all public institutions.

20 ft

a. What is the area of the garden in square feet? b. What is the area of the garden in square yards? c. Marie wants to put a stone border around the outside of the garden. What is the perimeter of the garden?

8. The table below shows the speeds of several animals. Animal

Speed (mph)

Cheetah Zebra Reindeer Cat (domestic) Wild Turkey

70 40 32 30 15

Source: World Book

a. What is the rate of the cat in feet per minute? b. How many times faster is the cheetah than the wild turkey? Round to the nearest tenth.

826


Year

Cost ($)

1997–1998 1998–1999 1999–2000 2000–2001

$6813 $7107 $7310 $7621

Source: World Book

a. What was the rate of change between the 1999–2000 school year and the 2000–2001 school year? b. What was the rate of change in dollars per year between the 1997–1998 school year and the 2000–2001 school year? c. Between what two school years was the rate of change the greatest? d. Find the mean annual increase in tuition cost between 1997 and 2001. Round to the nearest dollar. e. Based on your answer in part d, what might you expect the cost of education to be in the 2009–2010 school year if the average increase remains constant?

Glossary/Glosario Cómo usar el glosario en español: 1. Busca el término en inglés que desees encontrar. 2. El término en español, junto con la definición, se encuentran en la columna de la derecha.

A mathematics multilingual glossary is available at www.math.glencoe.com/multilingual_glossary. The glossary includes the following languages. Arabic Bengali Cantonese English

Haitian Creole Hmong Korean

Russian Portuguese Spanish

Tagalog Urdu Vietnamese

English

A

Español

absolute value (80) The distance a number is from zero on the number line.

valor absoluto Distancia que un número dista de cero en la recta numérica.

acute angle (477) An angle with a measure greater than 0° and less than 90°.

ángulo agudo Ángulo con una medida mayor que 0° y menor que 90°.

acute triangle (478) A triangle that has three acute angles.

triángulo acutángulo Triángulo que posee tres ángulos agudos.

Addition Property of Equality (137) If you add the same number to each side of an equation, the two sides remain equal.

Propiedad de adición de la igualdad Si sumas el mismo número a ambos lados de una ecuación, los dos lados permanecen iguales.

additive inverses (88) An integer and its opposite.

inverso aditivo Un entero y su opuesto.

adjacent angles (513) Two angles that have the same vertex, share a common side, and do not overlap.

ángulos adyacentes Dos ángulos que poseen el mismo vértice, comparten un lado y no se traslapan.

algebra (37) A branch of mathematics dealing with symbols.

álgebra Rama de las matemáticas que emplea números, letras y signos.

algebraic expression (37) An expression that contains sums and/or products of variables and numbers.

expresión algebraica Expresión que contiene sumas y/o productos de números y variables.

algebraic fraction (198) A fraction with one or more variables in the numerator or denominator.

fracción algebraica Fracción con una o más variables en el numerador o denominador.

alternate exterior angles (512) Nonadjacent exterior angles found on opposite sides of the transversal. In the figure below, ∠1 and ∠7, ∠2 and ∠8 are alternate exterior angles.

ángulos alternos externos Ángulos exteriores no adyacentes que se encuentran en lados opuestos de una transversal. En la siguiente figura, ∠1 y ∠7, ∠2 y ∠8 son ángulos alternos externos.

5 6 8 7

alternate interior angles (512) Nonadjacent interior angles found on opposite sides of the transversal. In the figure above, ∠4 and ∠6, ∠3 and ∠5 are alternate interior angles.

Glossary/Glosario

1 2 4 3

1 2 4 3 5 6 8 7

ángulos alternos internos Ángulos interiores no adyacentes que se encuentran en lados opuestos de una transversal. En la figura anterior, ∠4 y ∠6, ∠3 y ∠5 son ángulos alternos internos. Glossary/Glosario

R1

altitude (545) A line segment that is perpendicular to the base of a figure with endpoints on the base and the side opposite the base.

altura Segmento de recta perpendicular a la base de una figura y cuyos extremos yacen en la base y en el lado opuesto de la base.

angle (757) Two rays with a common endpoint form an angle. The rays and vertex are used to name an angle. The angle below is ∠ABC.

ángulo Dos rayos con un punto común forman un ángulo. Los rayos y el vértice se usan para identificar el ángulo. El siguiente ángulo es ∠ABC.

A

A

B

C

B

C

area (163) The measure of the surface enclosed by a geometric figure.

área Medida de la superficie que encierra una figura geométrica.

arithmetic sequence (158) A sequence in which the difference between any two consecutive terms is the same.

sucesión aritmética Sucesión en que la diferencia entre dos términos consecutivos cualesquiera es siempre la misma.

Associative Property (44) The way in which numbers are grouped when added or multiplied does not change the sum or product.

Propiedad asociativa La forma en que se suman o multiplican dos números no altera su suma o producto.

average (108) The sum of data divided by the number of items in the data set, also called the mean.

promedio Suma de los datos dividida entre el número de elementos en el conjunto de datos. También llamado media.

Glossary/Glosario

B back-to-back stem-and-leaf plot (627) Used to compare two sets of data. The leaves for one set of data are on one side of the stem and the leaves for the other set of data are on the other side.

diagrama de tallo y hojas consecutivo Se usa para comparar dos conjuntos de datos. Las hojas de uno de los conjuntos de datos aparecen en un lado del tallo y las del otro al otro lado de éste.

bar graph (722) A graphic form using bars to make comparisons of statistics.

gráfica de barras Tipo de gráfica que usa barras para comparar estadísticas.

bar notation (759) In repeating decimals, the line or bar placed over the digits that repeat. For −− example, 2.63 indicates the digits 63 repeat.

notación de barra En decimales periódicos, la línea o barra que se escribe encima de los −− dígitos que se repiten. Por ejemplo, en 2.63 la barra encima del 63 indica que los dígitos 63 se repiten.

base (180) In 24, the base is 2. The base is used as a factor as many times as given by the exponent (4). That is, 24 = 2 × 2 × 2 × 2.

base En 24, la base es 2. La base se usa como factor las veces que indique el exponente (4). Es decir, 24 = 2 × 2 × 2 × 2.

R2 Glossary/Glosario

base (545) The base of a parallelogram or a triangle is any side of the figure. The bases of a trapezoid are the parallel sides.

base La base de un paralelogramo o de un triángulo es cualquier lado de la figura. Las bases de un trapecio son los lados paralelos.

base (576) The bases of a prism are any two parallel congruent faces.

base Las bases de un prisma son cualquier par de caras paralelas y congruentes.

base or whole (322) In a percent proportion, the whole quantity, or the number to which the part is being compared.

base o entero En una proporción porcentual, toda la cantidad o número al que se compara la parte. por ciento parte _ =_

part percent _ =_ whole

100

entero

100

biased sample (344) A sample that is not representative of a population.

muestra sesgada Una muestra que no es representativa de una población.

binomial (701) A polynomial with exactly two terms.

binomio Polinomio con exactamente dos términos.

box-and-whisker plot (638) A diagram that divides a set of data into four parts using the median and quartiles. A box is drawn around the quartile values and whiskers extend from each quartile to the extreme data points.

diagrama de caja y patillas Diagrama que divide un conjunto de datos en cuatro partes usando la mediana y los cuartiles. Se dibuja una caja alrededor de los cuartiles y se extienden patillas de cada uno de ellos a los valores extremos.

C celda Casilla dentro de una hoja de cálculos.

center (551) The given point from which all points on the circle are the same distance.

centro Punto dado del cual equidistan todos los puntos de un círculo.

center of dilation (524) A fixed point used for measurement when dilating a figure.

centro de dilación Punto fijo que se usa para medir cuando se dilata una figura.

center of rotation (531) A fixed point around which shapes move in a circular motion to a new position.

centro de rotación Punto fijo alrededor del cual una figura gira con un movimiento circular hasta alcanzar una nueva posición.

central angle (557) An angle whose vertex is the center of a circle and whose sides intersect the circle.

ángulo central Ángulo cuyo vértice es el centro de un círculo y cuyos lados intersecan el círculo.

circle (551) The set of all points in a plane that are the same distance from a given point called the center.

círculo Conjunto de todos los puntos del plano que están a la misma distancia de un punto dado del plano llamado centro.

circle graph (760) A type of statistical graph used to compare parts of a whole.

gráfica circular Tipo de gráfica estadística que se usa para comparar las partes de un todo.

circumference (551) The distance around a circle.

circunferencia Longitud del contorno de un círculo.

Glossary/Glosario

Glossary/Glosario

cell (42) A box within a spreadsheet.

R3

Glossary/Glosario

coefficient (129) The numerical part of a term that contains a variable.

coeficiente Parte numérica de un término que contiene una variable.

combination (677) An arrangement or listing in which order is not important.

combinación Arreglo o lista en que el orden no es importante.

common difference (158) The difference between any two consecutive terms in an arithmetic sequence.

diferencia común Diferencia entre dos términos consecutivos cualesquiera de una sucesión aritmética.

common multiples (257) Multiples that are shared by two or more numbers. For example, some common multiples of 4 and 6 are 0, 12, and 24.

múltiplos comunes Múltiplos compartidos por dos o más números. Por ejemplo, algunos múltiplos comunes de 4 y 6 son 0, 12 y 24.

Commutative Property of Addition (43) The order in which numbers are added does not change the sum.

Propiedad conmutativa de la adición El orden en que se suman los números no altera su suma.

Commutative Property of Multiplication (43) The order in which numbers are multiplied does not change the product.

Propiedad conmutativa de la multiplicación El orden en que se multiplican los números no altera su producto.

compatible numbers (746) Numbers that have been rounded so when the numbers are divided by each other, the remainder is zero.

números compatibles Números redondeados de modo que cuando se dividen, el residuo es cero.

complementary (513) Two angles are complementary if the sum of their measures is 90°.

complementarios Dos ángulos son complementarios si la suma de sus medidas es 90°.

composite event (682) Two or more simple events.

evento compuesto Dos o más eventos simples.

composite figure (558) A figure that is made up of two or more shapes.

figura compleja Figura compuesta de dos o más formas.

composite number (186) A whole number that has more than two factors.

número compuesto Número entero que posee más de dos factores.

compound interest (337) Interest paid on the initial principal and on interest earned in the past.

interés compuesto Interés que se paga sobre el capital inicial y sobre el interés que se haya ganado en el pasado.

cone (576) A three-dimensional figure with one circular base. A curved surface connects the base and vertex.

cono Figura tridimensional con una base circular, la cual posee una superficie curva que une la base con el vértice.

congruent (518) Line segments that have the same length, or angles that have the same measure, or figures that have the same size and shape.

congruentes Segmentos de recta que tienen la misma longitud o ángulos que tienen la misma medida o figuras que poseen la misma forma y tamaño.

conjecture (27) An educated guess.

conjetura Suposición informada.


constant (129) A term without a variable.

constante Término sin variables.

constant of proportionality (298) A constant ratio or unit rate of a proportion.

constante de proporcionalidad La razón constante o tasa unitaria de una proporción.

constant of variation (378) The slope, or rate of change, in the equation y = kx, represented by k.

constante de variación La pendiente, o tasa de cambio, en la ecuación y = kx, representada por k.

constant rate of change (376) The rate of change between any two data points in a linear relationship is the same or constant.

tasa constante de cambio La tasa de cambio entre dos puntos cualesquiera en una relación lineal permanece constante o igual.

converse (487) The statement formed by reversing the phrases after if and then in an if-then statement.

recíproca Un enunciado que se forma intercambiando los enunciados que vienen a continuación de si-entonces en un enunciado si-entonces.

converse of the Pythagorean Theorem (487) The reversal of the if and then statement that forms the Pythagorean Theorem.

recíproco del Teorema de Pitágoras El intercambio de las frases del enunciado sientonces que forman el Teorema de Pitágoras.

coordinate (79) A number that corresponds with a point on a number line.

coordenada Número que corresponde a un punto en la recta numérica.

coordinate plane (54) Another name for the coordinate system.

plano de coordenadas Otro nombre para el sistema de coordenadas.

coordinate system (54) A coordinate system is formed by the intersection of two number lines that meet at right angles at their zero points, also called a coordinate plane.

sistema de coordenadas Un sistema de coordenadas se forma de la intersección de dos rectas numéricas perpendiculares que se intersecan en sus puntos cero. También llamado plano de coordenadas.

corresponding angles (512) Angles that have the same position on two different parallel lines cut by a transversal. In the figure, ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 are corresponding angles.

ángulos correspondientes Ángulos que tienen la misma posición en dos rectas paralelas distintas cortadas por una transversal. En la figura, ∠1 y ∠5, ∠2 y ∠6, ∠3 y ∠7, ∠4 y ∠8 son ángulos correspondientes.

1 2 4 3 5 6 8 7

1 2 4 3 5 6 8 7

partes correspondientes Partes de figuras congruentes o semejantes que se corresponden mutuamente.

counterexample (45) An example that shows a conjecture is not true.

contraejemplo Ejemplo que muestra que una conjetura no es verdadera.

b , then ad = bc. If cross products (302) If _a = _

b , entonces productos cruzados Si _a = _

b . ad = bc, then _a = _ c

d

c

d

c

Glossary/Glosario

corresponding parts (518) Parts of congruent or similar figures that match.

d

b . ad = bc. Si ad = bc, entonces _a = _ c

d

Glossary/Glosario

R5

cubic function (722) A function that can be described by an equation of the form y = ax3 + bx2 + cx + d, where a ≠ 0.

función cúbica Función que puede describirse por una ecuación de la forma y = ax3 + bx2 + cx + d, donde a ≠ 0.

cylinder (576) A solid that has two parallel, congruent bases (usually circular) connected with a curved side.

cilindro Sólido que posee dos bases congruentes y paralelas (por lo general circulares) unidas por un lado curvo.

Glossary/Glosario

D deductive reasoning (45) The process of using facts, properties, or rules to justify reasoning or reach valid conclusions.

razonamiento deductivo Proceso de usar hechos, propiedades o reglas para justificar un razonamiento o para sacar conclusiones válidas.

defining a variable (38) Choosing a variable and a quantity for the variable to represent in an equation.

definir una variable Seleccionar una variable y una cantidad para la variable que represente en la ecuación.

degree (702) The sum of the exponents of the variables of a monomial.

grado Suma de los exponentes de las variables de un monomio.

degree (757) The most common unit of measure for angles.

grado La unidad de medida angular más común.

dependent events (683) Two or more events in which the outcome of one event does affect the outcome of the other event(s).

eventos dependientes Dos o más eventos en que el resultado de uno de ellos afecta el resultado del otro o de los otros eventos.

diagonal (540) A line segment that joins two nonconsecutive vertices of a polygon.

diagonal Segmento de recta que une dos vértices no consecutivos de un polígono.

diameter (551) The distance across a circle through its center.

diámetro Distancia de un lado a otro de un círculo medida a través de su centro.

dilation (524) A transformation that alters the size of a figure but not its shape.

dilatación Transformación que altera el tamaño de una figura, pero no su forma.

dimensional analysis (241) The process of including units of measurement when computing.

análisis dimensional Proceso que incorpora las unidades de medida al hacer cálculos.

direct variation (378) A special type of linear equation that describes rate of change. A relationship such that as x increases in value, y increases or decreases at a constant rate.

variación directa Tipo especial de ecuación lineal que describe tasas de cambio. Relación en que a medida que x aumenta de valor, y aumenta o disminuye a una tasa constante.

discount (333) The amount by which the regular price of an item is reduced.

descuento Cantidad por la que se reduce el precio normal de un artículo.


Distance Formula (492) The distance between two points, with coordinates (x1, y1) and (x2, y2), is given by d = √ (x2 - x1)2 + (y2 - y1)2 .

Fórmula de la distancia La distancia entre dos puntos, con coordenadas (x1, y1) y (x2, y2), (x2 - x1)2 + (y2 - y1)2 . se calcula con d = √

Distributive Property (124) To multiply a sum by a number, multiply each number in parentheses by the number outside the parentheses.

Propiedad distributiva Para multiplicar una suma por un número, multiplica cada número en paréntesis por el número fuera del paréntesis.

divisible (740) A number is divisible by another if, upon division, the remainder is zero.

divisible Un número es divisible entre otro si, al dividirlos, el residuo es cero.

Division Property of Equality (141) When you divide each side of an equation by the same nonzero number, the two sides remain equal.

Propiedad de igualdad de la división Cuando divides ambos lados de una ecuación entre el mismo número no nulo, los dos lados permanecen iguales.

domain (56) The domain of a relation is the set of all x-coordinates from each pair.

dominio El dominio de una relación es el conjunto de coordenadas x de todos los pares.

E arista Recta en donde se intersecan dos planos.

empty set (426) A set with no elements shown by the symbol { } or Ø.

conjunto vacío Conjunto que carece de elementos y que se denota con el símbolo { } o Ø.

equation (49) A mathematical sentence that contains an equals sign (=).

ecuación Enunciado matemático que contiene el signo de igualdad (=).

equilateral triangle (478) A triangle with all sides congruent.

triángulo equilátero Un triángulo cuyos lados son todos congruentes.

equivalent equations (136) Two or more equations with the same solution. For example, x + 4 = 7 and x = 3 are equivalent equations.

ecuaciones equivalentes Dos o más ecuaciones con las mismas soluciones. Por ejemplo, x + 4 = 7 y x = 3 son ecuaciones equivalentes.

equivalent expressions (124) Expressions that have the same value.

expresiones equivalentes Expresiones que tienen el mismo valor.

evaluate (32) Find the numerical value of an expression.

evaluar Calcular el valor numérico de una expresión.

experimental probability (666) What actually occurs in a probability experiment.

probabilidad experimental Lo que realmente sucede en un experimento probabilístico.

exponent (180) In 24, the exponent is 4. The exponent tells how many times the base, 2, is used as a factor. So, 24 = 2 × 2 × 2 × 2.

exponente En 24, el exponente es 4. El exponente indica cuántas veces se usa la base, 2, como factor. Así, 24 = 2 × 2 × 2 × 2.

Glossary/Glosario

Glossary/Glosario

edge (575) Where two planes intersect in a line.

R7

ángulos exteriores Cuatro de los ángulos formados por una transversal y dos rectas paralelas. Los ángulos exteriores yacen fuera de las dos rectas paralelas.

exterior angles (512) Four of the angles formed by the transversal and two parallel lines. Exterior angles lie outside the two parallel lines.

F face (575) A flat surface, the side or base of a prism.

cara Superficie plana, el lado o la base de un prisma.

factor (188) To write a number as a product of its factors.

factorizar Escribir un número como el producto de sues propios factores.

factors (180) Two or more numbers that are multiplied to form a product.

factores Dos o más números que se multiplican para formar un producto.

factor tree (187) A way to find the prime factorization of a number. The factors branch out from the previous factors until all the factors are prime numbers.

árbol de factores Forma de encontrar la factorización prima de un número. Los factores se ramifican de los factores anteriores hasta que todos los factores son números primos.

formula (162) An equation that shows a relationship among certain quantities.

fórmula Ecuación que muestra la relación entre ciertas cantidades.

frequency table (644) A chart that indicates the number of values in each interval.

tabla de frecuencias Tabla que indica el número de valores en cada intervalo.

function (359) A function is a special relation in which each element of the domain is paired with exactly one element in the range.

función Una función es una relación especial en que a cada elemento del dominio le corresponde un único elemento del rango.

Fundamental Counting Principle (671) If event M can occur in m ways and is followed by event N that can occur in n ways, then the event M followed by event N can occur in m · n ways.

Principio fundamental de contar Si el evento M puede ocurrir de m maneras y lo sigue un evento N que puede ocurrir de n maneras, entonces el evento M seguido del evento N puede ocurrir de m · n maneras.

Glossary/Glosario

G graph (55) A dot at the point that corresponds to an ordered pair on a coordinate plane.

gráfica Marca puntual en el punto que corresponde a un par ordenado en un plano de coordenadas.

greatest common factor (GCF) (191) The greatest number that is a factor of two or more numbers.

máximo común divisor (MCD) El número mayor que es factor de dos o más números.


H histogram (644) A histogram uses bars to display numerical data that have been organized into equal intervals.

histograma Un histograma usa barras para exhibir datos numéricos que han sido organizados en intervalos iguales.

hypotenuse (485) The side opposite the right angle in a right triangle.

hipotenusa Lado opuesto al ángulo recto en un triángulo rectángulo.

hypotenuse

hipotenusa

I identidad Ecuación que es verdadera para cada valor de la variable.

image (524) Every corresponding point on a figure after its transformation.

imagen Todo punto correspondiente de una figura después de ser transformación.

independent events (682) Two or more events in which the outcome of one event does not influence the outcome of the other event(s).

eventos independientes Dos o más eventos en que el resultado de uno de ellos no afecta el resultado del otro o de los otros eventos.

indirect measurement (499) Using the properties of similar triangles to find measurements that are difficult to measure directly.

medición indirecta Uso de las propiedades de triángulos semejantes para hacer mediciones que son difíciles de realizar directamente.

inductive reasoning (27) Conjecture based on a pattern of examples or past events.

razonamiento inductivo Conjetura basada en un patrón de ejemplos o sucesos pasados.

inequality (79, 430) A mathematical sentence that contains , ≠, ≤, or ≥.

desigualdad Enunciado matemático que contiene , ≠, ≤, o ≥.

integers (78) The whole numbers and their opposites. . . . , -3, -2, -1, 0, 1, 2, 3, . . .

enteros Los números enteros y sus opuestos. . . . , -3, -2, -1, 0, 1, 2, 3, . . .

interior angle (540) An angle inside a polygon.

ángulo interno Ángulo ubicado dentro de un polígono.

interior angles (512) Four of the angles formed by the transversal and two parallel lines. Interior angles lie between the two parallel lines.

ángulos interiores Cuatro de los ángulos formados por una transversal y dos rectas paralelas. Los ángulos interiores yacen entre las dos rectas paralelas.

interquartile range (634) The range of the middle half of a set of data. It is the difference between the upper quartile and the lower quartile.

amplitud intercuartílica Amplitud de la mitad central de un conjunto de datos. Es la diferencia entre el cuartil superior y el inferior.

Glossary/Glosario

Glossary/Glosario

identity (426) An equation that is true for every value of the variable.

R9

inverse operations (136) Operations that undo each other, such as addition and subtraction.

operaciones inversas Operaciones que se anulan mutuamente, como la adición y la sustracción.

irrational number (469) A number that cannot be expressed as _a , where a and b are integers and b b does not equal 0.

número irracional Número que no puede escribirse como _a , donde a y b son enteros y b b no es igual a 0.

isosceles triangle (478) A triangle that has at least two congruent sides.

triángulo isósceles Triángulo que posee por lo menos dos lados congruentes.

L lateral area (597) The sum of the areas of the lateral faces of a solid.

área lateral Suma de las áreas de las caras laterales de un sólido.

lateral faces (597) The lateral faces of a prism, cylinder, pyramid, or cone are all the surface of the figure except the base or bases.

caras laterales Las caras laterales de un prisma, cilindro, pirámide o cono son todas las superficies de la figura, excluyendo la base o las bases.

least common denominator (LCD) (257) The least common multiple of the denominators of two or more fractions.

mínimo común denominador (mcd) El mínimo común múltiplo de los denominadores de dos o más fracciones.

least common multiple (LCM) (257) The least of the nonzero common multiples of two or more numbers. The LCM of 4 and 6 is 12.

mínimo común múltiplo (MCM) El menor de los múltiplos comunes no nulos de dos o más números. El MCM de 4 y 6 es 12.

leaves (626) In a stem-and-leaf plot, the next greatest place value of the data after the stem forms the leaves.

hojas En un diagrama de tallo y hojas, las hojas las forma el segundo valor de posición mayor después del tallo.

legs (485) The sides that are adjacent to the right angle of a right triangle.

catetos Lados adyacentes al ángulo recto de un triángulo rectángulo.

Glossary/Glosario

legs

catetos

like terms (129) Expressions that contain the same variables to the same power, such as 2n and 5n or 6xy2 and 4xy2.

términos semejantes Expresiones que tienen las mismas variables elevadas a los mismos exponentes, como 2n y 5n ó 6xy2 y 4xy2.

line (757) A never-ending straight path.

recta Trayectoria rectilínea interminable.


line graph (722) A type of statistical graph used to show how values change over a period of time.

gráfica lineal Tipo de gráfica estadística que se usa para mostrar cómo cambian los valores durante un período de tiempo.

line of fit (403) On a scatter plot, a line drawn that is very close to most of the data points. The line that best fits the data.

recta de ajuste En una gráfica de dispersión, una recta que está muy cercana a la mayoría de los puntos de datos. La recta que mejor se ajusta a los datos.

line of symmetry (524) Each half of a figure is a mirror image of the other half when a line of symmetry is drawn.

eje de simetría Cuando se traza un eje de simetría, cada mitad de una figura es una imagen especular de la otra mitad.

line segment (476) Part of a line containing two endpoints and all the points between them.

segmento de recta Parte de una recta que contiene dos extremos y todos los puntos entre éstos.

linear equation (365) An equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. The graph of a linear equation is a straight line.

ecuación lineal Ecuación en que las variables aparecen en términos separados y en la cual ninguna de ellas tiene un exponente distinto de 1. La gráfica de una ecuación lineal es una recta.

linear relationships (376) Relationships that have straight line graphs.

relación lineal Relación que al ser graficada forma una línea recta.

lower quartile (634) The median of the lower half of a set of data, indicated by LQ.

cuartil inferior Mediana de la mitad inferior de un conjunto de datos, se denota con CI.

M media Suma de los datos dividida entre el número de elementos en el conjunto de datos. También llamada promedio.

measures of central tendency (274) For a list of numerical data, numbers that can represent the whole set of data.

medidas de tendencia central Números que pueden representar todo el conjunto de datos en una lista de datos numéricos.

measures of variation (633) Used to describe the distribution of statistical data.

medidas de variación Se usan para describir la distribución de datos estadísticos.

median (274) In a set of data, the middle number of the ordered data, or the mean of the two middle numbers.

mediana En un conjunto de datos, el número central de los datos ordenados numéricamente o la media de los dos números centrales.

mixed number (228) The indicated sum of a 1 whole number and a fraction. For example, 3_ .

número mixto Suma de un entero y 1 . una fracción. Por ejemplo, 3_

mode (274) The number or numbers that occur(s) most often in a set of data.

moda Número o números de un conjunto de datos que aparecen más frecuentemente.

2

Glossary/Glosario

mean (108, 274) The sum of data divided by the number of items in the data set, also called the average.

2

Glossary/Glosario

R11

monomial (188) An expression that is a number, a variable, or a product of numbers and/or variables.

monomio Expresión que es un número, una variable y/o un producto de números y variables.

Multiplication Property of Equality (143) When you multiply each side of an equation by the same number, the two sides remain equal.

Propiedad de multiplicación de la igualdad Cuando multiplicas ambos lados de una ecuación por el mismo número, los dos lados permanecen iguales.

multiplicative inverses (245) Two numbers whose product is 1.

inversos multiplicativos Dos números cuyo producto es igual a uno.

multiple (257) The product of a number and a whole number.

múltiplo Producto de un número por un número entero.

mutually exclusive events (684) Two or more events that cannot happen at the same time.

eventos mutuamente exclusivos Dos o más eventos que no pueden ocurrir simultáneamente.

N negative number (78) A number less than zero.

número negativo Número menor que cero.

nets (597) A two-dimensional pattern for a three-dimensional figure.

redes Patrón bidimensional de una figura tridimensional.

nonlinear function (720) A function with a graph that is not a straight line.

función no lineal Función cuya gráfica no es una recta.

nonproportional relationship (297) A relationship in which two ratios are not equal.

relación no proporcional Relación en la que dos razones no son iguales.

null set (426) A set with no elements shown by the symbol { } or Ø.

conjunto vacío Conjunto que carece de elementos y que se denota con el símbolo { } o Ø.

numerical expression (32) A combination of numbers and operations such as addition, subtraction, multiplication, and division.

expresión numérica Combinación de números y operaciones, como adición, sustracción, multiplicación y división.

Glossary/Glosario

O obtuse angle (477) An angle with a measure greater than 90° but less than 180°.

ángulo obtuso Ángulo que mide más de 90°, pero menos de 180°.

obtuse triangle (478) A triangle with one obtuse angle.

triángulo obtusángulo Triángulo que posee un ángulo obtuso.

obtuse angle


triángulo obtusángulo

open sentence (49) An equation that contains a variable.

enunciado abierto Ecuación que contiene una variable.

opposites (88) Two numbers with the same absolute value but different signs.

opuestos Dos números que tienen el mismo valor absoluto, pero que tienen distintos signos.

ordered pair (54) A pair of numbers used to locate any point on a coordinate plane.

par ordenado Par de números que se usa para ubicar cualquier punto en un plano de coordenadas.

origin (54) The point at which the number lines intersect in a coordinate system.

origen Punto de intersección de las rectas numéricas de un sistema de coordenadas.

outcomes (665) Possible results of a probability experiment.

resultado Resultados posibles de un experimento probabilístico.

outliers (634) Data that are more than 1.5 times the interquartile range beyond the quartiles.

valores atípicos Datos que distan de los cuartiles más de 1.5 veces la amplitud intercuartílica.

P parabola (731) The graph of a quadratic function.

parábola La gráfica de una función cuadrática.

parallel lines (512) Two lines in the same plane that do not intersect.

rectas paralelas Dos rectas en el mismo plano que no se intersecan.

parallelogram (533) A quadrilateral with opposite sides parallel and congruent.

paralelogramo Cuadrilátero con lados opuestos congruentes y paralelos.

part (322) In a percent proportion, the number being compared to the whole quantity.

parte En una proporción porcentual, el número que se compara con la cantidad total.

percent (313) A ratio that compares a number to 100.

por ciento Razón que compara un número con 100.

percent equation (332) An equivalent form of percent proportion, where % is written as a decimal.

ecuación porcentual Forma equivalente a la proporción porcentual en la cual el % se escribe como decimal.

Part = Percent × Whole

Parte = Por ciento × Entero

porcentaje de cambio Razón del aumento o disminución de una cantidad a la cantidad original.

percent of decrease (340) The ratio of an amount of decrease to the previous amount, expressed as a percent. A negative percent of change.

porcentaje de disminución Razón de la cantidad de disminución a la cantidad original, escrita como por ciento. Un por ciento de cambio negativo.

Glossary/Glosario

R13

Glossary/Glosario

percent of change (338) The ratio of the increase or decrease of an amount to the original amount.

percent of increase (339) The ratio of an amount of increase to the original amount, expressed as a percent.

porcentaje de aumento Razón de la cantidad de aumento a la cantidad original, escrita como por ciento.

percent proportion (322)

proporción porcentual

p percent part a _ = _ or _ = _ whole

100

b

p por ciento parte _ a _ = or _ = _

100

todo

b

100

perfect squares (464) Rational numbers whose square roots are whole numbers. 25 is a perfect 25 = 5. square because √

cuadrados perfectos Números racionales cuyas raíces cuadradas son números racionales. 25 es 25 = 5. un cuadrado perfecto porque √

perimeter (163) The distance around a geometric figure.

perímetro Longitud alrededor de una figura geométrica.

permutation (676) An arrangement or listing in which order is important.

permutación Arreglo o lista en que el orden es importante.

perpendicular lines (514) Lines that intersect to form a right angle.

rectas perpendiculares Rectas que se intersecan formando un ángulo recto.

pi, ␲ (551) The ratio of the circumference of a circle to the diameter of the circle. 22 Approximations for ␲ are 3.14 and _ .

pi, ␲ Razón de la circunferencia de un círculo al 22 diámetro del mismo. 3.14 y _ son 7 aproximaciones de ␲.

plane (575) A two-dimensional flat surface that extends in all directions and contains at least three noncollinear points.

plano Superficie plana bidimensional que se extiende en todas direcciones y que contiene por lo menos tres puntos no colineales.

polygon (539) A simple closed figure in a plane formed by three or more line segments.

polígono Figura simple y cerrada en el plano formada por tres o más segmentos de recta.

polyhedron (575) A solid with flat surfaces that are polygons.

poliedro Sólido con superficies planas que son polígonos.

polynomial (701) An algebraic expression that contains the sums and/or products of one or more monomials.

polinomio Expresión algebraica que contiene sumas y/o productos de uno o más monomios.

population (343) A larger group used in statistical analysis.

población Grupo grande que se utiliza en análisis estadísticos.

power (180) A number that is expressed using an exponent.

potencia Número que puede escribirse usando un exponente.

prime factorization (187) A composite number expressed as a product of prime factors. For example, the prime factorization of 63 is 3 × 3 × 7.

factorización prima Número compuesto escrito como producto de factores primos. Por ejemplo, la factorización prima de 63 es 3 × 3 × 7.

prime number (186) A whole number that has exactly two factors, 1 and itself.

número primo Número entero que sólo tiene dos factores, 1 y sí mismo.

7

Glossary/Glosario

100


principle (334) The amount of money in an account.

capital Cantidad de dinero en una cuenta.

prism (576) A polyhedron that has two parallel, congruent bases in the shape of polygons.

prisma Poliedro que posee dos bases congruentes y paralelas en forma de polígonos.

rectangular prism

triangular prism

probability (665) The ratio of the number of ways a certain event can occur to the number of possible outcomes. P(event) = ___ number of favorable outcomes number of possible outcomes

prisma triangular

prisma rectangular

probabilidad La razón del número de maneras en que puede ocurrir el evento al número de resultados posibles. P(evento) = ___ número de resultados favorables número de resultados posíbles

properties (43) Statements that are true for any numbers.

propiedades Enunciados que son verdaderos para cualquier número.

proportion (302) A statement of equality of two or more ratios.

proporción Enunciado de la igualdad de dos o más razones.

proportional relationship (297) The ratios of related terms are equal.

relación proporcional Relación en la que la razón entre los términos relacionados permanece igual.

protractor (757) An instrument used to measure angles.

transportador Instrumento que se usa para medir ángulos.

pyramid (576) A polyhedron that has a polygon for a base and triangles for sides.

pirámide Poliedro cuya base es un polígono y cuyos lados son triángulos.

Pythagorean Theorem (485) If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs or c2 = a2 + b2.

Teorema de Pitágoras Si un triángulo es rectángulo, entonces el cuadrado de la longitud de la hipotenusa es igual a la suma de los cuadrados de las longitudes de los catetos, o c2 = a2 + b2.

a

c c

Glossary/Glosario

a b

b

Pythagorean Triple (484) The sides of right triangle represented by a, b, and c, when a2 + b2 = c2.

Triple pitagórico Los lados de un triángulo rectángulo representados por a, b y c, cuando a2 + b2 = c2.

Glossary/Glosario

R15

Q quadrants (112) The four regions into which the x-axis and y-axis separate the coordinate plane.

cuadrantes Las cuatro regiones en que los ejes x y y dividen el plano de coordenadas.

quadratic function (703) A function that can be described by an equation of the form y = ax2 + bx + c, where a ≠ 0.

función cuadrática Función que puede describirse por una ecuación de la forma y = ax2 + bx + c, donde a ≠ 0.

quadrilateral (532) A closed figure with four sides and four vertices, including squares, rectangles, and trapezoids.

cuadrilátero Figura cerrada de cuatro lados y cuatro vértices, incluyendo cuadrados, rectángulos y trapecios.

quartiles (634) The values that divide a set of data into four equal parts.

cuartiles Valores que dividen un conjunto de datos en cuatro partes iguales.

R radical sign (464) The symbol √ used to indicate a nonnegative square root.

signo radical El símbolo √ que se usa para indicar la raíz cuadrada no negativa.

radius (551) The distance from the center to any point on the circle.

radio Distancia del centro a cualquier punto de un círculo.

range (56) The range of a relation is the set of all y-coordinates from each ordered pair.

rango El rango de una relación es el conjunto de coordenadas y de todos los pares.

range (633) A measure of variation that is the difference between the least and greatest values in a set of data.

amplitud Medida de variación que es la diferencia entre los valores máximo y mínimo de un conjunto de datos.

rate (293) A ratio of two measurements having different units.

tasa Razón de dos medidas que tienen unidades distintas.

rate of change (371) A change in one quantity with respect to another quantity.

tasa de cambio Cambio de una cantidad con respecto a otra.

ratio (292) A comparison of two numbers by division. The ratio of 2 to 4 can be stated as 2 2 . out of 4, 2 to 4, 2 : 4, or _

razón Comparación de dos números mediante división. La razón de 2 a 4 puede escribirse 2 . como 2 de cada 4, 2 a 4, 2 : 4 ó _

rational number (234) A number that can be written as a fraction in the form _a , where a and b b are integers and b ≠ 0.

número racional Número que puede escribirse como una fracción de la forma _a donde a y b b son enteros y b ≠ 0.

ray (757) A part of a line that extends indefinitely in one direction.

rayo Parte de una recta que se extiende indefinidamente en una dirección.

real numbers (469) The set of rational numbers together with the set of irrational numbers.

números reales El conjunto de los números racionales junto con el de números irracionales.

reciprocal (245) Another name for a multiplicative inverse.

recíproco Otro nombre del inverso multiplicativo.

Glossary/Glosario

4


4

rectangle (533) A parallelogram with four right angles.

rectángulo Paralelogramo con cuatro ángulos rectos.

reflection (524) A transformation where a figure is flipped over a line. Also called a flip.

reflexión Transformación en que una figura se voltea a través de una recta.

regular polygon (541) A polygon having all sides congruent and all angles congruent.

polígono regular Polígono cuyos lados son todos congruentes y cuyos ángulos son también todos congruentes.

relation (56) A set of ordered pairs.

relación Conjunto de pares ordenados.

repeating decimal (229) A decimal whose digits repeat in groups of one or more without end. Examples are 0.181818… and 0.8333… .

decimal periódico Decimal cuyos dígitos se repiten en grupos de uno o más. 0.181818… y 0.8333… son ejemplos de este tipo de decimales.

rhombus (533) A parallelogram with four congruent sides.

rombo Paralelogramo con cuatro lados congruentes.

right angle (477) An angle that measures 90°.

ángulo recto Ángulo que mide 90°.

right triangle (478) A triangle with one right angle.

triángulo rectángulo Triángulo que tiene un ángulo recto.

rotation (531) A transformation where a figure is turned around a fixed point. Also called a turn.

rotación Transformación en que una figura se hace girar alrededor de un punto fijo. También se llama vuelta.

S muestra Subgrupo o subconjunto de una población que se usa para representarla.

sample space (666) The set of all possible outcomes.

espacio muestral Conjunto de todos los resultados posibles.

scale (308) The relationship between the measurements on a drawing or model and the measurements of the real object.

escala Relación entre las medidas de un dibujo o modelo y las medidas de la figura verdadera.

scale drawing (308) A drawing that is used to represent an object that is too large or too small to be drawn at actual size.

dibujo a escala Dibujo que se usa para representar una figura que es demasiado grande o pequeña como para ser dibujada de tamaño natural.

scale factor (308) The ratio of a length on a scale drawing or model to the corresponding length on the real object.

factor de escala Razón de la longitud en un dibujo a escala o modelo a la longitud correspondiente en la figura verdadera.

Glossary/Glosario

Glossary/Glosario

sample (343) A subgroup or subset of a population used to represent the whole population.

R17

scale model (308) A model used to represent an object that is too large or too small to be built at actual size.

modelo a escala Modelo que se usa para representar una figura que es demasiado grande o pequeña como para ser construida de tamaño natural.

scalene triangle (478) A triangle with no congruent sides.

triángulo escaleno Triángulo que no tiene lados congruentes.

scatter plot (61) A graph that shows the relationship between two sets of data.

gráfica de dispersión Gráfica en que se muestra la relación entre dos conjuntos de datos.

scientific notation (214) A number in scientific notation is expressed as a × 10n, where 1 ≤ a < 10 and n is an integer. For example, 5,000,000 = 5.0 × 106.

notación científica Un número en notación científica se escribe como a × 10n, donde 1 ≤ a < 10 y n es un entero. Por ejemplo, 5,000,000 = 5.0 × 106.

sequence (158) An ordered list of numbers, such as, 0, 1, 2, 3, or 2, 4, 6, 8.

sucesión Lista ordenada de números, como 0, 1, 2, 3 ó 2, 4, 6, 8.

sides (757) The two rays that make up an angle.

lados Los dos rayos que forman un ángulo.

similar figures (497) Figures that have the same shape but not necessarily the same size.

figuras semejantes Figuras que tienen la misma forma, pero no necesariamente el mismo tamaño.

similar solids (608) Solids that have the same shape but not necessarily the same size.

sólidos semejantes Sólidos que tienen la misma forma, pero no necesariamente el mismo tamaño.

simple event (665) One outcome or a collection of outcomes.

evento simple Resultado o colección de resultados.

simple interest (334) The amount of money paid or earned for the use of money.

interés simple Cantidad que se paga o que se gana por usar el dinero.

Glossary/Glosario

I = prt (Interest = principal × rate × time)

I = crt (Interés = capital × rédito × tiempo)

simplest form (130) An algebraic expression in simplest form has no like terms and no parentheses.

forma reducida Una expresión algebraica reducida no tiene ni términos semejantes ni paréntesis.

simplest form (196) A fraction is in simplest form when the GCF of the numerator and the denominator is 1.

forma reducida Una fracción está reducida si el MCD de su numerador y denominador es 1.

simplify (45) To write an expression in a simpler form.

reducir Escribir una expresión en forma más simple.

simplify the expression (130) To use distribution to combine like terms.

reducir la expresión Usar la distribución para combinar términos semejantes.

simulation (688) The process of acting out a situation to see possible outcomes.

simulación Proceso de representación de una situación para averiguar los resultados posibles.


slant height (602) The length of the altitude of a lateral face of a regular pyramid or cone.

altura oblicua En una pirámide regular, la longitud de la altura de una cara lateral.

slope (384) The ratio of the rise, or vertical change, to the run, or horizontal change. The slope describes the steepness of a line.

pendiente Razón de la elevación o cambio vertical al desplazamiento o cambio horizontal. La pendiente describe la inclinación de una recta.

slope = _ run rise

pendiente = __ elevación desplazamiento

forma pendiente-intersección Una ecuación lineal de la forma y = mx + b, donde m es la pendiente y b es la intersección y.

solid (575) Three-dimensional figure.

sólido Figura tridimensional.

solution (49) A value for the variable that makes an equation true. For x + 7 = 19, the solution is 12.

solución Valosss y que posee cuatro ángulos rectos.

solving the equation (49) The process of finding a solution to an equation.

resolver la ecuación Proceso de hallar una solución a una ecuación.

solving a right triangle (486) Using the Pythagorean Theorem to find the length of the third side of a right triangle, if the lengths of the other two sides are known.

resolver un triángulo rectángulo Uso del Teorema de Pitágoras para hallar la longitud de un tercer lado de un triángulo rectángulo, si se conocen las longitudes de los otros dos lados.

sphere (590) The set of all points in space that are a given distance, r, from the center.

esfera El conjunto de todos los puntos en el espacio que se hallan a una distancia r del centro.

spreadsheet (42) A table that performs calculations.

hoja de cálculos Tabla que realiza cálculos.

square (533) A parallelogram with all sides congruent and four right angles.

cuadrado Paralelogramo cuyos lados son todos congruentes y que posee cuatro ángulos rectos.

square root (464) One of the two equal factors of a number. The square root of 25 is 5 since 52 = 25.

raíz cuadrada Uno de los dos factores iguales de un número. Una raíz cuadrada de 25 es 5 porque 52 = 25.

standard form (214) A number is in standard form when it does not contain exponents. The standard form for seven hundred thirty-nine is 739.

forma estándar Un número está en forma estándar si no contiene exponentes. Por ejemplo, la forma estándar de setecientos treinta y nueve es 739.

stem-and-leaf plot (626) A system used to condense a set of data where the greatest place value of the data forms the stem and the next greatest place value forms the leaves.

diagrama de tallo y hojas Sistema que se usa para condensar un conjunto de datos, en que el valor de posición máximo de los datos forma el tallo y el segundo valor de posición máximo forma las hojas.

stems (626) The greatest place value common to all the data values is used for the stem of a stem-and-leaf plot.

tallos Máximo valor de posición común a todos los datos que se usa como el tallo en un diagrama de tallo y hojas.

Glossary/Glosario

R19

Glossary/Glosario

slope-intercept form (391) A linear equation in the form y = mx + b, where m is the slope and b is the y-intercept.

straight angle (477) An angle with a measure equal to 180°.

ángulo llano Ángulo que mide 180°.

Subtraction Property of Equality (136) If you subtract the same number from each side of an equation, the two sides remain equal.

Propiedad de sustracción de la igualdad Si restas el mismo número de ambos lados de una ecuación, los dos lados permanecen iguales.

supplementary (514) Two angles are supplementary if the sum of their measures is 180°.

suplementarios Dos ángulos son suplementarios si sus medidas suman 180°.

surface area (597) The sum of the areas of all the surfaces (faces) of a 3-dimensional figure.

área de superficie Suma de las áreas de todas las superficies (caras) de una figura tridimensional.

Glossary/Glosario

T term (129) When plus or minus signs separate an algebraic expression into parts, each part is a term.

término Cada una de las partes de una expresión algebraica separadas por los signos de adición o sustracción.

term (158) Each number within a sequence is called a term.

término Cada número de una sucesión se llama término.

terminating decimal (228) A decimal whose digits end. Every terminating decimal can be written as a fraction with a denominator of 10, 100, 1000, and so on.

decimal terminal Decimal cuyos dígitos terminan. Todo decimal terminal puede escribirse como una fracción con un denominador de 10, 100, 1000, etc.

theoretical probability (666) What should occur in a probability experiment.

probabilidad teórica Lo que debería ocurrir en un experimento probabilístico.

transformation (524) A movement of a geometric figure.

transformación Desplazamiento de una figura geométrica.

translation (524) A transformation where a figure is slid from one position to another without being turned. Also called a slide.

translación Transformación en que una figura se desliza sin girar, de una posición a otra. También se llama deslizamiento.

transversal (512) A line that intersects two parallel lines to form eight angles.

transversal Recta que interseca dos rectas paralelas formando ocho ángulos.

trapezoid (533) A quadrilateral with exactly one pair of parallel sides.

trapecio Cuadrilátero con sólo un par de lados paralelos.

tree diagram (670) A diagram used to show the total number of possible outcomes.

diagrama de árbol Diagrama que se usa para mostrar el número total de resultados posibles.

triangle (476) A figure having three sides.

triángulo Figura de tres lados.

trinomial (701) A polynomial with three terms.

trinomio Polinomio de tres términos.


ecuación de dos pasos Ecuación que contiene dos operaciones.

two-step equation (147) An equation that contains two operations.

U unbiased sample (343) A random sample that is representative of a larger sample.

muestra insesgada Muestra aleatoria que es representativa de una muestra más grande.

unit rate (293) A rate simplified so that it has a denominator of 1.

tasa unitaria Tasa reducida que tiene denominador igual a 1.

upper quartile (634) The median of the upper half of a set of data, indicated by UQ.

cuartil superior Mediana de la mitad superior de un conjunto de datos, denotada por CS.

V variable (37) A placeholder for any value.

variable Marcador de posición para cualquier valor.

Venn diagram (191) A diagram that is used to show the relationships among sets of numbers or objects by using overlapping circles in a rectangle.

diagrama de Venn Diagrama que se usa para mostrar las relaciones entre conjuntos de números o elementos mediante círculos, que pueden traslaparse, dentro de un rectángulo.

vertex (476) A vertex of a polygon is a point where two sides of the polygon intersect.

vértice El vértice de un polígono es un punto en que se intersecan dos lados del mismo.

vertex (575) Where three or more planes intersect in a point.

vértice Punto en que se intersecan tres o más planos.

vertex (757) The common endpoint of the rays forming an angle.

vértice Extremo común de los rayos que forman un ángulo.

vertical angles (513) Two pairs of opposite angles formed by two intersecting lines. The angles formed are congruent. In the figure, the vertical angles are ∠1 and ∠3, ∠2 and ∠4.

ángulos opuestos por el vértice Dos pares de ángulos opuestos formados por dos rectas que se intersecan. Los ángulos que resultan son congruentes. En la figura, los ángulos opuestos por el vértice son ∠1 y ∠3, ∠2 y ∠4.

4

1 3

2 4

1 3

2

prueba de la recta vertical Si todas las rectas verticales trazadas en la gráfica de una relación no pasan por más un punto para cada valor de x en el dominio, entonces la relación es una función.

volume (583) The measure of space occupied by a solid region.

volumen Medida del espacio que ocupa un sólido.

Glossary/Glosario

R21

Glossary/Glosario

vertical line test (360) If any vertical line drawn on the graph of a relation passes through no more than one point on the graph for each value of x in the domain, then the relation is a function.

W whole or base (322) In a percent proportion, the whole quantity, or the number to which the part is being compared.

entero o base En una proporción porcentual, toda la cantidad o número al que se compara la parte. porciento parte _ _ =

part percent _ =_ whole

100

todo

100

X x-axis (54) The horizontal number line which helps to form the coordinate system.

eje x Recta numérica horizontal que forma parte de un sistema de coordenadas.

x-coordinate (54) The first number of an ordered pair.

coordenada x El primer número de un par ordenado.

x-intercept (54) The x-coordinate of a point where a graph crosses the x-axis.

intersección x La coordenada x de un punto en que una gráfica interseca el eje x.

Y y-axis (54) The vertical number line which helps to form the coordinate system.

eje y Recta numérica vertical que forma parte de un sistema de coordenadas.

y-coordinate (54) The second number of an ordered pair.

coordenada y El segundo número en un par ordenado.

y-intercept (391) The y-coordinate of a point where a graph crosses the y-axis.

intersección y La coordenada y de un punto en que una gráfica interseca el eje y.

Z

Glossary/Glosario

zero pair (84) A positive tile paired with a negative tile.


par nulo Ficha positiva apareada con una negativa.

Selected Answers

Selected Answers Chapter 1 The Tools of Algebra Page 25

Chapter 1

Get Ready

1. 14.8 3. 2.95 5. 3.55 7. $3.70 9. Sample answer: 1200 11. Sample answer: 20,000 13. Sample answer: 150 miles 15. Sample answer: $5 17. Sample answer: 100 19. Sample answer: 10

Pages 29–30

Lesson 1-1

1. 1:33 P M. 3. 17 5. 3072 7. $1119.33 9. 146 beats per min 11. 24 13. 486 15. 37 17. 33 21. 2 quarters, 1 dime, 5 pennies 19. 23. 80,000,000 jelly beans 25. Sample answer: 0, 4, 8, 12, … 27. Yes; In the addition table, there is only one more even number than odd. 29. D 31. C 33. 5 35. 50

Pages 34–36

Lesson 1-2

1. 20 3. 6 5. 66 7. 42 9. 4 11. 6 × 8 13. 11 + 16 15. 14 17. 25 19. 38 21. 2 23. 55 25. 24 27. 7 + 2 29. 9 × 5 31. 24 ÷ 6 33. 3 × $6 35. 4(8) + 3(5) + 4 37. (3 × 16) + (2 × 9) is 66. So, Joshua’s luggage is within the limit. 39. 1554 41. 5 43. 12 × 3 ÷ (1 + 2) = 12 45. (5 + 2) ․ (9 - 3) = 42 49. Emily; she followed the order of operations and divided first. 51. Sample answer: 111 - (1 + 1 + 1) × (11 + 1) 53. A 55. 64 57. 85 59. 32 61. $275 63. about 31 million 65. 126 67. 496

21. Associative Property (+) 23. Additive Identity 25. Associative Property (×) 27. 55; (17 + 33) + 5 = 50 + 5 = 55 29. 420; (2 · 30) · 7 = 60 · 7 = 420 31. 0; 0 times any number equals 0 33. 58; 8 + (23 + 27) = 8 + 50 = 58 35. no; 9 - 3 ≠ 3 - 9 37. no; 1 + 1 = 2 39. a + 27 41. k + 37 43. 28d 45. 15w 47. No; marinating meat must be done before cooking meat. 49. Sample answer: 3 × 4 = 4 × 3 51. The set of whole numbers is not closed under subtraction and division. 2 - 3 = -1 and -1 is not a whole number. 1 ÷ 2 = 0.5 and 0.5 is not a whole number. 53. B 55. 7 57. 15 59. 74 61. 240 63. 192 65. 1378 Pages 51–53

Lesson 1-5

1. 6 3. 37 5. C 7. Let n = the number; 25 = n - 10; 35 9. 9 h 11. 11 13. 12 15. 5 17. 15 19. Let w = the number; 9 + w = 36; 27 21. Let z = the number; z - 12 = 54; 66 23. Let x = the number; 3x = 45; 15 25. 12 27. 10 29. Let h = the increase in height; 65 + h = 68; 3 31. Sample answer: b + 7 = 12 and 8 - h = 3 33. Sample answer: paper/pencil; you can write an equation to represent the situation. 650 = x + 439; x = 211 35. D 37. 23 + d 39. k + 57 41. 32.50 + 4.95c 43. 17 45. 2 47. 42 49. 18 51. 95 Pages 57–59

1–4.

Lesson 1-6

5. (2, 3) 7. (4, 3)

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H Pages 39–41

W

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Lesson 1-3

1. 18 3. 21 5. 3 7. s + 8 9. k ÷ 4 - 5 or k - 5 4 11. 16p 13. 11 15. 38 17. 2 19. 15 21. 3 23. 56 25. s + $200 27. h - 6 29. 3b 31. 71˚F 33. s = c + m - d 35. 71 37. 36 39. 48 41. x + 3 43. p - 4 45. 2x - 8 ÷ 2x 47. 6a - b; a + 3b, 4b (a + 1), and 12 + b have a value of 17 and 6a - b has a value of 7. 49. Expressions show relationships, and the variables in the relationships are placeholders for numbers. For example, x + y = 2. This shows that the sum of two numbers is equal to 2. 51. J 53. 5 55. 9 57. 48 59. 960 61. 69 63. 29

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Pages 46–47

Lesson 1-4

1. Commutative Property (+) 3. Commutative Property (×) 5. Commutative Property (×) 7. 28; (13 + 7) + 8 = 20 + 8 = 28 9. 45; (8 + 22) + (11 + 4) = 30 + 15 = 45 11. n + 13 13. 27w 15. $42; To find the total cost, add the three costs together. Since the order in which the costs are added does not matter, the Commutative Property of Addition holds true and makes the addition easier. By adding 4 and 26, the result is 30, and 30 + 12 is 42. 17. Commutative Property (×) 19. Multiplicative Identity

R24 Selected Answers

11.

x

y

2

8

4

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5

20

13–18.

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19. (1, 7) 21. (0, 4) 23. (2, 1)

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x

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y

Temperature (°C)

25.

Selected Answers

100 95 90 85 80 75 70 65 60 55

x

50 45

27.

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y

7

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35. Science Experiment

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37.

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51. (4, 2), (4, 8), (10, 2) 53. The figures have the same shape but not the same size. 55. Ordered pairs can be used to graph real-world data by expressing the data as ordered pairs and then graphing the ordered x pairs. The x- and y-coordinate of an ordered pair specifies the point on the graph; longitude and latitude lines 57. J 59. 41 61. 9 63. 13 65. 12 67. 21 - 15 69. 60 12 71. Multiply 8000 by 365 days per year; 2,920,000 73. 12 75. 28 77. 7 79. 9 y

Selected Answers

R25

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25. positive 27. Sample answer: By studying scatter plots and identifying the type of relationship the scatter plot shows, you can determine trends. A positive relationship means that as one set of data increases, the other set also increases. A negative relationship means that as one set of data increases, the other set decreases. No relationship means that one set of data is not related to the other set of data; The amount of money spent and the amount of money saved would be an example of a negative relationship. The number of hours spent studying and test scores would represent a positive relationship. The number of days in a month and the number of rainy days is an example of no relationship. 29. F

Lesson 1-7

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1. false; algebraic expression 3. false; range 5. true 7. false; conjecture 9. $30.66 11. 12 13. 486 15. 22 17. 16 19. 68 21. $130 23. 11 25. 15 27. Let B = Brandon’s time for climbing rope; 5 + B 2 29. Add. Identity 31. Assoc.(×) 33. 72d 35. 10 37. 22 39. 6 41. d = 72r 43.

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9. no; hair color is not related to height. 11. Positive; As speed increases, distance traveled increases. 13. Negative; candle burn time increases as the height of the candle decreases 15. Yes; positive; as the number of minutes increases, the number of field goal attempts increases. 19. The number decreases. 21. Sample answer: Perhaps government regulation or private protection was targeted at eagles in the early 1970s. The number of eagles is rising. 23. Sample answer: No; the trend would not continue indefinitely. There is a maximum score of 800, so the average scores cannot go above 800. Also, even though the scores have been increasing, the data are taken from the past and do not necessarily predict future events; the average score dropped from 519 in 2003 to 518 in 2004.


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Pages 63–66

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Pages 81–83

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y

1

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2

600

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3

900

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4

1200

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2

4

Pages 89–90

Lesson 2-1

1. -8

2

3. -7 < 3; 3 > -7 87654 32 1 0

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23. -5 > -10; -10 < -5 25. -4 < 0; 0 > -4 27. > 29. > 31. > 33. > 35. {-15, -4, -2, -1} 37. {-23, -20, -14, 41, 50} 39. {-60, -57, 38, 98, 188} 41. 46 43. -5 45. 7 47. 2 49. 9 51. -6 53. 4 55. 40 57. 3 59. 61. -54 > -70 54 90 70 50 30 10

63. 4321 0 1 2 3 4

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67. A: 1, B: -5, C: 3, D: -2 71. Sample answer: The distance between -4 and 5 can be found by evaluating |-4| + |5|.|-4| is the distance between -4 and 0, and |5| is the distance between 5 and 0. The sum of these two distances can be represented by |-4| + |5|, which equals 9. 73. Sometimes true; if A and B are both positive, both negative, or one is 0, it is always true. If one number is negative and the other is positive, it is never true. 75. Sample answer: Integers describe real-world situations involving measures above and below zero. Rainfall above normal is described using positive integers, and rainfall below normal is described using negative integers. Some examples of negative numbers used in real life are: accounting uses + for deposits and - for withdrawals; gains are represented by +, losses by -; in golf, scores above par are +, below par are -; temperatures above zero are +, below zero are -. 77. 7 79. positive; As height increases, so does arm length.

Lesson 2-2

1. -6 3. -18 5. 8 7. -14 9. -11 11. 4 13. -5 15. -10 17. -15 19. -23 21. -26 23. 3 25. -4 27. -9 29. -14 31. -1500 + (-1250); -2750 points 33. 6 35. -3 37. 11 39. -26 41. -$7; 50 + (-25 + -30) = -5; The answer is reasonable. 43. 5 45. 37 47. 7747 49. False; sample counterexample: If n = -2, then -(-2) is positive. 51. Multiplicative Property of 0 53. B 55. -219°C 57. negative 59. 7 61. 14 63. 5 Pages 95–97

Lesson 2-3

1. -3 3. -24 5. 17 7. -2 9. 3 11. 20 13. 21 15. -1 17. -3 19. -9 21. -12 23. 10 25. 12 27. 3 29. -9 31. 98 33. 14,776 ft 35. 24 37. -36 39. -9 41. -20 43. -23 45. -17 47. -$623 million 49. expenses > income 51. -214 53. -$2.71, $0.45, $0.93, -$1.82 57. False; (5 -2) -1 ≠ 5 -(2 -1) 59. Sample answer: 5, -5; -9, 9 61. Sample answer: Addition and subtraction of integers are related because a subtraction problem can be rewritten as an addition problem. For example, 4 -1 is the same as 4 + (-1); both have an answer of 3. 63. F 65. 2 67. t + 9 69. s - 8 71. 1 73. 75 75. 120 Pages 102–104

Lesson 2-4

1. -24 3. -22 5. 28 7. -48 9. -540 11. A 13. -21y 15. 120 17. -12 19. -32 21. -36 23. 114 25. 88 27. 120 29. -168 31. 1440 33.-35x 35. -48a 37. 56st 39. -42ab 41. -4pq 43. -54cd 45. 28 47. 72 49. 320 51. -15 ft 53. 3(-5) = -15 55. 248 ft 57. Sample answer: (-4)(9)(2) 59. 22 61. When you multiply two integers with the same sign, the product is positive; when you multiply two integers with different signs, the product is negative. Sample answer: If the positive integer represents the number of groups made up of the negative integer, the total will be negative. 63. 56 65. 4 67. 10 69. 2 71. -3 73. (3, 4) 75. (8, 0) 77. (0, 3) 79. 6 81. 13 Pages 109–110

Lesson 2-5

1. 11 3. 9 5. -10 7.-13 9. 0°F 11. 9 13. 8 15. 10 17. -50 19. -15 21. -5 23. -12 25. -13 27. 16 29. 49 points 31. 21 33. -184.4°C 35. Sample answer: x = -144; y = 12; z = -12 37. Sample answer: When the signs of the integers are the same, both a product and a quotient are positive; when the signs are different, the product and quotient are negative. Sample answer: 4 · (-6) = -24 and -24 ÷ 4 = -6; -3 · 2 = -6 and -6 ÷ (-3) = 2 39. G 41. -39 43. -50cd 45. -5, -10 47. F 49. A Selected Answers

R27

Selected Answers

Page 77

81. {(3, 2), (3, 4), (2, 1), (2, 4)} 83. Commutative Property (×) x y 85. Commutative 3 2 Property (×) 87. 388 89. 17 3 4 91. 1049

Pages 113–115

Lesson 2-6

37.

Selected Answers

1. (-4, 5) 3. (-3, -2) 5–8.

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21–32.

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x

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51.

47. Yes; total spending increased from $2232 in 2002 to $2798 in 2003. 49. 3a + 3b 51. 4j - 4k 53. -8u + 8w 55. 8(20 + 3) = 184 57. 16(10 + 1) = 176 59. Sample answer: 2(3 + 4) = 2 · 3 + 2 · 4 61. No; 3 + (4 · 5) = 23, (3 + 4)(3 + 5) = 56 63. D Y 67. 6 69. 5 + (-3) 65. 71. 10 + (-14) X

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Chapter 2


1. negative number 3. additive inverses 5. integers 7. inequality 9. opposites 11. = 13. > 15. 25 17. 22 19. -9 21. 5 23. 8 25. +1 27. -8 29. 5 31. -4 33. -45 35. 28 37. -15 39. 13 41. -22 y

43. K (1, 3)

I 45. II 47. Andy: I, Owen: II, Cheryl: IV, Denyce: III, and Fabio: IV.

A(4, 3)

O R (3, 0)

Lesson 3-2

1. terms: 4x, 3, 5x, y ; like terms: 4x, 5x; coefficients: 4, 5, 1; constant: 3 3. terms: 4y, - 2x, -7; like terms: none; coefficients: 4, -2; constant: -7 5. 10x + 3 7. 3x + 6y 9. -2x + 3 11. -12e - 8f 13. 2x + 20 15. terms: y, 3y, 8y, 2; like terms: y, 3y, 8y; coefficients: 1, 3, 8; constant: 2 17. terms: 5c, -2d, 3d, -1d; like terms: -2d, 3d, -1d; coefficients: 5, -2, 3, -1; constant: none 19. terms: 7x, -3y, 3z, -2; like terms: none; coefficients: 7, -3, 3; constant: - 2 21. 9b 23. 6y 25. 3m + 4 27. 10x + 12 29. 8b 31. -2r 33. -x + 4 35. 5x + 48 37. 2y - 5 39. 16m + 2n 41. -9c + 2d 43. 13x + 15 45. -6 47. 6a + 14 49. 6x + 2 51. 72(38 - 18) 53. Koko; 5x + x = 6x, not 5x. 55. Distributive 57. Substitution 59. Sample answer: You can use algebra tiles to simplify an algebraic expression by grouping the tiles with the same size and shape together. 61. F £ £ 63. -2y + 16 X X X X X X X 65. -3x + 3 £ {X

x

Ó

ÎX

J (2, 5)

£

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X

Get Ready

1. -6 3. 10 5. 88 7. -$10.25 9. 6 + (-10) 11. 11 + (-10) 13. -19 + (-10) 15. -3 17. 0 19. -21 21. -22 ft Pages 126–128

X ÇX

Lesson 3-1

1. 5 · 7 + 5 · 8; 75 3. 2 · 6 + 4 · 6; 36 5. $6.25(4 + 5), $6.25(4) + $6.25(5) 7. 4x + 12 9. 3n + 6 11. 8y - 16 13. -6x + 30 15. 2 · 6 + 2 · 1; 14 17. 4 · 9 + 6 · 9; 90 19. 9 · 4 + 2 · 4; 44 21. 7 · 3 + 7(-2); 7 23. -3 · 9 + (-3)(-2); -21 25. -5 · 8 +(-5)(-4); -20 27. 4($7 + $3), 4($7) + 4($3); $40 29. 2x + 6 31. 3n + 3 33. 4x + 12 35. 18 + 6y 37. 3x - 6 39. 8z - 24 41. 6r - 30 43. -2z + 8 45. $5596

67.

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Chapter 3

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Chapter 3 Equations Page 123

£

Ç È x { Î Ó £ ä ½Ç ½n ½ ½ää ½ä£ ½äÓ ½äÎ ½ä{ 9i>À

Selected Answers

R29

Selected Answers

The graph can include any integer pairs where x < 4 and x > -4 and y < 3 and y > -3. 53. Keisha; a point in x Quadrant I has two positive coordinates. Interchanging the coordinates will still result in two positive coordinates, and the point will be in Quadrant I. 55. III 57. Sample answer: y y

Pages 139–140

Lesson 3-3

separating tiles into groups models the Division Property of Equality;

Selected Answers

1. 11 È

Ç

n

£ä ££ £Ó £Î

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27. 15 = x - 19; 34 31. 18 million 33. -34.9 35. -33.07 37. 103.19 39. Sample answer: equivalent: x + 3 = 5, x = 2; When x + 3 = 5, x = 2; not equivalent: x + 5 = 2, x + 1 = 5; When x + 5 = 2, x = -3. When x + 1 = 5, x = 4. 41. Sample answer: x - 2 = -7, x + 2 = -3 43. B 45. -2x - 10 47. -4x + 8 49. 6x + 12 51. 6 years 53. -10 55. 7 Pages 143–145

Lesson 3-4

1. 6 3. 6 5. 27 7. 66 9. 9 11. 36 13. 84 15. -19 17. 8 19. -140 21. 17 23. -9 25. -18 27. 1452 29. -504 31. 17 33. 60p = 960; 16 min 35. 40x = 720; 18 cows 37. 0

1

2

3

4

5

6

7

0

1

2

3

4

39. 3 2 1

41. 3y = 15; 5 yd 43. 6p = 24, 4 painters 45. Sample answer: -5x = -20 47. 223 49. C 51. B 53. 25 55. 11x 57. -2y - 5 59. 10 61. -2 63. -36 Pages 149–151

Lesson 3-5

1. 8 3. 24 5. -3 7. -91 9. 10 11. 2 13. 4 15. 8 17. 13 19. 3 21. 28 23. 64 25. 65 27. 21 29. 3 h 31. 33 33. -13 35. 5 37. 5 39. -2 41. 10 43. 11 45. -1 47. 9x + 16 = 169; $17 49. 2y 51. Sample answer: You can add or remove tiles from each side of a mat. This models the Addition and Subtraction Properties of Equality. Also,


x x

1

1

1

1

;x=2 53. C 55. -7 57. -2 59. -2 61. -5y - 15 63. -9y + 36 65. -8r + 40 67. no 69. 2x - 6 71. 2x - 8 73. 9x + 4 Pages 155–157

Lesson 3-6 n

1. 4n + 3 = 23; 5 3. - 8 = 16; 72 5. 2x + 5 = 37; 3 21 yr 7. 3n + 20 = -4; -8 9. 10n - 8 = 82; 9 n 11. _ - 8 = -42; 136 13. 8 - 5x = -7; 3 h 15. 2x + -4 320 = 1772; 726 ft 17. 17 - 2n = 5; 6 19. 4n + 3n + 5 = 47; 6 21. 3 ft 23. Ben; Three less than means that three is subtracted from a number. 25. Sample answer: By 2030, Texas is expected to have 1.3 million more people age 65 or older than New York will have. Together, they are expected to have 9.1 million people age 65 or older. Find the expected number of people age 65 or older in New York by 2030. 27. D 29. -2 31. 4 33. n(t + h + s) 35. (2, -3) 37. (-3, -4) 39. 5 41. -11 43. 21 45. 40 Pages 160–161

Lesson 3-7

1. The terms have a common difference of 1. A term is 1 more than the term number. t = 1 + n 3. The terms have a common difference of 3. A term is 3 times the term number. t = 3n 5. t = 9 + n; 19 7. t = 3n - 1; 59 9. 15 11. The terms have a common difference of 1. A term is 7 more than the term number. t = 7 + n 13. The terms have a common difference of 1. A term is 14 more than the term number. t = 14 + n 15. The terms have a common difference of 8. A term is 8 times the term number. t = 8n 17. The terms have a common difference of 20. A term is 20 times the term number. t = 20n 19. The difference of the term numbers is 1. The terms have a common difference of 2. A term is 2 times the term number, plus 2. t = 2n + 2 21. The difference of the term numbers is 1. The terms have a common difference of 4. A term is 4 times the number, minus 1. t = 4n - 1 23. t = 13 + n; 29 25. t = 11n; 275 27. t = 2n + 5; 71 29. t = 5n - 2; 348 31. 13 33. 665 35. 34 37. 81 39. A n 41. 3n + 5 = 20 43. _ -3 = -63 45. 7 -10

47. 10(12 + 7 + 15), 10(12) + 10(7) + 10(15); $340 49. -4 51. 6

Pages 164–167

Lesson 3-8

29.

4 ft

31.

3 ft

4 cm

Pages 188–190

4 cm

33. 2 bags 35. 30 units2 answer: 5 in.

37. 25.1 mph 39. Sample

4 in.

41. P = 6w - 2 43. B 45. t = 10 + n; 70 47. 14 49. domain: {1870, 1881, 1910, 2000}, range: {14, 600, 1000, 1500} Pages 169–172

Chapter 3


1. like terms 3. coefficient 5. inverse operations 7. area 9. 3h + 18 11. -5k - 5 13. -2b + 8 15. 3(3) + 3(5) = 3(3 + 5); $24 17. 4y + 7 19. 4m - 14 21. -2n - 8 23. 3 25. 4 27. 5 29. x + 13 = 37; 24 pages 31. -10 33. -66 35. 1 37. -6 39. 153 n 41. _ - 6 = -3, 12 43. The terms have a difference of 4 1. A term is 4 more than the term number; t = 4 + n. 45. The terms have a difference of 6. A term is 6 times the term number; t = 6n. 47. t = n + 7; 26 49. t = 7n; 490 51. 36 in.; 81 in2 53. 10 yd; 6.25 yd2 55. 95 words per minute

Chapter 4 Factors and Fractions Page 179

Chapter 4

Get Ready

1. 14 3. 28 5. 63 7. 30 9. 295 11. 3n - 3 13. -4x + 20 15. -15s + 5t 17. 9b - 18c 19. 45 21. 78 23. 0.005 25. $15.90 Pages 182–184

factors of 2 in the number of megabytes increases. In describing the amount of memory in modern computers, it would be impractical to list all the factors of 2. Using exponents is a more efficient way to describe and compare computer data. 69. 512 71. c = r + 2 73. 4 75. 1, 11 77. 1, 2, 4, 8, 16

Lesson 4-1

1. n3 3. (-4)3 5. (y - 3)3 7. 16 9. 50 11. 45 13. 132 15. 61 17. (-8)4 19. (-t)3 21. m4 23. 2x2y2 25. 9(p + 1)2 27. 1000 29. -32 31. 128 33. 2430 35. 256 37. 64 39. -9 41. 70 43. 14 45. 8 47. 121 = 112; 100 = 102 49. 7 · 7 · 7 · x · x 51. 960, 96, 962, 965, 9610; As the exponents increase, the additional factors of 96 increase. Since 960 = 1, it has the least value. 53. 21, 22, 23, 24, 25 55. After 10 folds, the noodles are 5(210) = 5(1024) or 5120 feet long, which is slightly less than a mile. So, after 11 folds the length of the noodles will be greater than a mile. 57. = 59. 6 · 32 cm2 61. No; the surface area is multiplied by 4. The volume is multiplied by 8. 63. No; The perimeter is doubled and the area is four times the area of the original square. 65. original volume: n3, new volume: (3n)3 or 27n3 67. As the capacity of computer memory increases, the number of

Lesson 4-2

1. prime 3. composite 5. 2 · 32 7. 2 · 52 9. 5 · a · a · b 11. composite 13. prime 15. prime 17. prime 19. 2 · 13 21. 2 · 3 · 11 23. 23 · 13 25. 23 · 72 27. 2 · 7 · w 29. -1 · 7 · c · c 31. 2 · 2 · 5 · s · t 33. 2 · 2 · 7 · x · x · y 35. 13 · q · q · r · r 37. -1 · 3 · 5 · 5 · a · b · b 39. prime 41. prime 43. 52 45. 1 × 1 × 36, 1 × 2 × 18, 1 × 3 × 12, 1 × 4 × 9, 1 × 6 × 6, 3 × 3 × 4, 2 × 3 × 6, 2 × 2 × 9 47. Francisca; 4 is not prime. 49. 211 51. D 53. (-5)3h2k 55. 9 P M. 57. EST is 3 hours ahead of PST 59. -9 61. -28 63. negative 65. 2n + 8 67. -3t - 12 69. -2b + 6 Pages 193–195

Lesson 4-3

1. 2 3. 8 5. 12 7. 12 students 9. 14n 11. 3(n + 3) 13. 5(3 + 4x) 15. 3 17. 7 19. 6 21. 5 23. 1 25. 6 27. 6-in. squares; 20 tiles 29. 8 31. 2s 33. x 35. 5 37. 2(x + 4) 39. 8(1 + 5a) 41. 3(5f + 6) 43. 7; Sample answer: 7

14

21

28

35

↓

↓

↓

↓

↓

7(1)

7(2)

7(3)

7(4)

7(5)

The terms increase by 7. 45. 6a2b 47. yes 49. no 51. k(k2 + k + 5) 53. 5(n - 2m + 5) 55. Sample answer: = x, w = x + 2 57. Sample answer: 12 and 24 59. Yes; the GCF of 2 and 8 is 2. 61. B 63. 3 · 3 · n 65. -1 · 5 · j · k 67. 92 69. $13 71. -6 73. -10 75. 36 77. 100 Pages 198–200 1 1. _ 7

Lesson 4-4

16 3. simp. 5. _

4c 1 7. _ 9. _ 2

5m 5 11. _ 13. _

6 13 2 y 9 2 1 15. _ 17. _ 19. simp. 21. _ 23. _ or y2 9 22 3 1 20 38 3 7z2 1 25. _ 27. _ 29. _ 31. 2004: _ ; 2001: _ ; 55 21 16t 4 31 1 330 3 264 12 1999: _ ; 1996: _ 33. yes; _ =_ 35. yes; _ 51 5 528 440 4 10 1 =_ 39. _ ; the others are in simplest form. 2 12 17

x

5d

41. Fractions represent parts of a whole. So, measurements that contain parts of units can be represented using fractions. Answers should include the following. • Measurements can be given as parts of a whole because smaller units make up larger units. For example, inches make up feet. 3 • Twelve inches equals 1 foot. So, 3 inches equals _ 12 1 or _ foot. 4 43. J 45. 3 47. 8x 49. prime 51. comp. 1 53. n + 9 = -2; -11 55. _ h(b1 + b2) 2

57. (s · s)(t2 · t) 59. (3 · -5)(x4 · x2) 61. (a · a5)(c3)(12 · 15 · 9) Selected Answers

R31

Selected Answers

1. 14 h 3. 34 km, 30 km2 5. 4 in. 7. 165 mi 9. 10 mi, 6 mi2 11. 34 ft, 60 ft2 13. 46 m, 102 m2 15. 96 m, 380 m2 17. 18 cm 19. 24 m 21. 11 in. 23. 390 yd, 9000 yd2 25. 14 bricks 27. d = 2r

Lesson 4-5

1. 95 3. 67 5. n8 7. 33 9. a4 11. 35 13. 99 15. d10 17. t6 19. 18y5 21. 40x8 23. 53 25. (-2)1 or -2 27. b3 29. (-x)4 31. 97 33. 106 or 1,000,000 times 35. 3 37. 23 or 8 times 39. 16 41. 79 43. 224 45. 8a3b10 47. n6 49. Sample answer: 5 · 52 = 53 51. false; If a = 3, then (-3)2 = 9 but -32 = -9. 3 53. B 55. _ 10

n 57. _

59. 4 61. 6 63. -144

4

65. Positive; as the high temperature increases, the amount of electricity that is used also increases. 1 1 67. _ 69. _ 25

64

Pages 211–213

Lesson 4-6

1 1 1 1. _ 3. _ 5. 3-4 7. 7-2 9. 10-3 11. -_ 27 52 t6 1 1 1 1 1 1 13. _ 15. _ 17. _ 19. _ 21. _ 23. _ 8 53 (-3)3 104 a10 q4 1 25. _ 27. 5-5 29. 13-2 31. 9-2 or 3-4 33. 2-4 r20 1 1 or 4-2 35. __ m; 10-9 m 37. _ 1,000,000,000 81 1 1 41. 2-4 or _ ; 0.0625 43. 10-2 or 100-1 39. _ 64 16

45. 10-5 47. penicillin; 105 times greater 49. r4 8

3t 51. y16 53. 3s-3t8 or _ 55. In 2-3, the negative sign 3 s

indicates that the factor 2 is repeated three times in the denominator, or 1 is divided by 2 three times. So, 2-3

1 =_ . If the exponent were -4, then 1 would 2·2·2

be divided by 2 four times, and so on. So, as the value 1 decreases. 57. (5 × 103) + of n increases, the value of _ 2n (9 × 102) + (3 × 101) + (1 × 100) 59. (1 × 10-1) + (7 × 10-2) + (3 × 10-3) 61. Exponents are used to represent repeated factors in a multiplication. If the exponent is a negative number, then the repeated factors act as repeated divisors in the denominator. 1 1 Answers should include the following: 4-2 = _ =_ 4·4 42 1 _ 6 7 or 63. H 65. x 67. y 69. 18 shelves 16 71. 8y + 48 73. -18 - 2k 75. 77. 720 Mr. Stanley’s Class 79. 40.5 y 81. 0.038 4.0 3.8 83. 0.924

GPA

Selected Answers

Pages 205–207

3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 0

x 59 60 61 62 63 64 65 66 67 68 69 70 Height (in.)

Pages 216–218

81

49

Pages 219–222

Chapter 4


1. true 3. true 5. true 7. false; 1 9. 41 11. f 2g4 13. 25 15. 144 17. 32 · 5 19. 22 · 17 21. 2 · 3 · 3 · x 23. 2 · 2 · 2 · 2 · 2 · p · q 25. 8; 1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 × 4 27. 8 29. 4cd 2 31. 3(x + 8) 33. 5(9s + 5) 35. 4(16 - 15k) 37. _ 7 m 14 1 39. simplified 41. _ 43. _ 45. _ 47. 35 49. k

1 51. _ 2 7

2n

19w

1 53. _ 3 (-4)

3

55. 2-6 or 8-2 or 4-3

57. 106 59. 0.0029 61. 70,450 63. 8.0 × 10-3 65. 4.571 × 107

Chapter 5 Rational Numbers Page 227

Chapter 5

1. 0.6 3. 0.1 5. 0.2 7. -1.7 18 15. simplified 17. -5 13. _ 25

Getting Ready 3 9. 33 11. _ 5

19. -11

21. -7 ft Pages 231–233

Lesson 5-1

1. 0.875 3. 0.3125 5. 2.4 7. 0.6 9. 4.18 11. = 13. > 15. 0.2 17. -0.32 19. 7.3 21. -5.125 23. 0.1 25. -0.45 27. 0.883 29. > 31. = 33. < 1 35. < 37. < 39. = 41. More than; _ = 0.25 and 4 2 = 0.28 > 0.25 43. > 45. > 47. > 49. Japanese; _ 5 3 1 2 _ _ _ 53. -1 , -1.1 , 0.4 and 0.4 > 0.26 51. -0.29, - , 8 7 11 1 55. 0.83 57. Sample answer: fractional form: -1_ 10 customary measurement; decimal form: stock price 59. 2 = 2, 3 = 3, 4 = 22, 5 = 5, 6 = 2 · 3, 8 = 23, 9 = 32, 1 10 = 2 · 5, 12 = 22 · 3, 15 = 3 · 5, 20 = 22 · 5; _ = 0.5, 2

_1 = 0.3, _1 = 0.25, _1 = 0.2, _1 = 0.16, _1 = 0.125, _1 = 0.1, 3

Lesson 4-7

105

1. 0.000308 3. 849,500 5. 6.97 × 7. 517 s 9. 424 11. 0.3347 13. 0.00015 15. 139,900 17. 61,000 19. 4.99 × 105 21. 1.25 × 10-2 23. 3.956 × 104 25. 4.25 × 10-4 27. 343 s 29. Arctic,


Indian, Atlantic, Pacific 31. -3.14 × 102, -3.14 × 10-2, 3.14 × 10-2, 3.14 × 102 33. 9.05 × 10-6, 905,000, 9.5 × 106, 9,562,301 35. 20 times 37. 2.0 × 106; 2,000,000 39. 1.6575 × 10-5; 0.000016575 41. 3.8 × 106 43. Sample answer: 3.8 million is easier to read and understand than the standard form (3,800,000) or scientific notation (3.8 × 106) of the number. 45. Scientific notation is a shorthand way of writing very large or very small numbers. Answers should include the following. • Examples of data that can be written in scientific notation, are the surface area of planets and the diameter of atoms. • Scientific notation is useful because you can compare quantities by simply looking at the exponent of the power of 10, rather than counting decimal places. 1 1 47. 130 49. _ 51. _ 53. 411 55. c3 57. 107

5

4

6

8

9

1 1 1 1 _ = 0.1, _ = 0.083, _ = 0.06, _ = 0.05; Sample 10

12

15

20

answer: Fractions whose denominators have only 2 or 5 as prime factors are terminating decimals. Fractions whose denominators have 3 as a prime factor are

shaded and another fraction represents the fraction of rows that are shaded.

1 1 71. _ 69. _ (-2)7 y3 7 2 81. _ 79. _ 8 3

60 in2 81. 2 83. 3y + 8

Pages 236–238 10 7 1. -_ 3. _ 1 3

2 75. _ 15

73. 29 mi/gal

3 77. _ 5

Lesson 5-2

5 4 4 5. _ 7. 3_ 9. _ 11. integer, 5 25 11 60 9 11 _ 17. _ 19. _ rational 13. rational 15. 7 1 100 5 17 1 2 2 1 21. 1_ 23. 8_ 25. _ 27. -_ 29. 5_ 31. 2_ 25

250

25

3

3

99

33. integer, rational 35. natural, whole, integer, 9 rational 37. rational 39. not rational 41. Fate: _ ; 500

1 21 ; Rockwall: _ ; Royse City: McLendon-Chisholm: _ 50

50

7 _ 43. = 45. < 47. < 49. greater than; 100 22 _ ≈ 3.142857 > 3.141592 7

51. Let N = 0.999. . . and let 10N = 10(0.999. . .) 10N = 9.999. . . -(N = 0.999. . .) 9N = 9 9N 9 _ =_ 9

9

N=1 1 and 2 are both rational numbers, but 53. sometimes; _ 2 only 2 is an integer. 55. The set of rational numbers includes the set of whole numbers and integers. In the same way, natural numbers are part of the set of whole numbers and the set of whole numbers is part of the set of integers. Answers should include the following. • The number 5 belongs to the set of natural numbers, whole numbers, integers, and rational numbers. 1 • The number _ belongs only to the set of rational 2 numbers. 57. G 59. -7.8 61. 2.5 63. 3,050,000 65. 0.01681 67. 46 in.; 112 in2 69. 18 71. -5 · 4 = -20 73. 7 · 2 = 14 Pages 242–244 Lesson 5-3 3 3 5 3 1 _ _ _ 1. 3. 5. 7. -_ 9. -_ 11. 18 20 32 9 12 40 5a 1 7 mi _ _ _ · h or 227_ mi 13. 140 15. c 17. d = 65 2 h 2 8 6 9 1 1 7 21. -_ 23. _ 25. _ 27. -_ 29. -1_ 19. _ 27 7 7 5 20 20 8 3 33. 8_ 35. -14 37. 1596 ft 39. 27 31. -1_ 27 4 8c xz 1 43. _ 45. 5 bags 47. _ 49. baseball 41. _ 3 11 4

51. 4400 53. 40 55. 12.7 57. 10.257 59. 774.72 1 _ 61. 210.105 63. Sample answer: _ , 1 65. Sample 2 3 8 3 ×_ 67. Fractions can be used to answer: _ 5 4 represent parts of rectangles. The product of the fractions equals a portion of the rectangle’s area. The product of the fractions is the overlapping shaded area. Answers should include the following. • A rectangle that is divided into rows and columns; one fraction represents the fraction of columns that are

7 1 73. _ 75. 3.0 × 109 77. x6 79. 34 in., 69. F 71. -_ 5

9

85. b 87. 2p

Pages 248–249 Lesson 5-4 5 8 29 4 7 1 _ _ _ 1. 3. 5. 7. -_ 9. 1_ 11. -1_ 13. 14 25 5 36 4 15 18 5 2x 3 11 4 1 15. _ 17. _ 19. -_ 21. _ 23. _ 25. -_ 5a 6 21 12 7 5 5 2 2 4 _ _ _ _ 27. 29. 1 31. 1 33. 27 35. 37. 1 3 25 32 5 2 5 8k 2 _ _ _ 39. -1 41. 6 43. 45. 47. 11 hamburgers 3 3 8d 2 1 _ _ 51. $6/hour 53. 1 , 3, 6, 9; The quotient 49. -2 7 2 1 increases. The quotient decreases. 55. D 57. _ 5 1 _ 59. -1 61. natural, whole, integers, rational 20 1 4 numbers 63. rational numbers 65. 0.85 67. 1_ 69. 4 _ 7 5 Pages 252–254 Lesson 5-5 6 8r 5 1 4 2 1 _ _ 1. 3. 5. -3 _ 7. _ 9. -_ 11. _ 13. _ 7 5 7 11 7 9 2 3 1 4 1 2 2 15. _ 17. -1_ 19. 11_ 21. 9_ 23. _ 25. _ 5 2 5 3 3 11 3 6 3 5x 3 1 _ _ _ _ _ 27. 29. 4 31. -5 33. 2 35. 37. 39. 2_ c 7 8 7 4 4 4 3 5 1 _ _ _ 41. 3 in. 43. 3 lb 45. 8 47. 38 ft 49. Sample 8 8 8 19 1 1 1 1 1 ,C=_ ,D=_ , answer: _ - _ 51. A = _, B = _ 25 8 4 4 16 25 1 1 1 2 1 _ _ _ _ _ E= ,F= ,G= 53. or 55. A and B 8 8 8 16 16 3 2 4 1 57. D 59. _ 61. _ 63. _ 65. 1_ 67. 32y7 9 3 15 10

69. 71 min 71. 52 · 7 73. 22 · 3 · n 75. 2 · 3 · 7 · a2 · b Pages 259–261

Lesson 5-6

1. 24 3. 70 5. 60x2y2 7. 8 9. 40 11. = 13. Frankfort, Lexington, Bowling Green, Louisville 15. 60 17. 48 19. 84 21. 100 23. 96 25. 112a2b 27. 75n4 29. 60 s 31. 15 33. 35 35. 24 37. 18 23 1 4 7 2 39. < 41. > 43. = 45. -_ , -_ , -_ , -_ 47. 1_ , 8 5 30 3 10 5 3 1 _ _ _ 49. GCF; the question asks for the 1 ,1 ,1 24 3 4 “greatest” number of teams; 12 teams with 15 players (5 six-year-olds, 8 seven-year-olds, 2 eight year olds) on each team 51. 180 53. every fifth day 57. sometimes; Sample answer: the LCM of 2, 4, and 8 is 8, which is one of the numbers. The LCM of 2, 3, and 4 is 12, which is not one of the numbers. 59. always; Sample answer: the GCF of 6 and 12 is 6 and the LCM is 12. The GCF of 3 and 5 is 1 and the LCM is 15. In each case, LCM > GCF. 61. The LCM of two numbers is the least number into which both of the numbers will divide evenly. The LCM involves the product of prime factors. You can find the LCM of two or more numbers by following the procedure: Write the prime factorization of each number in exponential form. Multiply each factor the greatest number of times that it appears in either factorization. If the same factors appear in more than one number, multiply the greatest power of the factor 3 4 67. 7 69. _ 71. 4k that appears. 63. H 65. 3_ 11 4 73. 0 + 1 = 1 75. 1 + 2 = 3 Selected Answers

R33

Selected Answers

repeating decimals. 61. 0.5 = 0.50 and 0.5 = 0.55. . .; 0.5 is greater because in the hundredths place, 5 > 0. 63. G 65. 7.7 × 10-2 67. 9.25 × 105

Selected Answers

Pages 265–267 Lesson 5-7 19 5 2 1 1 2 _ _ _ 1. 3. 5. 14 7. -_ 9. _ 11. 4 _ 9 2 24 24 3 12 5 15 1 1 11 1 yd 15. _ 17. _ 19. _ 21. 4_ 23. -1_ 13. 20_ 8 8 28 12 12 26 19 5 1 11 27. -_ 29. 2_ 31. 10_ 33. Subtraction; 25. _ 6 40 15 15

the question asks how long the page is inside the margin given the length of the page and the lengths of 3 in. 35. Addition; the the top and bottom margins. 8_ 4 question asks for the sum of the average precipitation

8 5 in. 37. -14_ for August, September, and October. 6_ 25

3

6

5 1 +_ = • For example, the LCM of 4 and 6 is 12. So, _ 6 4 3 10 13 1 _ _ _ _ + = or 1 . 12

12

• Writing the prime factorization of the denominators is the first step in finding the LCM of the denominators, which is the LCD. Then the fractions can be added or subtracted. 4 7 45. H 47. 56 49. 6n3 51. 2_ 53. 14_ 55. 8.5 5 12 57. 2.7 59. 6 Pages 270–272

Lesson 5-8

13 1 1. 11.4 3. -2_ 5. 1_ 7. 1.4 9. -8.5 11. 12 30 12 13 31 1 _ 19. 1_ 13. x = 54; 324 15. -2.4 17. -_ 6 18 36 1 2 21. 12.24 23. -_ 25. 13_ 27. -5 29. 45 6 15 1 1 off 35. -1_ 31. -10 33. 24.99x = 16.66; _ 3

4

3 _ 1 x = 3_ ; 11c 37. x - 0.36 = 28.79; 29.15 in. 39. 2_ 2

2

4

1 in. 43. Ling; dividing 0.3 by 3 does not isolate 41. 7_ 2

3 the variable on the left side. 45. _ 5 7 53. -5_ 51. -4_ 8

12

7

55. >

57. =

3

15 3 61. _ 8

Lesson 5-9

Pages 281-284

6

40

Chapter 5

2


1. rational 3. algebraic fraction 5. LCD 7. median 23 1 1 9. 0.875 11. 3.6 13. 8.12 15. _ 17. _ 19. -8_ 100

8

7 4 2 7 1 21. -1_ 23. 3_ 25. _ 27. 6_ 29. 7_ lb 9 5 11 10 12


1 39. 5_ mi 2

4

6

6

12

1 57. 12.4; 8; none 59. Mean; there 53. 3.25 55. -2_ 2 are no extreme values or numbers that are identical; 135.5 min.

Chapter 6 Numbers Patterns and Functions Chapter 6

Get Ready

5

2

23. 1.5 25. 129 Pages 294–296 Lesson 6-1 5 2 1. _ 3. _ 5. $45.75/ticket 7. 6.9 feet/second 18 5

9. Alicia ran 16 ft in 2 s or 8 ft/s, Jermaine ran 12 ft in 2 2 7 4 2 s or 6 ft/s 11. 1760 13. _ 15. _ 17. _ 19. _ 9 5 18 1 21. $1.25/can 23. 23.3 mi/gal 25. 7.6 yd/min 27. 52.6 mi/day 29. Since the 12-issue subscription costs $2.80 per issue, and the 18-issue subscription costs $2.25 per issue, the 12-issue subscription costs more per issue. 31. 66 33. 15.6 35. 37.5 37. 60 3 5 31 41. _ 43. _ 45. about 579 mi/h 39. _ 5 41 15 47. California; The population density of California is 230 people/sq mi. The population density of all the other states are less than 230 people/sq mi. 49. Number sense; there are no numbers with which to calculate so mental math and estimation cannot be used. Since the cafeteria workers cut the pies into less slices today than they did yesterday means that each slice was larger today. Therefore, Ishi received more apple pie today than yesterday. 51. B 53. A 1 55. -_ 12

1 57. 13 _ 3

59. 4.224 × 104

61. 2.1 × 10-4

Lesson 6-2

1. no 3. c = 2.49g; $46.07 5. no 7. yes 9. no 11. p = $18l; $126 13. no

1. 12.4; 8; none 3. 3.6; 3.5; 4 5. Mean; the data has no extreme values. 54.9 7. 40; 41; 43 9. 9; 8.5; none 11. 18.3; 18; 16, 18, and 20 13. Mean; there are no extreme values. 788.8 15. Mean; the height of the tallest basketball player increases the mean height of the team. 17. 4.5 19. Sample answer: The median home price would be useful because it is not affected by the cost of the very expensive homes. The cost of half the homes in the county would be greater than the median cost and half would be less. 21. C 23. C 5 23 1 1 27. _ 29. 9_ 31. 3_ yd 33. 6.25 × 106 25. -8_ 4

1 37. -3_

5 1 1 49. 2_ 51. 6_ c 45. = 47. 1_

Pages 298–300

63. 0 65. -38 Pages 277–279

11

63. 9 65. 48

14 47. 8 49. _ 2 59. 8_

41. 120 43.
Þ iÜÃ«>«iÀÃ /ViÌÃ -` Ã®

1.

Lesson 7-8

3. Sample answer: Using (6, 502.5) and (4, 479), y = 11.75x + 432.

Óä £{ än äÓ nÈ ä

15. Yes; each year is only paired with one value for the price of gas. 17. y

y x 4

¼Ç ¼n ¼ ¼ää ¼ä£ ¼äÓ ¼äÎ 9i>À

x

£Èää £xää £{ää £Îää £Óää ££ää ä

y

19.

y x x ¼Ç ¼n ¼ ¼ää ¼ä£ ¼äÓ ¼äÎ ¼ä{ 9i>À

7.

y Barometric Pressure (in. mercury)

Selected Answers

Pages 405–407

35

21. Sample answer: (0, 4) means she can buy 0 regular smoothies and 4 large smoothies with $12; (6, 1) means she can buy 6 regular smoothies and 1 large smoothie -1 with $12. 23. dec. of 2°F/h 25. $8.25 27. _ 4 Y 29.

30 25 20 15 10 5 0

Y X 10 20 30 40 50 60 70 Altitude (1000s ft)

x

9. No, the equation gives a negative value for barometric pressure, which is not possible. Also, the data in the scatter plot do not appear to be linear. 11. Sample answer: 238 in. 13. As latitude increases, temperature decreases. 15. Sample answer: 64°F 17. Sample answer: The slopes are the same because the rate of change in the percent of schools is the same per year whether you use the year or the number of years since 1995. The y-intercepts are different, depending on which scale you use. 19. D 21. y = -2x + 2 23. y = -4 y 25.

X

"

31.

y

x

y 1x 2 3

y x 3

33. slope = -100, the descent in ft per min; y-intercept = 400, initial altitude 35. y = 6x – 3 37. y = 3x + 2 39. $220 41. Sample answer: 3800 43. $240,000

x

16 59 4 27. $6,600,000 29. 30 31. _ 33. _ 35. _ 200 25 45

Pages 408–412

Chapter 7


1. function 3. line of fit 5. linear equation 7. vertical line test 9. constant of variation 11. Yes; each x-value is paired with only one y-value. 13. Yes; each x-value is paired with only one y-value.


Chapter 8 Equations and Inequalities Page 417

Chapter 8

Get Ready

1. 4 3. 24 5. -7 7. $75 9. -8 11. 18 13. -23 15. 6 17. -48 19. 150 21. -2 23. 4 25. -36°F Pages 422–423

Lesson 8-1

1. -8 3. 4 5. 3 7. 1.84 9. 75 miles 11. -7 15. 0.875 17. -6 19. -2 21. 3y – 14 = y; 7

3 13. _ 2

23. 10 songs 25. 4.2 27. 0.3 29. 3.4 31. 72 33. 29 1 35. 144 s 37. B 39. A 41. y = -_ x + 4 43. 2.4c + 28 Pages 426–428

Lesson 8-2

i`> *ÀVi ÌÕÃ>`Ã®

1. 5 3. 11 5. 4.5 7. = 7 ft; w = 3 ft 9. 11. all numbers 13. 13.2 15. 18 17. 3 19. 35 21. 23. all numbers 25. Kim: 6; Lynn: 11; Camilla: 33 27. 70 yd by 150 yd 29. -0.5 31. 0 33. -0.775 35. triangle: 7, 8, 9; rectangle, 4, 8; perimeter: 24 37. Sample answer: 3x + 2 = 3x + 1 has no solution; 2x + 4 = 2(x + 2) is an identity. 39. draw a model; There’s a lot of information presented in this problem. A model will help you keep all of that information straight and allow you to see information without having to reread the problem.; w = 55 feet, = 95 feet, s = 100 feet. 41. Sample answer: Many equations include grouping symbols. You must use the Distributive Property to correctly solve the equation. Answers should include the following. • The Distributive Property states that a(b + c) = ab + ac. • You use the Distributive Property to remove the grouping symbols when you are solving equations. 43. H 45. 0.4 47. ; $205,000 fÓ£x f£ä

Lesson 8-4

1. x < 5 3. b ≤ -22 5. c ≤ 1 7. h > 0 4 32 1 0 1 2 3 4

9. x ≤ 1.25 ä°Óx ä ä°Óx ä°x ä°Çx £ £°Óx £°x £°Çx Ó Ó°Óx

11. $37.50 13. t > -9 15. k < 27 17. n > 37 19. a < 19 21. h < -7 23. d ≥ -25 25. t > 5 1 2 3 4 5 6 7 8 9

27. z > 5 1 2 3 4 5 6 7 8 9

f£Èx

29. a ≤ -27

f£{ä

30 28 26 24 22

2 31. s ≤ 4_

f££x fä

3

£ä £x Óäää Óääx Óä£ä 9i>À

49. -3.7 × 104 51. -2.03 × 10-3 53. -4 Pages 432–434

Pages 437–439

Lesson 8-3

1. x ≤ 45 3. f > 8000 5. false 7. x È Ç n £ä ££ £Ó £Î £{ £x

9. £ä n Ç È x { Î Ó £ ä

3

4

5

33. 980 lb 35. 42 + x ≥ 74; x ≥ 32; at least 32 mph 37. 74 + y ≥ 110; y ≥ 36; at least 36 mph 39. c > -4.1 5 41. z > -1.8 43. g > 3_ 45. Sample answer: x + 6 < 6 28 47. Always; subtracting x gives -1 < 0, which is always true. 49. C 51. false 53. false 55. 8 + 32 57. 5x – 17.5 59. -2 61. 18 Pages 444–445

1. x < 4

Lesson 8-5 21 3. x < 1 5. y ≥ _ 20

7. C

9. z ≥ 3 1 2 3 4 5 6 7 8 9

11. x ≥ -20 13. m ≤ 60 15. s < 500 17. j ≤ 6.8 19. true 21. false 23. true 25. 3 4 5 6 7 8 9 10 11

11. a ≥ -2 13. d > 12 15. p > 30 17. m < 11 19. 24 + s < 100; less than $76 21. w ≤ 2

27. 2 3 4 5 6 7 8 9 10

4 3 2 1 0 1 2 3 4

23. t ≥ -16 20 18 16 14 12

29. 2 1 0 1 2 3 4 5 6

31. 6 7 8 9 10 11 12 13 14

33. 6 54 32 1 0 1 2

25. y ≤ 9.6

27. x > -42

50 46 42 38 34

5

37. x ≤ -3 39. x > 13 43. Sample answer: x < 9 means that the value of x is less than 9; x > 9 means that the value of x is greater than 9; x ≤ 9 means that the value of x is 9 or less; x ≥ 9 means that the value of x is 9 or greater. 45. Inequalities describe numbers

29. 40m ≥ 2000; at least 50 min 31. y > 5.7

35. 6 5 4 32 1 0 1 2

5.2

5.4

5.6

5.8

33. b < 72 Èx ÈÈ ÈÇ Èn È Çä Ç£ ÇÓ ÇÎ Ç{ Çx

35. k > -18 20 18

16 14

12

37. m ≥ -8.4 9

8.6 8 2 7.8 7.4

Selected Answers

R41

Selected Answers

2

that are greater than or less than a given number. Answers should include the following. • Sample answer: My brother’s age is less than () from school. I drink at least 2 (≥) glasses of milk each day. • Sample answer: My brother is younger than I am. The other students are my age or older. I live farther than 2 miles from school. I drink 2 or more glasses of milk each day. 15 1 55. 5_ 47. F 49. 5 51. 4n – 6 = 3n + 2; 8 53. _ 2 4 57. -10 59. 18 61. 1

Pages 448–450

2 43. a ≥ -2_

5 4

5 6 7 8 9 10 11 12 13

2

4

6

4

Page 461

3. n > 2

Chapter 9

Get Ready

1. < 3. < 5. > 7. < 9. 0.601, 0.594, 0.546, 0.523, 0.509 11. 7 13. 9 15. 12 17. 13 19. 8 21. 34 23. 97 25. 128

1 2 3 4 5 6 7 8 9

5. c > 1 4 32 1 0 1 2 3 4

7. b < -5

Pages 466–468

7 6 5 4 3 2 1 0 1

Lesson 9-1

43 2 1 0 1 2 3 4

1. 5 3. -6, 6 5. -5.7 7. -10 9. 4 11. -1 13. -10, 10 15. 5.5 17. -6.6 19. 13.4 21. -0.9, 0.9 23. 9 25. -7 27. -14, 14 29. 12 mi; 14.8 mi 31. Kinda Ka, Six Flags Great Adventure in Jackson, 64 NJ: 456 ft tall; 26.1 mi 33. 9; Since 64 < 65 < 81, 兹 65 < 兹 81. Thus, it follows that 8 < 兹 65 < 9. So, 9 < 兹 65. 35. Sample answer: 兹 300 is greater than 兹 37. 10.4 in.; 41.6 in. 39. 8.4 m; 33.6 m 41. 70.7 43. 180 m; 182.4 m 45. Sample answer: A number that has a rational square root will have an ending digit of 0, 1, 4, 5, 6, or 9. The last digit is the ending digit in one of the squares from 1-100. There are just six ending digits. 47. 100 49. a 51. C 53. y > 6 55. k ≥ -9 57. t ≥ 15 59. positive slope 61. 390 mi 63. It can 3 8 . 65. It can be written as _ . 67. It can be written as _ 27 9 be written as -_.

1 2 3 4 5 6 7 8 9

Pages 472–474

9. 700 clicks 11. q ≤ 4 1 2 3 4 5 6 7 8 9

13. t ≤ -1 43 2 1 0 1 2 3 4

15. p ≤ 3 4 3 2 1 0 1 2 3 4

17. n < -1 4 3 2 1 0 1 2 3 4

19. a < -4 10

8

6

4

2

21. d > 7 1 2 3 4 5 6 7 8 9

23. 3(x - 1) ≥ 18; at least 7h 25. c > -3

1

27. a ≤ 7 29. m ≥ -8 10

8

6

4

2

31. n < 44 33. at least 3 hours and 45 minutes 35. at least 340 37. more than 93 minutes 39. Sample answer: 10y - 3 ≤ -12 41. Jerome is correct. By the Distributive Property 2(2y + 3) = 4y + 6 not 4y + 3. 43. D 45. x < -4.5 47. q < 24 49. k > 29 51. z ≤ -4 53. 60% 55. $2.50 a loaf 57. 6 mps 59. 6.4 ft Pages 451–454

Chapter 8


1. true 3. false; identity 5. false; inequality 7. false; greater than or equal to 9. -2 11. 2 13. -1.2 15. 5 17. 9 19. 16 21. all numbers 23. true 25. false 27. true 29. ≥ 3 31. b ≥ 17 £{

£È

£n

Óä

ÓÓ

Î

Ó{

ä

Ó°È Ó Ó £°n £°{ £

Ó

Î

{

£ x

37. 920 + n ≤ 1800; The elevator can carry at most an additional 880 pounds. 39. k ≤ 5.1

Lesson 9-2

− 10 _ 1. N, W, Z, Q 3. Q 5. < 7. _ , 3 3 , 兹 13, 3.7 3 5 9. 5, -5 11. 13.5 ft 13. Q 15. N, W, Z, Q 17. Q 19. Q 21. Z, Q 23. Z, Q 25. < 27. > 29. = −− 2 1 17 31. 兹 16, 4.23, 兹 18, 4_ 33. -兹 14, -3.8, -4_ , -_ 3 10 4 35. 17.3, -17.3 37. 0.1, -0.1 39. 9, -9 41. 1.3, -1.3 43. 3.1 s 45. never 47. sometimes 49. Sample answer: 4 and 兹 49 51. 5.6 or -5.6 53. Sample answer: 兹 The length of the skid marks show that the car was traveling at least 49 mph before the driver applied the brake. The car could have slid further if it wasn’t for the collision. Therefore, the skid marks would have been longer and thus, the car would have been traveling at a higher rate than 55 mph. 55. Sample 25 57. 兹 50 is irrational, the other numbers answer: 兹 are rational 59. B 61. D 63. -11 65. -4 67. i>Ì ìÝ >Ì Çxc

33. x ≤ -2.8 2

0

Chapter 9 Real Numbers and Right Triangles

Lesson 8–6

1. x ≤ 9

1 35. t < 3 _

2

45a. 8h ≥ 1200; 45b. h ≥ 150; 45c. at least 150 h 1 47. n ≤ 4 49. t > -12 51. b > -4_

i>Ì ìÝ ®

Selected Answers

39. 13 41. Inequalities can be used to compare the weights of objects on different planets. Comparing the weight of two astronauts on Mars to the same astronauts on the Moon: 113 > 50 and 106 > 46.9 3 1 43. 40 45. x > 27 47. n < 16 49. _ 51. -2_ 32 12 53. 3 55. -6 57. 20

ÇÇ ÇÈ Çx Ç{ ÇÎ ÇÓ Ç£ Çä È

Y

X

£ ä £ Ó Î { x È Ç n

ä¯ £ä¯ Óä¯ Îä¯ {ä¯ xä¯

41. y ≤ -7 £Ó £ä


n

È

{

,i>ÌÛi Õ`ÌÞ

69. $0.53/cupcake 71. 0.6 ft/h 73. 105 75. 71 77. 20 Lesson 9-3

1. 25; acute 3. 105; obtuse 5. acute 7. right isosceles 9. acute isosceles 11. 55; acute 13. 102; obtuse 15. 66; acute 17. 10°, 10°, 160° 19. acute 21. acute 23. obtuse 25. obtuse 27. obtuse 29. acute equilateral 31. right scalene 33. acute scalene 35. 89˚

39˚

closer to the cup. 27. B 29. 9.2 31. 35.1 33. 47.6 35. 28 37. 21 Pages 500–502

Lesson 9-6

31.5 ft 126 ft 1. 10 3. 6 km 5. 8 7. 9 9. 5 11. _ =_ ; 76 ft 19 ft

IN 52˚

9

37. not possible 39. obtuse 41. acute 43. 20°; 60°; 100° 45. 30°; 45°; 105° 47. 10, 15, 21 49. The angles of a triangle have a sum of 180°. The angles of a triangle can be folded to form a straight angle. Since the measure of a straight angle is 180°, the sum of the angles of a triangle is 180°. Sample answer:

:

IN

21. The scale factors of both the side lengths and the 3 perimeters are _ . 23. The scale factor of the side

2

2

3 3 2 and the scale factor of the areas is _ lengths is _ or _9 . 2

25. Sample answer:

120˚

61˚

! IN

IN

51. G 53. 14, -14 55. 10.1, -10.1 57. -4 59. 12 61. 0.47 million or 470,000 metric tons 63. 144 65. 324 67. 576 Lesson 9-4

1. 25 3. 15 5. 37.7 7. yes 9. 26 11. 50 13. 18.9 15. 12 17. 12.1 19. 22.2 21. yes 23. no 25. no 50 27. about 23 in. 29. 19.8 31. 37.3 33. 28 35. 兹 41 39. 933 ft 41. 32 in. 43. Marcus; Allyson 37. 兹 incorrectly substitutes 15 for b in the equation. Since the side measuring 15 units is opposite the right angle, it is the hypotenuse. So, 15 should be substituted for c in the equation. 45. The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Answers should include the following. hypotenuse leg

IN

"

IN

;

#

3 XY YZ XZ 2 4 _ =_ =_ =_ =_ =_ 27. No; triangles ABC 9 6 12 AB BC AC

and DEF are not similar since corresponding sides are not proportional. 29. C 31. 3.6 33. 7.1 35. 33.0 37. 29 ft

Pages 503–506

Chapter 9


1. obtuse 3. hypotenuse 5. perfect square 7. Congruent 9. legs 11. 10 13. ±11 15. -15 17. 3.14 s 19. 6.2, -6.2 21. 2.7, -2.7 23. $2775 25. obtuse isosceles 27. obtuse scalene 29. 13.7 LM MN 31. 15.9 33. 2034 ft 35. 9.4 37. 5 39. _ =_ ; TU UV 7 m 41. 36 in. or 3 ft

Chapter 10 Two-Dimensional Figures Page 511

Chapter 10

Getting Ready

1. 44 3. 51 5. 39 7. 30 mi 9. 25.5 11. 10 23 1 7 2 13. 19.5 15. 10_ 17. 3_ 19. 6_ 21. 20_ lb 6

leg

• An example of a set of numbers that represents the measures of the lengths of the legs and hypotenuse of a right triangle is 15, 20, 25. 47. 84.9 49. 90; right 51. Z, Q 53. I 55. 98 57. 5 Pages 494–496

IN

9 IN :

35˚

29˚

4

8

25˚

Pages 488–490

x

13. 68 yd 15. Always; corresponding angles in similar figures are congruent. 17. 27 in. 19. 8

Lesson 9-5

1. 12.7 3. 18.6 5. 9.2 7. 10.2 9. 16.6 11. 19.2 −−− −−− 13. 7.3 ft 15. Yes; PM and MN have equal measures. 17. 8.8 19. 4.6 21. 7.3 in. 23. (1, 3) and (5, 6) 25. Mental math; The cup is located at (0, 0) on a coordinate system with points (-2, -3) and (1, 4); you can use mental math to find the distance between the (-2)2 + (-3)2 = 兹 13; cup and the two balls. 兹

Pages 515–517

8

9

24

Lesson 10-1

1. 56° 3. 124° 5. 28 7. 108°; 72° 9. 127° 11. 53° 13. 127° 15. 148 17. 175 19. 9 21. 41° 23. 73° 25. 18 27. 55° 29. Sample answer: 6:15, 20 s 31. Sample answer: Both graphs intersect to form right angles. 33. Sample answer: The slopes of the graphs of perpendicular lines are negative reciprocals of each other unless the perpendicular lines are horizontal and vertical. 35. They are supplementary. 37. D 39. 6 ft 41. m < 1 43. x ≤ 4 45.

45˚

Selected Answers

R43

Selected Answers

Pages 479–481

(1)2 + (4)2 = 兹 17; Since 兹 13 < 兹 17, Joan’s ball is 兹

Selected Answers

47.

9. R’(-4, -3), S’(0, -3), T’(-4, 1) 145˚

Pages 521–523

y

R

S T'

Lesson 10-2

−−− −− −− −− 1. ∠J ∠C, ∠K ∠B, ∠M ∠G, KM BG, MJ GC, −− −− KJ BC, BGC 3. yes; ∠A ∠D, ∠B ∠E, ∠C ∠F, −− −− −− −− −− −− AB DE, BC EF, AC DF; ABC DEF −− 5. 15 ft 7. ∠K ∠N, ∠J ∠P, ∠KMJ ∠NMP, KJ −− −− −−− −−− −−− NP, JM PM, KM NM, NPM 9. ∠Z ∠S, ∠W −− −−− −− −− −− −−− ∠T, ∠Y ∠R, ZW, ST, WY TR, ZY SR, STR 11. no 13. yes; ∠J ∠K, ∠JHT ∠KTH, −− −−− −− −− −− −− ∠JTH ∠KHT, JT KH, JH KT, TH HT; JTH KHT 15. 30° 17. 20 19. false; for ABC and DEF, AB = 2 ft, BC = 3 ft. CA = 4 ft, DE = 3 ft, EF = 3 ft, and FD = 3 ft; perimeters of ABC and DEF are both 9 ft but ABC DEF. 23. ABH IJG; ACG IEF 25. D 27. 51° 29. Sample answer: triangle rate of change = 3; square rate of change = 4; The perimeter of a square increases at a faster rate than the perimeter of a triangle. A steeper slope on the graph indicates a greater rate of change for the square. 31–36. y H

x

T S'

R'

11. A’(-1, 3), B’(-4, 4), C’(-4, 1), D’(-2, -1)

B'

B

y

A' A

C

C' D

D'

x

A J

D

O

13. P’(4, 4), Q’(8, 12), R’(12, 4);

x

Y

1g

W

Pages 527–530

Lesson 10-3 1

1. B 3. reflection 5. A’(-2, 5), B’(1, 3), C’(-2, 0)

A'

y

A

B'

0g

0

2g

2 X

"

C'

B

x

C 15. X’(1.5, 1.5), Y’(2, 2.5), Z’(2.5, 0.5)

) (

(

) (

) (

Y

)

7. D’ 1, 3 1 , E’ 3, 4 1 , F’ 3, -1 , G’ 1, -11 2 2 2 2 y

D

G

E

F

D'

G'

9

E' 8

F'


x

9g 8g

: :g

"

X

17. X’(2, 2), Y’(4, -2), Z’(1, 1) Y

50

: X

"

Pages 541–543 9g 9

19. 0.0192 mm 21. Reflection; the image is flipped over the line of reflection (mirror). 23. 6 yd 25. translation 27. Sample answer: Y

Lesson 10-5

1. pentagon; not regular 3. square, hexagon, 12-gon 5. 128.6° 7. heptagon; not regular 9. pentagon; regular 11. decagon; not regular 13. 1080° 15. 720° 17. 3780° 19. 108° 21. 144° 23. 165.6° 25. No, it is not a polygon since it has curved sides. 27. hexagons, triangles, squares 29. 180 cm 31.

!

!g

" #

#g "

"g X

33. 120° 35. In tessellations, polygons are fit together to create a pattern such that there are no gaps or spaces. Answers should include the following. •

29. G(-3, -4) G’(-3, 4) is a reflection over the x-axis; to reflect over the y-axis, multiply the x-coordinate by -1. 31. The image is in Quadrant III. It has also been −− turned upside down. 33. D 35. DF 37. ∠F 39. 44.9 mi 41. 113 43. 47 • Pages 534–536

Lesson 10-4

1. 110; 110° 3. rectangle 5. trapezoids, parallelograms 7. 102; 102° 9. 60; 60°; 60°; 120° 11. 70; 70°; 80°; 120° 13. square 15. parallelogram 17. rhombus 19. Sample answer: A chessboard; it is a square because it is a parallelogram with 4 congruent sides and 4 right angles. 21. always 23. sometimes 25. See students’ drawings; quadrilateral 27. Sample answer: The artist used triangles, quadrilaterals, and a few shapes having five and six sides. Some of the quadrilaterals are trapezoids. 29. Yes; a rectangle is equiangular but may not be equilateral.

37. G 39. quadrilateral 41. y

M M'

31. Many designs contain patterns formed by using shapes such as quadrilaterals. Answers should include the following. • Sample answer: Many quilt designs contain quadrilaterals. One such example is shown.

Pages 548–550

N'

N

43. 14.4 45. 28.52.

P

x

P'

Lesson 10-6

1. 8 ft2 3. 60 m2 5. 78 km2 7. 108,756 mi2 9. 11 m2 11. 21.6 in2 13. 212.8 ft2 15. 9 cm2 17. 14.4 ft2 19. 25.2 km2 21. about 51,113 mi2 23. 16 in. 25. 40 units2 27. 147 km2 29. 57 ft2 31. Sample answer: 3 in. 8 in.

Selected Answers

R45

Selected Answers

8g :g

• The quadrilaterals used are trapezoids, squares, and parallelograms. 23 33. 50 35. no 37. 70 ft 39. _ 41. 540 43. 1440

8

35. H 37. 144° 45. 50.2

Pages 554–556

39. 156°

41. 24; 24°; 96° 43. 13.5

Lesson 10-7

1. 12.6 in.; 12.6 in2 3. 31.4 mi; 78.5 mi2 5. 19.2 cm; 29.2 cm2 7. 18.8 cm; 28.3 cm2 9. 62.8 m; 314.2 m2 11. 79.8 m; 506.7 m2 13. 28.3 m; 63.6 m2 15. 24.5 ft; 47.8 ft2 17. 452.4 ft2 19. c 21. b 23. 8.2 in. 25. 2513 ft 27. 28.3 in2 29. The area of the new circle graph is 9 times greater than the area of the original circle graph. The diameter was increased by a factor of 3; the area was increase by a factor of 32 or 9. 31. 51.4 ft; 157.1 ft2 Y 33.

ÀVÕviÀiVi

Selected Answers

33. The area of a parallelogram is found by multiplying the base and the height of the parallelogram. The area of a rectangle is found by multiplying the length and the width of the rectangle. Since in a parallelogram, the base is the length of the parallelogram, and the height is the width of the parallelogram, both areas are found by multiplying the length and the width. Answers should include the following. • Parallelograms and rectangles are similar in that they are quadrilaterals with opposite sides parallel and opposite sides congruent. They are different in that rectangles always have 4 right angles.

X

41. J 43. 48 km2 45. 360° 47. x < -7 49. 6:00 P M. Tuesday 51. 150.5 53. 774.86

Pages 560–562

Lesson 10-8

1. 49.5 yd2 3. 2 5. 72 ft2 7. 56.1 cm2 9. 18.3 in2 11. 45 ft2 13. 86.8 yd2 15. 144 bags 17. Sample answer: 62,500 mi2 19. 167.9 units2 21. 331.1 units2 23.

; Break the first figure into two rectangles and find the area of both rectangles. In the second figure, make a trapezoid and a square. Then find the area of each figure. 25. You can use polygons to find the area of a composite figure by finding the area of each individual polygon and then finding the total area of the composite figure. Answers should include the following. •

• To find the area of the composite figure shown above, the figure can be separated into two rectangles and a triangle. 27. D 29. 44.0 cm, 153.9 cm2 31. 27 in2 33. 37 min 30 s ,

>iÌiÀ

; Since the formula for the circumference of a circle, C = d, is in the form of y = mx, is the slope. 35. Sample answer: 1.5

37. 50 yd2 39. The circumference of a circle is about 3 times its diameter. Answers should include the following. • Since the circumference is about 3 times the diameter, the ratio describing the relationship would be about 3 to 1. • As the diameter increases, the circumference increases. As the diameter decreases, the circumference decreases.


Pages 564–568

Chapter 10


1. adjacent 3. corresponding angles 5. parallel 7. altitude 9. 109° 11. 71° 13. 71° 15. ∠F ∠L, −− −−− −−− −−− −− −− ∠G ∠M, ∠H ∠N; FG LM, GH MN, HF NL; FGH LMN Y 17. # #g X "

$

'

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&g

19. Y

19. £ { Î £

1 0g 1g

.

21. 7.5 in. by 10.5 in. 23. 70; 70°; 70° 25. hexagon; 720° 27. decagon; 1440° 29. 135° 31. 11 yd2 33. 8 m2 35. 31.4 cm; 78.5 cm2 37. 56.5 ft; 254.5 ft2 39. about 11,781 ft 41. 863.4 cm2

Chapter 11 Three-Dimensional Figures Chapter 11

Get Ready

1. yes; triangle 3. pentagon 5. 102.4 7. 4 3 1 2 4 =_ =_ =_ 9. yes 11. yes 13. yes; _ 1.25

Pages 578–581

60

49. 6.5 cm2 X

"

2

33 41 45. chocolate; _ >_

4

.g

Page 573

6

3 43. y < 1_

2.5

3.75

5

Pages 586–588

50

47. 14 in2

Lesson 11-2

cm3

1. 183.6 3. 1608.5 ft3 5. 700 cm3 7. at least 198.9 ft 9. 512 in3 11. 748 cm3 13. 88.0 ft3 15. 225 mm3 17. 18.6 m3 19. 6.2 m 21. 972 m3 23. 300.8 yd3 25. 37.5 m3 27. 1000 29. 240 g 31. 11.5 in. 33. Marissa; a cube has three dimensions and when each dimension is multiplied by 2, the volume is multiplied by 2 × 2 × 2 or 8. 35. Sample answer: The area of a prism involves multiplying the length times the width. The volume V of a solid involves multiplying the area of the base B times the height of the solid h. Therefore, V = Bh where B = · w. 37. 13.4 39. D 41. 12.9 yd2 43. x > -8 45. y ≤ -3.2 47. 12 49. 154

Lesson 11-1

1. triangular pyramid; any one of the following faces −−− can be considered a base: RST, QRS, QST, QRT; QR, −− −− −− −− −− QS, QT, RT, RS, ST; Q, R, S, T 3. cone; base: circle Y 5. Ó Ó { { Ó Ó { { Ó Ó { { Ó Ó { {

7. rectangular prism; LMNP, QRST or LPTQ, MNSR or PNST, LMRQ; LMNP, QRST, LPTQ, MNSR, PNST, −−− −−− −− −− −− −−− −− −− −−− −− −− LMRQ; LM, MN, NP, PL, LQ, MR, NS, PT, QR, RS, ST, −− TQ; L, M, N, P, Q, R, S, T 9. cylinder; circles M and N 11. rectangular pyramid; BCDE; ABC, ACD, ADE, ABE, −− −− −−− −− −− −−− −− −− BCDE; AB, AC, AD, AE, BC, CD, DE, EB; A, B, C, D, E 13. Top Side Front

Pages 592–594

Lesson 11-3

1. 100 m3 3. 160 in3 5. 378 ft3 7. 523.6 in3 9. 32 ft3 11. 124.5 mm3 13. 628.3 in3 15. 268.1 mi3 17. 4188.8 ft3 19. 270.8 cm3 21. 565.5 m3 23. 3.1 mm3 25. about 75 min or 1 h 15 min 27. 33,614.6 lb 29. 44.0 m3 31. 1072.3 in3 33. 4 in. 35. Sample answer: 8 cm

4 cm

37. The volume doubles. 39. The volume increases 4 3 by a factor of 8. Since the volume of a sphere is _ ␲r , 3 the volume of a sphere whose radius doubles is

_4 ␲(2r)3 or 8 _4 ␲r3 . 41. B 43. 64 cm3 45. triangular 3 3

(

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17.

x x x x x

x x x x x

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top

)

pyramid; any one of the following faces is a base: DCG, −−− −−− −− −− −− −− DGF, DFC, CGF; DC, DG, DF, CG, GF, FC; C, D, F, G 7 47. (-3, 2), (-1, 4), (0, 5), (1, 6) 49. _ 51. Sample 8 answer: 75 53. Sample answer: 42

15. 30

È È È È È

Pages 600–601

Lesson 11-4

1. 100 ft2; 150 ft2 side

front

3. 263.9 in2; 571.8 in2

Selected Answers

R47

Selected Answers

21. Sample answer: using angles and shading to portray depth 23. never; The bases of a cylinder have the same radii and are congruent. 25. sometimes; Three planes may intersect in a line. Or, the planes may be parallel and −−− not intersect at all. 29. WR 31. skew 33. B 6 x 35. 30 cm2 37. _ = _; 18 39. c < 8 41. n > -11

0

Selected Answers

5. 40.2 mm2; 140.7 mm2

5. 208 m2

7. 114 in2; 282 in2 7. 110.4 ft2 9. 216 m2; 264 m2

11. 1256.6 ft2; 1885.0 ft2

13. 196 ft2; 294 ft2

15. 471.2 in2; 628.3 in2

9. 235.6 cm2 11. 506.6 mm2 13. 85.3 in2 15. 614,992 ft2 17. 578 ft2; 6 squares 19. 471.2 cm2 21. 275 in2 23. Sinkers: about 251 in2, original bar: 82 in2; the surface area is about tripled. The volume of the sinkers is about 26.2 in3 and the volume of the bar is exactly 26 in3. The volumes are about the same. 25. 480 in.2; 736 in.2 27. Surface area is used in covering building exteriors and in designing interiors. Answers should include the following. • Contractors and architects use surface area when designing buildings. It is important to know surface areas so the amounts and costs of building materials can be estimated because many building materials are priced and purchased by square footage. 29. F 31. 447.4 cm2 33. 50° 35. $9 37. 3 39. 2.88

Pages 611–613

17. 2 19. 10 ft2 21. 397.3 in2 23. Sample answer: find the surface area of a cedar chest and a cylindershaped potato chip container. 25. Sample answer: The can contains less surface area than the box so it would cost less to package juice in a can if the amount of liquid is the same. Other factors include the additional costs associated with manufacturing, packaging, and distributing the juice. For example, the cost of the packaging (paper or aluminum) would also have to be considered and how it would affect the cost. 27. F 29. 70 ft3 31. 82.4 33. 678.96 1 35. 1.3 37. 3_ 4

Pages 604–606

1. 66.4

Lesson 11-5

ft2

21. always; same shape, lengths are proportional 23. never; different shapes 25. 887.8 m2 27. Fred; Cassandra did not convert 5 meters to centimeters. 29. The surface area quadruples. 31. If the ratios of corresponding linear dimensions are equal, the solids are similar. Answers should include the following. • For example, if the ratio comparing the heights of two cylinders equals the ratio comparing their radii, then the cylinders are similar. • A cone and a prism are not similar. 33. G 35. 184 ft2 37. 14.8 m2 39. 65° 41. 18, -18 43. 1.5

Pages 615–618

3. 282.7 cm2


Lesson 11-6

1 1 1. no 3. x = 33_ ft 5. 450 ft2 7. _ 9. no 11. no 3 150 200 13. x = 5.1 ft 15. 4800 mm2 17. _ 19. 34,686.7 cm3 1

Chapter 11


1. false; lateral area 3. false; volumes 5. true 7. true 9. rectangular prism; QRST, UVWX or QTXU, RSWV or QRVU, TSWX; QRST, UVWX, QTXU, RSWV, −−− −− −− −− −−− −−− −−− −−− −−− QRVU, TSWX; QR, RS, ST, TQ, UV, VW, WX, XU, QU, −− −−− −− TX, SW, RV; Q, R, S, T, U, V, W, X 11. 217.9 m3 13. 192.5 cm3 15. 18.9 in3 17. 12.8 m3 19. 469.5 mm2; 548.7 mm2 21. 78.5 ft2 23. 1411.8 cm2 25. 308.8 cm2 27. x = 36 in. 29. 10 ft2

should include the following. A presidential candidate might use the display to determine the importance of each state when campaigning. Stem

Page 625

Chapter 12

Get Ready 1 2 3. 5000, 4000, 3000 5. 2_ 7. 8_ 2 5 5

0 1 2 3 4 5

1. 16; 15; none 1 1 11. _ 13. _ 9. _ 6

6

8

Pages 628–631

1. Stem

Leaf

0

67

1

255

2

0

3

5

4

01

9. Stem

Lesson 12-1

2|0 = 20

Leaf

3. 50%, 99% 5. Sample answer: The lowest score was 50%. The highest score was 99%. Most of the scores were in the 70–79% interval. 7. Chicken; whereas chicken sandwiches have 8–20 grams of fat, burgers have 10–36 grams of fat. 11. Stem Leaf

1

8

0

5555668

2

3

1

01468

3

156

2

02

4

479

5

01

0|6 = 6

2|3 = 23 wins

13. 73 15. 49 17. 28 19. 12 23. never 25. Stem Leaf 43 44 45 46 47 48 49 50 51 52 53 54 55 • • 62 • • 76 77 78 79

5 0 0 4 0447 2 0 0 0

Leaf 333333334444455555666777788999 00001111235557 0117 14 5

2|1 = 21 electors

33. no 35. 351.9 mm2 37. 6 ft 39. 247% 41. 0.65% 43. 57.1% 45. 1.6% 47. 28 49. 1.1

Pages 635–637

Lesson 12-2

1. 56; 15; 42 3. Earth 5. Sample answer: Since the length of a day ranges 5832 - 10 or 5822 hours, the lengths of days for the planets vary greatly. 7. 24; 5.5; 25 9. $21; $6; $10 and $12 11. 29; 11.5; no outliers 13. Sample answer: July; the temperatures are more tightly clustered around the median. 15. Sample answer: The winners of the Cotton Bowl scored more points on average than the winners of the Rose Bowl. The number of points scored by the Cotton Bowl winners varies greater than the number of points scored by the Rose Bowl winners. The Rose Bowl data in the middle are more spread out than the Cotton Bowl data. 17. Sample answer: {8, 9, 13, 25, 26, 26, 26, 27, 28, 30, 35, 40} 19. {15, 18, 20, 20, 44, 60, 60, 64, 70, 70, 75, 79} 21. They allow us to see how the data are distributed. Answers should include the following. • 155; 39; 17; no outliers • The median speed was 155 miles per hour. The speeds varied by 39 miles per hour. The speeds of the cars in the middle of the data set are close in value. There are no outliers so the data is clustered around the median. 23. G 25. no 27. 461.8 cm3 29. 1.6 ft 31. 4.3, 4.8, 4.9, 5.0, 5.3, 5.6

99 Pages 640–642 5

1.

4 9 2 45|0 = 450

27. Sample answer: It would be easier to find the median in a stem-and-leaf plot because the data are arranged in order from least to greatest. 29. Sample answer: It would be easier to find the mode in a stem-and-leaf plot because the value or values that occur most often are grouped together. 31. Stem-andleaf plots can help you understand an election by allowing you to see how the number of electors in the U.S. is distributed. Most of the states in the U.S. have 0 to 10 electors. However, there are some states that have 10–19, 20–29, 30–39, and 50–59 electors. Answers

Lesson 12-3

19 21 23 25 27 29 31 33 35 37 39 41 43

3. £xÈ £Èä £È{ £Èn £ÇÓ £ÇÈ £nä £n{ £nn £Ó £È Óää Óä{ Óän

5. Since SUVs average the least miles per gallon, they tend to be less fuel-efficient. 7. 35 40 45 50 55 60 65 70 75 80 85 90 95100

9. Óä Óx Îä

Îx {ä

{x xä xx Èä

Èx Çä Çx

nä nx

Selected Answers

R49

Selected Answers

Chapter 12 More Statistics and Probability

11.

Number of Players

17. 8 21. 28, 29, 30, 30, 31, 35, 38, 39, 41, 42, 42, 47, 48 23. A box-and-whisker plot would clearly display any upper and lower extreme temperatures and the median temperature. Displaying data in a box-andwhisker plot allows you to more easily see how the temperatures vary. 25. C 27. Stem Leaf 2

147

3

9

4 5

4 2 0

Number of Touchdowns

13. 13 15. 37.5% 17. Los Angeles £ä 19.

8 5 2|4 = 2.4

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29. 25

Pages 646–649

1.

n Ç È x { Î Ó £ ä

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Lesson 12-4

Pet Survey 22 20 18 16 14 12 10 8 6 4 2 0

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21. Since exact data values are not listed, you are not able to find any measure of central tendency. However, the median is located in the 41–50 minute time period by counting the number of students in each time period. 25. Sample answer: 10, 12, 15, 19, 20, 25, 26, 31, 31, 31, 33, 36, 50, 52, 55, 57, 58, 59; Ç È x

1–3

4–6 7–9 10–12 13–15 Number of Pets

3. the number of states that have a certain number of roller coasters; most states have less than 9 roller coasters 5. The numbers between 10 and 30 are omitted. 7. 2 or more national monuments 9. Weekly Study Time 12 11 10 9 8 7 6 5 4 3 2 1 0

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Number of Participants

6

1 18 5– 4 16 16 8– 14 7 14 1– 13 0 13 4– 11 3 1 –1 97 6 –9

4

8

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Touchdowns in a Season 10

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Number of Students

Selected Answers

11. 96 13. Sample answer: Based on the plot, the students’ overall scores are between 63 and 96. 15.

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27. D 29. 75% 31. 36 ft2 33. 21.6 35. 6.1 × 10-5, 0.0061, 6.1 × 10-2, 6100, 6.1 × 104 37. 19 39. 8 Pages 654–656

Lesson 12-5

1. line graph; shows change over a period of time 3. A 5. histogram; data that can be organized into equal age groups 7. Venn diagram; shows a relationship between the three groups of people 9. box-and-whisker plot; shows how the data are spread out 11. Sample answer: box-and-whisker plot 0–3

4–6

7–9 Time (h)


10–12 13–15 ä x £ä £x Óä Óx Îä Îx {ä {x xä xx Èä Èx Çä Çx nä nx ä

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histogram:

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• The stem-and-leaf plot displays the data in a succinct form. Also, the stem-and-leaf plot is the only graph of the three that displays each piece of data. The histogram groups the data by interval and displays the data the frequency of data in each interval. You can quickly see how the class generally performed on the exam. The circle graph is similar to the histogram in that it divides the data into the same intervals. However, the circle graph compares the data in each interval when compared to the whole. 21. >ÃiÌL> /i>

19. Each type of graph analyzes and displays the data in different ways. Certain graphs display the data in a way that other graphs can’t. Answers should include the following. • Exam Scores 71

65

68

80

78

79

77

95

84

75

79

93

95

85

79

86

91

86

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23. 62.8 ft; 314.2 ft2 Pages 661–663

stem-and-leaf plot:

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25. Brand D Lesson 12-6

1. Graph A has a break in the vertical scale. 3. No, the statement is not accurate. The vertical scale does not start at 0 so there could be between 1 and 25 sophomores who received a text message. 5. Graph A 7. The vertical scale of graph A goes from 0% to 10.0% whereas the vertical scale of graph B goes from 4.0% to 8.0%. In addition, the actual distance between vertical scales is less than the distance used in Graph B. 9. No, the statement is not accurate. There is a break in Selected Answers

R51

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Pages 667–669

1 1 1 2 1. _ ; 10% 3. _ ; 20% 5. _ ; 16_ %

1 7. _ or 20% 5 6 3 5 10 5 3 1 1 1 1 ; 37.5% 9. _; 12_% 11. _; 50% 13. _; 62_% 15. _ 8 2 2 8 2 8 7 2 _ _ 17. ; 87.5% 19. 0 21. 0.21; 21% 23. ; 22.2% 8 9 11 1 ; 61.1% 27. _ ; 33.3% 29. 1; 100% 25. _ 3 18 3 11 1 31. _ ; 61.1% 33. _ 35. _ 37. No; many of the 18 4 16

common two-letter words like at, in, of, and on do not contain the letter E. 39. D 41. Sample answer: Vertical scale that does not start at zero. 43. 12 or -12 45. 1890 47. 5643

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Pages 672–674

1.

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Lesson 12-8

Blue

Red

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15. The use of inconsistent scales, broken scales, expanded and shortened axes will make the graph misleading. Sample answer: • Cookie Sales

B

Y

R

BB

BY

BR

G

B

BG RB

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700 Amount Made (millions)

Lesson 12-7

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G

RY

RR

RG

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600

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LÕi

Book Survey 10 9 8

ÞiÜ

7 Frequency

Selected Answers

the vertical axis giving the impression that Dallas won 4 times as many games as Phoenix. However, Dallas won 41 games and Phoenix won 22 games; Dallas won less than twice as many games as Phoenix. 11. No, the statement is not accurate. The inconsistent horizontal scale causes the data to be misleading. The graph gives the impression that the year 1970–1971 was a down year and did not make as much money as in 1960–1961. However, the first bar represents 2 years (1960–1961 and 1961–1962). 13. >ÃV >Li -ÕLÃVÀLiÀÃ

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B2

B3

B4

B5

B6

Green 1

2

3

4

5

6

G1

G2

G3

G4

G5

G6

number of outcomes. Using the Fundamental Counting Principle is faster and uses less space; using a tree diagram shows what each outcome is. 31. Answers may include making a list pairing each deck choice with each wheel choice, making a tree diagram, multiplying the number of choices for deck times the number of choices for wheels. Answers should include the following. • The number of designs is the same as the product of the number of choices. • The number of skateboards would double. 33. F 35. Sample answer: The break is not shown on the y-axis.

Yellow 4

5

6

1 37. _ 7

Pages 678–680

1 39. -_

41. 10 43. 15

3

Lesson 12-9

1. P; 120 ways 3. C; 84 ways

Y1

Y2

Y3

Y4

Y5

Y6

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2

3

4

5

6

R1

R2

R3

R4

R5

R6

24 outcomes 11.

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91

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Lesson 12-10

3 10 1 1 1 1 1 1. _ 3. _ 5. _ 7. _ 9. _ 11. _ 13. _ 8 56 2 4 10 10 91 3 5 5 4 1 1 4 17. _ 19. _ 21. _ 23. _ 25. _ 27. _ 15. _

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39. y > -3 1 41. _

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9. C; 105 ways 11. P; 1320 ways 13. C; 364 ways 15. 36 handshakes 17. 12,870 ways 19. 3; 10; 35 1 21. _ 23. 10,000 25. Amanda; five CDs from a 10 collection of 30 is a combination because the order of the CDs is not important. 27. always 29. When order is not important, duplicate arrangements are not included in the number of arrangements. Answers should include the following. Order is important when you arrange things in a line; order is not important when you choose a group of things. 31. J 33. 28.6% 35. 11–20 37. a ≥ 6 Î

S1

23. 960 25. 60

512

1 29. Sample answer: Both methods find the 27. _ 20

2

3

S2

3 13. 16 15. 24 17. 32 19. _

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12 outcomes

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1

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M

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P2S1M P2S1S P2S2M P2S2S

7

7

8

4

9

364

29. 28% 31. Sample answer: 3 red, 2 white, and 4 blue 33. Shannon; rolling a difference of 0 or 1 are mutually exclusive events so the probability of rolling a difference of 0 must be added to the probability of rolling a difference of 1. 35. The probability of compound events is based on the probability of each simple event. Answers should include the following. Selected Answers

R53

Selected Answers

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£

Pages 690–694

Chapter 12


1. probability 3. range 5. simple event 7. measures of variation 9. theoretical probability 11. Stem Leaf 5

567899

6

0 1 1 2 2 5 6|1 = 61 in.

13. Frank; whereas Frank’s times range from 8–20 minutes, Shandra’s times range from 13–21 minutes. 15. 10; 4 17. 25% 19. Books Read in a Month 21 Number of People

Selected Answers

• The probability for two draws is the product of the probabilities of each single draw. • Independent events do not affect one another; dependent events do. 37. G 39. 6 ways 41. 32,768 43. 65° 45. 65° 47. 90° 49. 40.6 51. 16 53. 48.7 mi/h

Lesson 13-2

1. 3x + 2 3. 2x2 - 2x + 9 5. 6x + 24 7. -2b + 3 9. 4a3 + 3a2 + 8a - 2 11. 5x + 2 13. 11m - n 15. -x2 - 2x - 3 17. 3a2 + 2b2 + 3a 19. 825x - 17.5 21. 23x - 12 23. 42x - 22 25. -2a + 7b + 3c + 6; 46 27. 3a2 + 4b2 - 2; 89 29. Hai; the terms have the same variables, but in a different order. 31. Use algebra tiles to model each polynomial and combine the tiles that have the same size and shape. Algebra tiles that represent like terms have the same size and shape. When adding polynomials, a red tile and tile of any other color that have the same size and shape are zero pairs and may be removed. The result is the sum of the 3 2 39. _ 41. x2 + polynomials. 33. J 35. 2 37. _

Pages 712–713

12 9 6 3 0 0–1 2–3 4–5 Number of Books

21. line graph; show change over a period of time 23. box-and-whisker plot; divides data in four parts using the median and quartiles 25. 1 27. 2 29. 0 3 3 31. 16 outcomes 33. C; 364 ways 35. C; 21 ways 1 37. 64

Chapter 13 Polynomials and Nonlinear Functions Chapter 13

Get Ready

1. 1 3. 2 5. 0 7. 3 9. 6y - 16 11. 6x + 12y 13. 5q - 10r + 15s 15. yes 17. yes 19. yes 21. yes Pages 703–704

Pages 708–709

(-7) 43. 6b + (-3a2)

18 15

Page 699

35. Sometimes; x3 + xy + 5 has degree 3; x + xy + 5 has degree 2. 37. x + 3 39. D 41. B 2 43. exclusive; _ 45. -16 47. 4 49. 10 3 51. (x + 2x) + 4 53. (6n + 3n) + (2 + 5)

Lesson 13-1

1. Yes, there are no denominator variables and no variables under radical signs; monomial. 3. Yes, there are no denominator variables and no variables under radical signs; trinomial. 5. 4 7. 1 9. 5 11. Yes, there are no denominator variables and no variables under radical signs; monomial. 13. Yes, there are no denominator variables and no variables under radical signs; binomial. 15. No, the variable c is under a radical sign. 17. Yes, there are no denominator variables and no variables under radical signs; trinomial. 19. 0 21. 3 23. 1 25. 2 27. x2 - xy 29. 3z + 5x + yx 31. 70(1 + r)4 + 70(1 + r)3 + 70 (1 + r)2 + 70(1 + r) + 70 33. Sample answer: Using an interest rate of r = 0.025, Sarah would have $367.94.


13

26

45. (s + t) + (-2s)

Lesson 13-3

1. 4r 3. 5x + 2 5. 3x2 + 3x - 8 7. 2x units 9. -2n2 + 6n 11. y2 - 7y + 10 13. 15w2 - 2w + 10 15. 4x + 3 17. -2t + 2 19. 3a2 + 2b2 21. 9n2 - n - 12 23. 2x2 - 2xy + 3y2 25. a + 3b + b3 27. 120°F; -23°F 29. Sample answer: 3x2 + 4x + 1 and 2x2 + 2x + 5 31. Draw a model. A model can be used to show that you need to subtract the area of the pool from the area of the whole deck-pool combo. 5x + 15 33. A 35. B 37. 12x - 3y 39. -t2 + 12t + 2 41. Yes; there are no variables in the denominator and no variables under radical signs; binomial. 43. Stem Leaf 45. 27 47. 3x2 49. 6t3 5

49

6

468

7

0112

8

59

9

1 5 | 4 = 54

Pages 718–719

Lesson 13-4

1. 15y - 12 3. a2 + 4a 5. 12x2 - 28yx + 44x 7. A 9. 6 + 24b 11. x2 + 5x 13. 3y + 2y2 15. -18x2 + 12x 17. a3b + 7ab 19. 4m3 - 4m2 21. -18y + 27y2 - 12y3 23. -12x5 + 48x3 + 6x2 25. 56x2 + 16xy - 8y2; 82.48m2 27. y3 - 4y2 - 3y + 2 29. x4 - 1 31. -3 33. False; the order in which numbers or terms are multiplied does not change the product, by the Commutative Property of Multiplication; x(2x + 3) = 2x2 + 3x and (2x + 3)x = 2x2 + 3x. 35. To use the Distributive Property, multiply each term of the polynomial by the monomial. Answers should include the following. • The Distributive Property combines multiplication and addition when used to find the product of a polynomial and a monomial. • To find the product of 3x and x + 2, first find 3x · x. Then find 3x · 2. Finally, add the products, 3x2 + 6x. 37. F 39. 10a + 3a2 + 4

41. 7n2 + 6n 43.

4x

(x, y)

0

0

(0, 0)

1

4

(1, 4)

2

8

(2, 8)

x3 + 1

(x, y)

0

1

(0, 1)

1

2

(1, 2)

2

9

(2, 9)

x

49.

y

2

x

Pages 728–730

1. Pages 723–725

y 1x 3

Lesson 13-6 y

Lesson 13-5

y x2

1. Linear; graph is a straight line. 3. Nonlinear; equation cannot be written as y = mx + b 5. Nonlinear; rate of change is not constant. 7. No; y

x

3.

y

y x2 1

x

O

9. Linear; the graph is a straight line. 11. Nonlinear; the graph is a curve. 13. Nonlinear; the graph is a curve. 15. Nonlinear; the equation cannot be written in y = mx + b form. 17. Linear; the equation can be written in y = mx + b form. 19. Nonlinear; the equation cannot be written in y = mx + b form. 21. Nonlinear; rate of change is not constant. 23. Linear; rate of change is constant. 25. Nonlinear; the amount of change in price each year is not constant. 29. y2 = 3x is not a linear equation. 31. A function is linear if it has a constant rate of change. This can be determined from graphs, equations, or tables of values. Answers should include the following. • Functions can be represented using graphs, equations, or tables. • A graph that is a straight line represents a linear function. An equation that can be written in the form y = mx + b is a linear function. If a table of values shows a constant rate of change, the function is linear. 33. H 35. 4t + 9t2 37. a2b - 2ab2 39. -x 41. obtuse − 43. 3.7 45. -1.7 y 47. x

Selected Answers

45.

x

x

5.

26 24 22 20 18 16 14 12 10 8 6 4 2

S

0

7.

S 6a2

1

2

3

a

y y x 3 x

yx4

Selected Answers

R55

Selected Answers

9.

21.

y

Y

Y £ X Ó £ Ó x

X

"

y x 3 2

11.

23. 25 ft by 25 ft 25. V ≈ 0.6r2 V

y

y 0.5x 2

13.

11 10 9 8 7 6 5 4 3 2 1

x

y

0

V = 0.6r 2

1 2 3 4 5 6 7r

27. Similar shape; y = 2x3 is more narrow.

y

y 0.5x 2 y 2x 3

x

y 0.5x 3

15.

x

y y 3x

3

x

29. same shapes; They are the same graph only y = x3 - 3 is translated down 3 units.

y

x

y x3

y x3 3

17.

31. Make a table of values and plot the points. 33. Sometimes. For all x-values between 0 and -3 the y-values are negative. y is positive for all other values of x. Y 35. (0, -6)

y

y 0.1x 3 x

"

19.

X

y

O

x

37. C 39. Since the points on the graph do not lie on a straight line, the function is nonlinear. 41. n2 + 6n 43.

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£n

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Pages 732–734

Chapter 13


Selected Answers

1. trinomial 3. cubic 5. like terms 7. curve 9. yes; 2; binomial 11. no 13. 4 15. y2 - y - 1 17. -2a2 + 25 19. x - 23 21. 14x + 21y 23. -3y3 24y2 + 12y 25. Linear; the graph is a straight line. 27. Nonlinear; rate of change is not constant. 29. y

y x2 2 x

31.

y

x y x 3 2

33.

y y x 3 x

35. V = 8r2 "

Y

6 ûR

X

Selected Answers

R57

Index A Absolute value, 80 Accuracy, 614, 660 Activity. See Algebra Lab, Geometry Lab, Graphing Calculator Lab, Spreadsheet Lab Acute, 475 Acute angle, 477

Index

Acute triangle, 478 Addition algebraic fractions, 252 Associative Property, 44 Commutative Property, 43 decimals, 745 equations, 137, 268 fractions, 250, 263 inequalities, 435 integers, 84-85, 86-88 like fractions, 250 mixed numbers, 264 number line and integers, 86–87 polynomials, 706–707 Property of Equality, 137, 420 solving equations using, 137 solving inequalities, 435–436 table, 85 unlike fractions, 250, 263-264

input and output, 358 Juniper Green, 262 modeling multiplication, 715 modeling polynomials with algebra tiles, 705 multiplying integers, 99 percent model, 320-321 probability and Pascal’s triangle, 675 Pythagorean Theorem, 483-484 scatter plots, 60 simulations, 688–689 slope, 383 solving equations with algebra tiles, 134-135 squares and square roots, 462–463 subtracting integers, 92

Additive Identity, 44

Algebra tiles. See also Models adding integers, 84–85 algebra lab, solving equations, 134-135 distributive property, 125 division, 105 equations with variables on each side, 418-419 integer addition, 84 multiplication, polynomials by monomial, 716 polynomials, 705, 706 solving equations with, 134-135 solving two-step equations, 147 subtraction, 92

Additive Inverse, 88 See also Opposites

Algebraic equations. See Equations

Adjacent angles, 513, 515

Algebraic expressions, 37, 125 absolute value, 80 coefficient, 129 constant, 129 factoring, 193 identify parts, 129 like terms, 129 multiplication, 102 simplifying, 45, 126, 129, 130 subtraction, 126 term, 129

Addition Property of Equality, 137

Algebra, 37 Algebra Connection, 45, 80, 95, 102, 108, 125, 188, 193, 198, 204, 241, 247, 252, 258, 470 Algebra Lab adding integers, 84-85 algebra tiles and solving equations, 134-135 analyzing data, 273 base 2, 185 capture-recapture, 307 dividing integers, 105 equations, variables on each side, 418–419 graphing irrational numbers, 491 half-life simulation, 208

R58 Index

Algebraic fractions, 198 addition, 252 division, 247 multiplication, 241 simplifying, 198 Algebraic thinking. See Algebra Alternate exterior angles, 512, 513

Alternate interior angles, 512, 513 Altitudes, 545, 602 Analysis of data. See Data of errors. See Find the Error of graphs, 103, 127, 139, 161, 217, 223, 232, 271, 325, 346, 374, 388, 489, 517, 529, 555, 636, 641, 647, 648, 662, 663, 668, 686 of tables, 29, 35, 40, 82, 96, 151, 199, 212, 237, 243, 260, 266, 295, 330, 345, 346, 362, 368, 374, 388, 433, 629, 630, 636, 668, 673 Angle of rotation. See Rotation; Symmetry Angle relationships, 512–514 Angles, 757 acute, 477 adjacent, 513, 515 alternate exterior, 512, 513 alternate interior, 512, 513 central, 557 complementary, 513, 515 congruent, 513, 518 corresponding, 518 exterior, 512 interior, 512, 540 measures, 513, 533, 540 measuring and drawing, 757–758 obtuse, 477 quadrilateral, 533 right, 477 straight, 477 supplementary, 514, 515 vertical, 513, 515 Applications. See also Interdisciplinary connections; Real-life careers; Real World Examples; Real World Links accounting, 98 ages, 145, 155, 373, 658 air pressure, 57 aircraft, 199 airports, 481, 506 algebra, 39, 40, 46, 47, 51, 52, 59, 66, 71, 73, 80, 81, 82, 90, 95, 96, 97, 98, 102, 103, 104, 109, 113, 114, 115, 119, 126, 127, 128, 133, 139, 140, 143, 144, 145, 146, 149, 150, 151, 157, 161, 182, 183, 184, 188, 189, 190, 195, 198, 199, 200, 202, 207,

birdseed, 394 birthdays, 411, 691 blood types, 687 boating, 144 book survey, 692 books, 171, 277, 324, 729 bowling, 170 boxes, 718 bread, 242 breakfast, 231, 247, 619 bridges, 242 buildings, 138, 156, 309, 342, 578, 579, 619, 631, 656 business, 36, 53, 96, 150, 218, 232, 271, 335, 341, 392, 393, 448, 504 butterflies, 522 calendars, 189, 741 camping, 453 candles, 603 candy, 30, 409 canned foods, 599 capacity, 39, 56 car rental, 422, 449 car value, 62 car wash, 155 cards, 683 carousel, 554 carpentry, 213, 248, 253, 282, 387 carpeting, 560, 567 cars, 394, 473 caverns, 123 cell phones, 34, 154, 366, 422, 449 cheerleading, 247 chemistry, 90, 210, 417 chess, 336, 678 clocks, 482, 504 clubs, 352 coasters, 542 coin collecting, 238 coins, 29, 309 collections, 71 comics, 579 communication, 64, 66, 661 communications, 315 community service, 165 computers, 148, 205, 249, 265, 340, 401 concerts, 157, 319, 346, 452 construction, 156, 160, 402, 467, 516, 614, 729 containers, 618 conversions, 381 converting measures, 243, 587 cookies, 461 cooking, 248, 271, 284, 535 crafts, 206, 445, 600 cross country running, 562

cruises, 133, 421 currency, 52, 305 darts, 495 decorating, 133, 427 design, 194, 221, 309, 311 directions, 357 dirt bikes, 714 distance, 222 diving, 453 doors, 565 driver’s education, 431 e-mail, 220 earth science, 266, 372, 398, 406 earthquakes, 205 ecology, 702 education, 639 elections, 139 electricity, 708 elevators, 103, 117 energy, 109, 207 engineering, 586 entertainment, 57, 173, 381, 403, 405, 407 environment, 324, 669 escalators, 566 exercise, 254, 275, 283 fairs, 72 families, 685 farming, 362, 517 fashion, 132, 171 fences, 537 fencing, 190 Ferris wheel, 568 festivals, 349 field hockey, 461 films, 243 fish, 325 fitness, 102, 128, 260, 291, 357, 368, 440, 636, 640 flags, 548 flooring, 473 flowers, 677, 679 food, 41, 46, 70, 125, 157, 183, 192, 244, 248, 283, 298, 299, 303, 315, 316, 330, 335, 345, 511, 523, 595, 628, 667, 678 food costs, 380 food service, 672 football, 89, 119, 231 forestry, 331 fountains, 554 frames, 727 fruit, 144, 316, 319, 389, 410, 683 fruit drinks, 599 fuel, 339 functions, 555 fund-raising, 139, 146, 231, 413, 449, 605 furniture, 619 game shows, 89

Index

R59

Index

211, 213, 218, 223, 233, 238, 242, 243, 244, 248, 249, 252, 253, 254, 261, 272, 279, 296, 304, 305, 306, 331, 347, 428, 434, 439, 445, 450, 472, 477, 479, 480, 481, 482, 496, 507, 514, 515, 516, 517, 522, 530, 533, 534, 536, 543, 556, 569, 581, 594, 642, 704, 713 amusement parks, 125, 299, 304, 679 analyze graphs, 103, 127, 139, 161, 217, 223, 232, 271, 325, 346, 374, 388, 489, 517, 529, 555, 636, 641, 647, 648, 662, 663, 668, 686 analyze tables, 29, 35, 40, 82, 96, 151, 199, 212, 237, 243, 260, 266, 295, 330, 345, 346, 362, 368, 374, 388, 433, 629, 630, 636, 668, 673 anatomy, 319 animals, 25, 41, 65, 66, 77, 95, 110, 164, 202, 210, 212, 225, 232, 294, 312, 317, 319, 340, 410, 501, 680 apple cider, 51 aquariums, 584, 600, 625 arcade, 444 archaeology, 494, 517 architecture, 310, 514, 522, 577, 579, 595, 603, 604, 605, 611, 712 armed forces, 339 art, 305, 412, 489, 535, 542, 580 art museum, 319 arts, 206 astronomy, 88, 98 attractions, 304 auto maintenance, 591 auto racing, 260 automobiles, 232, 393 aviation, 146 baby-sitting, 132, 249 backpacking, 282 bake sale, 202 ballooning, 165 banking, 334, 335 baseball, 28, 349, 466, 480, 490, 505, 565, 724, 727 baseball cards, 117, 131 basketball, 47, 65, 109, 170, 278, 351, 427, 534, 593, 656, 658, 718 batteries, 587 bedrooms, 311 beverages, 616 bicycles, 71 bicycling, 166 biology, 63, 183, 461, 529 birds, 211, 325, 411

Index

games, 54, 118, 529, 535, 537, 673, 680, 683, 685, 707 gardening, 253, 413, 482, 504, 536, 625, 703 gardens, 613 gasoline, 298, 409 genetics, 173 geography, 82, 95, 242, 256, 316, 330, 351, 407, 422, 536, 549, 561, 638, 639 geology, 140, 593 geometry, 29, 48, 58, 113, 114, 132, 160, 164, 165, 166, 172, 183, 200, 206, 237, 244, 271, 272, 295, 298, 317, 331, 336, 368, 372, 388, 425, 426, 427, 439, 450, 452, 455, 467, 468, 473, 489, 493, 494, 495, 501, 502, 523, 606, 649, 674, 679, 702, 703, 707, 708, 712, 714, 717, 725, 728, 729, 733, 734, 735 gifts, 353 golf, 108, 117, 230, 454, 694 golf scores, 79 government, 632 grades, 490, 628 groceries, 319 gymnastics, 489 handshakes, 679 hang gliding, 393, 471 health, 272, 318, 346 health care, 63 height, 52 highway maintenance, 595 hiking, 89, 448, 507 history, 592, 612 hobbies, 667 hockey, 232, 432 home improvement, 560 home repair, 387 homework, 437, 704 hot-air balloons, 304 hotels, 605 housing, 412, 428 humidity, 477 hurricanes, 275, 438 hybrid cars, 243 ice cream, 581, 591 ice skating, 278, 699 insects, 265, 311, 382 interior design, 310 internet, 330, 686 internet shopping, 556 investments, 335, 336, 342, 351, 353 jobs, 413, 454, 662 kaleidoscope, 541 keyboarding, 63, 213 ladder, 507 landscape design, 560 landscaping, 165, 166, 227, 472, 495, 520, 714

R60 Index

lawn care, 243, 299, 550, 554 libraries, 655 life science, 222, 325 light bulbs, 167 literature, 202 longitude, 712 lunch, 25 machinery, 237 magazines, 295 manufacturing, 59, 256, 734 maps, 308, 326, 350, 499, 500 marathon, 357 marbles, 693 marine biology, 438 measurement, 144, 199, 211, 217, 223, 236, 243, 252, 361, 368 media, 316 medicine, 30, 149, 212, 652, 733 memorials, 499 mental math, 46, 127 metals, 587 meteorology, 271, 735 microwaves, 587 mileage, 183 military, 306 mirrors, 529 money, 25, 29, 34, 70, 73, 89, 95, 123, 126, 131, 140, 146, 171, 329, 330, 403, 448, 724 monuments, 499 movies, 52, 127, 267, 284, 662 moving, 453 mowing, 434 murals, 305 museum, 568 music, 34, 47, 64, 173, 199, 271, 282, 291, 334, 350, 352, 369, 552, 554, 573, 639, 652, 658, 741 music downloads, 422 newspapers, 405 number theory, 183, 188, 194 nutrition, 276, 431, 640 oceanography, 97 oceans, 217 Olympics, 64, 155, 305, 341, 626, 627, 630, 640, 646, 692 packaging, 189, 593 painting, 144, 600 parades, 193 parking spaces, 537 parks, 142, 501 part-time job, 51, 296, 382 party planning, 299 patents, 724 patterns, 110 patios, 310 patterns, 110, 184, 194, 580 paychecks, 708 performing arts, 330 perimeter and area, 501 personal care, 150

pets, 253, 260, 269, 344, 350, 648 phone cards, 150 phone charges, 77 phones, 711 photographs, 566 photography, 39, 220, 304, 578 physical education, 71 physical science, 206, 211, 380 physics, 473, 699 physiology, 699, 703 piano lessons, 399 picnic, 284, 400 pies, 319 pizza, 299, 677 planets, 318 plants, 56, 260, 299, 378 plumbing, 52, 601 pole vaulting, 406 pools, 150, 378, 584 population, 90, 156, 318, 645 postal service, 27 presentations, 529 Presidents, 627, 649 pressure, 405 probability, 674, 680 profits, 70, 165 publishing, 35, 266, 271 quilting, 194, 520 racing, 118, 691 radio, 283 rain, 375 ramps, 384 ranching, 144 real estate, 335, 349, 449 recycling, 189, 237, 413, 425, 511 rentals, 421 repairs, 449 reports, 170 research, 68, 109, 114, 139, 266, 325, 341, 362, 433, 467, 475, 542, 561, 579, 580, 614, 630, 703 restaurants, 159, 329 retail, 270, 276, 342, 595, 606, 645 rides, 500 roads, 384 roller coasters, 240, 311, 388, 467, 646, 655 routes, 671 running, 166, 447 safety, 515 sales, 77, 148, 195, 299, 361, 417, 448, 454, 455, 468 savings, 29, 334, 337, 382, 437, 457 scale factors, 501 school, 146, 213, 341, 449, 450, 628, 641, 645, 661, 734 school colors, 677 school supplies, 132

tessellations, 541, 542 test drives, 81 test scores, 61 testing, 687 tests, 278, 440 theaters, 160 theme parks, 128 ticket sales, 693 tides, 103 tiling, 514 time, 29, 199, 479, 516 time zones, 190 toasters, 353 tornadoes, 184 towers, 521 toys, 143 track and field, 192, 237, 473, 536 traffic, 217 traffic laws, 635 transportation, 233, 438 travel, 35, 36, 51, 103, 165, 166, 173, 205, 221, 242, 256, 259, 295, 304, 382, 449, 468, 489, 494, 505, 511, 556, 594, 637, 640, 659, 660, 661, 695 trees, 72, 552 trucks, 291 typing, 172 U.S. Presidents, 692 umbrellas, 537 unit cost, 379 vacation, 277 video games, 335 videos, 346 volunteering, 369 voting, 266 walkways, 561 watches, 345 water, 30, 353, 573 water parks, 48, 644 water usage, 231 weather, 25, 94, 98, 108, 109, 115, 119, 139, 146, 161, 232, 259, 275, 285, 341, 347, 474, 481, 592, 628, 674 wedding, 619 White House, 237 wildlife, 156 world records, 506 writing, 393 zoo, 35, 495 Area, 163 See also surface area circle, 553, 558 composite figures, 558, 559 dilations, spreadsheet lab, 563 parallelograms, 545, 558 rectangle, 163 spreadsheet lab, 168

trapezoid, 547, 558 triangle, 546, 558 Area models, 239, 242, 245, 246, 248 Arithmetic sequence, 158 common difference, 158 term, 158 Assessment. See also Check Your Progress, Choose Your Method, Extra Practice, Mixed Problem Solving, Prerequisite Skills, Review, Spiral Review Associative Property, 44 Associative Property of Addition, 44 Associative Property of Multiplication, 44, 204

Index

science, 40, 48, 58, 67, 162, 319, 325, 363, 367, 466, 554, 593, 635, 730 scooters, 660 scuba diving, 227, 233, 360 seismic waves, 471 sewing, 256, 265, 279, 285 shadows, 500, 631 shipping, 711, 733 shoes, 275, 335 shopping, 40, 46, 73, 132, 154, 293, 295, 334, 351, 366, 369, 455 signs, 505, 573 skateboarding, 480 skateboards, 333, 673 skateparks, 311 skiing, 387, 671 skydiving, 729 skyscrapers, 241, 530 sledding, 138 sleep, 72 snacks, 305 snow, 541 snowboarding, 118 soccer, 39, 161, 165, 432, 444 softball, 230, 677 solar system, 82 sound, 182, 400, 401 space, 48, 98, 109, 215, 216, 217, 222, 270, 319, 381, 496 space shuttle, 36 space travel, 241, 610 speed record, 241 spending, 405 sports, 127, 274, 294, 433, 504, 534, 561, 611, 658, 679 stamps, 131 state fairs, 437 statistics, 109, 669, 719 statues, 311, 579 stock market, 119, 123, 340 stocks, 89, 417 storage, 617 strawberries, 73 street signs, 566, 567 student council, 677 submarines, 88 Supreme Court, 651 survey, 227, 300 surveys, 280, 625 swimming, 104, 172, 404, 405, 444, 676 swimming pool, 119 taxi fares, 550 taxis, 353 technology, 189, 345, 363, 371, 377, 378, 661, 679 television, 374, 489, 617, 652 temperature, 98, 155, 156, 635, 712

Average, 108 See also mean

B Back-to-back stem-and-leaf plots, 627-628 Bar graphs, 651, 657, 759 misleading, 659 Bar notation, 229 Base, 180 parallelograms, 545 percent proportion, 322 Base two numbers, 185 Bases, 576 Benchmark. See Estimation Best-fit lines, 403 Bias, 664 Biased sample, 344 Binary, 185 Binomials, 701 Box-and-whiskers plot, 638, 639, 643, 651

C Calculator, See also Graphing Calculator Lab Careers. See Real World Careers Cells, 42 Center, 524 circle, 551 Center of rotation, 531 Index

R61

Central angle, 557

Common difference, 158

Central tendency. See measures of central tendency

Common multiples, 257

Challenge. See H.O.T. Problems Chapter Test. See Practice Test

Index

Change, rate of, 371, 373, 390 Check Your Progress, 27, 28, 33, 34, 37, 38, 39, 44, 45, 49, 50, 51, 55, 56, 61, 62, 63, 78, 79, 80, 86, 87, 88, 93, 94, 95, 100, 101, 102, 107, 108, 112, 113, 125, 126, 130, 131, 137, 138, 142, 143, 148, 149, 153, 154, 155, 158, 159, 162, 163, 164, 181, 182, 187, 188, 192, 193, 196, 197, 198, 204, 205, 209, 210, 214, 215, 216, 228, 229, 230, 231, 234, 235, 236, 239, 240, 241, 245, 246, 247, 250, 251, 252, 257, 258, 259, 263, 264, 265, 268, 269, 270, 275, 276, 277, 292, 293, 294, 298, 303, 304, 309, 310, 313, 314, 315, 323, 324, 328, 329, 332, 333, 334, 338, 339, 340, 344, 345, 359, 360, 361, 365, 366, 367, 371, 372, 373, 377, 378, 379, 384, 385, 386, 391, 392, 393, 397, 398, 399, 400, 403, 404, 405, 420, 421, 425, 426, 430, 431, 432, 436, 437, 442, 443, 446, 447, 448, 465, 466, 470, 471, 477, 478, 485, 486, 487, 493, 494, 498, 499, 513, 514, 515, 519, 520, 525, 526, 533, 534, 539, 540, 541, 546, 547, 548, 552, 553, 559, 560, 577, 578, 583, 584, 585, 589, 590, 591, 598, 599, 603, 604, 609, 610, 611, 627, 628, 633, 634, 635, 639, 640, 645, 646, 652, 653, 660, 661, 666, 667, 670, 672, 677, 678, 683, 684, 701, 702, 707, 710, 711, 716, 717, 720, 721, 722, 727, 728 Circle area, 553, 558 center, 551 curcumference, 551 diameter, 551 pi, 551 radius, 551 Circle graphs, 557, 760 Circumference, 551 Coefficient, 129, 149 algebraic expressions, 129 expressions, 129 negative, 149 Collect the Data, See H.O.T. Problems Combinations, 677 notation, 678

R62 Index

Commutative Property of Addition, 43 Commutative Property of Multiplication, 43, 100, 204 Compare and order decimals, 742 fractions and decimals, 230 integers, 78 numbers in scientific notation, 216 percents, decimals, and fractions, 315 real numbers, 469 Compatible numbers, 746 Complementary angles, 513, 515 Composite events, 682 Composite figures, 558, 559 Composite numbers, 186 Composite shapes. See Composite figures Compound events, 682 Compound interest, 337 Computation, choose the method, 28 Concept Summary, adjacent angles, 515 area formulas, 558 area of a circle, 558 area of a parallelogram, 558 area of a trapezoid, 558 area of triangle, 558 bar graph, 651 biased samples, 344 box-and-whisker plot, 651 circle graph, 651 complementary angles, 515 convenience sample, 344 cubic functions, 722 dilations, 527 exponential functions, 722 frequency table, 651 histogram, 651 inverse variation, 722 line and angle relationships, 515 line graph, 651 line plot, 651 mean, 722 median, 722 mode, 722 nonlinear functions, 722 operations with integers, 108 order of operations, 32, 181 parallel lines, 515

percent equation, 332 percent-fraction equivalents, 327 percents, 327, 332 perpendicular lines, 515 polyhedrons, 576 probability of two independent events, 682 proportional linear realtionships, 379 quadratic functions, 722 rates of change, 373 rectangular prism, 576 rectangular pyramid, 576 reflections, 527 representing functions, 367 simple random sample, 343 statistical graphs, 651 stem and lear plot, 651 stratified random sample, 343 supplementary angles, 515 systematic random sample, 343 table, 651 transformations, 527 translations, 527 triangular prism, 576 triangular pyramid, 576 types of percent problems, 324 unbiased samples, 343 using mean, median, and mode, 722 Venn diagram, 651 vertical angles, 515 voluntary response sample, 344 Cones, 576, 590 surface area, 603, 604 volume, 590 Congruent, 478 angles, 513 segments, 478 triangles, 518 Congruence statements, 519 Conjecture, 27 Conjectures, make, 42, 58, 60, 85, 92, 99, 105, 168, 203, 208, 307, 337, 376, 383, 390, 463, 476, 497, 517, 531, 583, 596, 607, 731 Connections. See Applications Constant, 129 algebraic expressions, 129 expressions, 129 proportionality, 298 rate of change, 376 Constant of proportionality, 298 Constant of variation, 378

Constant rate of change, 376 Contraction. See Dilations Convenience sample, 344

Cylinders, 576 lateral area, 596 surface area, 596, 598, 599 volume, 585

Converse, 487 Conversions, See also Metric system and Customary system Coordinates, 79

D

Corresponding parts, 497, 518

Decagon, 539

Counterexample, 45

Decimals adding, 744, 745 dividing, 746, 749–750 estimating, 744, 746, 751, 752 fractions, comparing, 230 multiplying, 746, 747–748 ordering, 742 repeating, 229, 235 rounding, 743 subtracting, 744, 745 terminating, 228, 235 writing as fractions or mixed numbers, 228, 229, 235

Coordinate grid. See Coordinate plane Coordinate plane, 54 dilations, 524, 527 ordered pair, 54 origin, 54 quadrants, 112 reflections, 524, 526, 527 rotations, 531 transformations, 524, 527 translations, 524, 527 x-axis, 54 x-coordinate, 54 y-axis, 54 y-coordinate, 54 Coordinate system, 45, 111 Coordinates. See ordered pairs

Counting outcomes, 670 Critical Thinking. See H.O.T. problems Cross product, 302 Cross-Curricular Project algebra and agriculture, 623, 648, 722, 730 algebra and architecture, 459, 500, 561, 591, 613 algebra and geography, 23, 65, 104, 166, 167 algebra and nutrition, 177, 276, 279 algebra and recreation, 289, 335, 404, 406, 423, 450

Deductive reasoning, 45

Direct variation, 378 Discounts, 333 Discrete mathematics. See Combinations; Counting; Permutations; Probability; Sequences; Statistics Distance, 162 Distance formula, 492 Distributive Property, 124, 125, 424, 716 Dividend, 106 Divisible, 740 Division algebra tiles, 105 algebraic fraction, 247 equations, 141, 269 fractions, 245, 246 inequalities, 441 integers, 105, 106–108 mixed number, 246 monomials, 204 notation, 33 whole number, 246 Division Property of Equality, 141 Divisor, 106 Domain, 56, 359 Double bar graphs, 759 Double box-and-whisker plots, 640 Double line graphs, 759

Degree, 475, 757 Dependent events, 683 probability, 683

Cubic, 722

Dependent variable, 365

Customary system, 755 converting measurements, 755–756

Dimensional analysis, 241, 247, 294, 424

Defining a variable, 38

Cross-Curriculum. See Applications Cubic functions, 722, 728 graphing, 728

Dilations, 524 area, 563 coordinate plane, 527 enlargements, 497, 524, 527, 563 perimeter, 563 reductions, 497, 524, 527, 563

Index

Corresponding angles, 512

Data, 759, See also Graphs, Statistics analysis, 273, 277 bar graph, 651 box-and-whisker plots, 638-640, 643, 651, 652, 759 circle graph, 557, 651, 760 collecting, 60, 208, 273 comparisons, 301 display, 651, 759-760 frequency tables, 644, 651 histograms, 644-646, 650, 651 line graph, 651, 657, 759 line plot, 275, 632, 651 measures of central tendency. See Measures of central tendency patterns, 27 scatter plots, 60, 61, 67-68 stem-and-leaf plots, 626-628, 632, 651 table, 651 Venn diagram, 191, 651

Coordinate graph. See Coordinate plane

Differences, 33 See also subtraction

Diagnose readiness. See Prerequisite skills Diagonals, 540, 580 Diameter, 551

E Edges, 575 Empty set, 426 Enlargements, 497 Equations addition, 137, 268 division, 141, 269 Index

R63

Index

grouping symbols, 424 parentheses, 424 identifying functions, 721, 722 like terms, 149 multiplication, 141, 269 negative coefficients, 149 open sentences, 49 prediction, 403 solutions, 365 solving addition, 137 division, 141 multiplication, 141 subtraction, 136 two-step, 147 variables on each side, 420 solving the, 49 subtraction, 136, 268 two-step, 147 verbal sentences, translating, 50 writing linear, 391, 392, 397-399 Equilateral triangle, 478 Equivalent equations, 136 Equivalent expressions, 124, 130 Estimation, 240, 322, 328, 465, 553 Evaluate, 32 Events, 665. See also Probability composite, 682-684 dependent, 683 independent, 682-683 mutually exclusive, 684 simple, 665-667 Experimental probability, 666 Exponential functions. See Functions Exponents, 180 negative, 209 positive, 180 Expressions, 32 coefficient, 129 constant, 129 evaluating, 32–34, 37–39 like terms, 129 simplest form, 130 simplifying, 129, 130 term, 129 translating into words, 31 variables, 37 verbal expressions, 31, 38 Exterior angles, 512 Extra Practice, 761-792 Extreme value, 276, 277

R64 Index

F Faces, 575 Factor, 188 expressions, 193 monomials, 188 Factoring, 188 Factors, 180, 188, 740 Factor tree, 187 Family of functions, 395 of linear graphs, 395 of quadratic functions, 731 Family of functions, 395

comparing, 230, 258 decimals, comparing, 230 division, 245 estimating products and quotients, 752 estimating sums and differences, 751 mixed numbers, 264 multiplication, 239, 240, 241 percents as, 313 simplest form, 196 simplifying, 196 subtraction, 251, 264 writing as decimals, 228, 229, 230 writing as repeating decimals, 229 writing as terminating decimals, 228

Find the Data, 35, 52, 65, 82, 96, 144, 199, 253, 261, 271, 300, 306, 369, 388, 438, 473, 495, 555, 562, 612, 648, 724. See also United States Data File

Frequency table, 644

Find the Error. See H.O.T. Problems

Function tables, 364

First power, 180 Foldables™ Study Organizer equations and inequalities, 416, 451 equations, 122, 169 factors and fractions, 178, 219 functions and graphing, 356, 408 integers, 76, 116 more statistics and probability, 624, 690 polynomials and nonlinear functions, 698, 732 ratio, proportion, and percent, 290, 348 rational numbers, 226, 281 real numbers and right triangles, 460, 503 three-dimensional figues, 572, 615 tools of algebra, 24, 69 two-dimensional figures, 510, 564 Formulas, 162 See also Rates; Measurement; Interest Four-step problem-solving plan, 26 Fraction bars, 32, 292 Fractions, 313. See also Rational numbers addition, 250, 263, 264

Functional notation, 370 Function machine, 358 Functions, 359-361, 370 family of, 395 graphing, 726-728 linear, 365–367 machine, 358 nonlinear, 720. See also Cubic, Functions, Quadratic order pairs, 359 quadratic and cubic, 726-728, 731 relations, 359 vertical line test, 360 Fundamental Counting Principle, 671

G GCF. See Greatest common factor (GCF) Geometry applications in architecture, 310, 514, 522, 577, 579, 595, 603, 604, 605, 611, 712 applications in art, 305, 412, 489, 535, 542, 580 angles acute, 477 adjacent angles, 513, 515 alternate exterior angles, 512, 513 alternate interior angles, 512, 513

Geometry Lab lateral area, 596

rotations, 531 similar solids, 607 surface area, 596 tessellations, 544 three dimensional figures, 574 volume, 582

Grouping symbols, 32 brackets, 32 fractions bars, 32 inequalities, 429, 431 parentheses, 32 solving equations, 424-426

Get Ready for the Chapter. See Prerequisite Skills Get Ready for the Lesson. See Prerequisite Skills

H

Get Ready for the Next Lesson. See Prerequisite Skills

Half-life simulation, 208

Golden ratio, 300, 707

Heptagon, 539, 540

Golden rectangle, 707

Hexagon, 539, 540

Graph, 55

Histograms, 644–646, 650

Graph of ordered pair, 55

Homework Help, 29, 34, 40, 46, 52, 57, 64, 81, 89, 95, 103, 109, 114, 127, 131, 139, 144, 150, 155, 160, 165, 182, 189, 194, 199, 206, 211, 217, 231, 237, 242, 248, 252, 260, 266, 270, 278, 295, 299, 305, 311, 316, 325, 330, 335, 341, 346, 361, 368, 374, 380, 387, 393, 400, 405, 422, 427, 433, 438, 444, 448, 467, 472, 479, 488, 494, 500, 516, 521, 528, 534, 542, 549, 554, 561, 579, 586, 592, 600, 604, 611, 629, 636, 641, 647, 654, 662, 668, 673, 679, 685, 703, 708, 712, 718, 723, 729

Graphing. See also Graphs algebraic relationships, 113 functions, 365–367, 726–728 geometric relationships, 807–808 linear equations, 366-367 Graphing Calculator Lab box-and-whiskers plots, 643 family of linear graphs, 395–396 family of quadratic functions, 731 function tables, 364 histograms, 650 line plots, 632 mean and median, 280 probability simulation, 681 rate of change, 390 scatter plots, 67–68 slope, 390 stem-and-leaf plots, 632 Graphs. See also Data, Statistics bar, 657, 759 box-and-whisker plots, 638-640, 643, 651 circle, 557, 760 histograms, 644-646, 650, 651 line, 657, 759 line plot, 632, 651, 760 relations as, 54-56 scatter plots, 60, 61, 67-68 stem-and-leaf plots, 626-628, 632, 651 writing equations, 397 Greater than (>), 79, 431 Greatest Common Factor (GCF), 191, 239 Griddable. See Standardized Test Practice Gridded response. See Griddable; Standardized Test Practice

Height of parallelograms, 545

H.O.T. Problems Challenge, 30, 36, 41, 47, 53, 58, 65, 83, 90, 96, 103, 110, 115, 128, 133, 140, 145, 150, 156, 161, 166, 184, 189, 195, 200, 206, 212, 218, 233, 238, 244, 249, 253, 261, 266, 272, 278, 300, 306, 312, 317, 326, 331, 336, 341, 347, 363, 369, 375, 388, 394, 402, 406, 423, 427, 434, 439, 444, 449, 468, 473, 480, 490, 495, 502, 517, 523, 529, 535, 543, 550, 555, 562, 580, 588, 594, 601, 606, 613, 630, 637, 642, 648, 655, 663, 669, 674, 680, 686, 704, 709, 713, 719, 724, 729, Collect the Data, 630, 642, 648, Find the Error, 35, 47, 96, 115, 128, 132, 156, 189, 195, 243, 253, 267, 271, 312, 341, 388, 394, 439, 445, 449, 490, 502, 523, 556, 587, 612, 630, 679, 686, 704, 709, Number Sense, 65, 82, 115, 145, 157, 189, 212, 218, 233, 317, 369, 422, 428, 434, 468, 556, 612, 642, 729

Index

R65

Index

central angles, 557 complementary angles, 513, 515 congruent, 513 corresponding, 512, 513, 518 exterior, 512 interior, 513, 540 measures, 512, 533, 540 obtuse, 477 quadrilateral, 533 right, 477 straight, 477 supplementary, 514, 515 vertical, 513, 515 area circles, 553, 558 composite figures, 558–560 parallelograms, 545, 558 trapezoids, 547, 558 triangles, 546, 558 circumference of a circle, 551–552 diameter of a circle, 551 equilateral triangle, 478 isosceles triangle, 478 length of a hypotenuse in right triangles, 484, 485 length of sides of golden rectangles, 707 obtuse angles, 477 pi, 551 radius of a circle, 551 right angles, 477 scalene triangle, 478 straight angles, 477 sum of angles of triangle, 476 surface area prisms and cylinders, 597-599 pyramids and cones, 602-604 three-dimensional figures, 575-578 triangles, 476-478, 485-487 acute, 477, 478 angle measures, 476 congruent, 518 equilateral, 478 isosceles, 478 obtuse, 477, 478 right, 477, 478 scalene, 478 similar, 497, 498 volume cones, 590 cylinders, 585 prisms, 583 pyramids, 589 spheres, 591

Index

Open Ended, 30, 35, 41, 47, 53, 58, 65, 82, 90, 96, 110, 115, 128, 103, 132, 140, 145, 150, 156, 161, 166, 184, 189, 195, 200, 206, 212, 218, 232, 237, 243, 249, 253, 261, 266, 278, 296, 300, 306, 312, 317, 326, 331, 336, 341, 347, 362, 369, 375, 381, 388, 394, 401, 406, 423, 427, 434, 439, 444, 449, 468, 473, 480, 490, 495, 502, 517, 529, 536, 543, 550, 555, 562, 580, 587, 593, 601, 605, 612, 636, 655, 663, 669, 674, 680, 686, 709, 713, 719, 724, 729, Reasoning, 35, 103, 167, 207, 212, 238, 331, 362, 381, 389, 468, 580, 648, 674, 719, 729 Research, 300 Select a Technique, 53, 97, 140, 233, 296, 428, 495, 593 Select a Tool, 104, 184, 401, 428, 480, 543, 655, 713 Which One Doesn’t Belong?, 41, 82, 132, 200, 272, 317, 341, 474, 529, 724 Writing in Math, 30, 36, 41, 47, 53, 58, 65, 83, 90, 97, 104, 110, 115, 128, 133, 140, 145, 150, 157, 161, 167, 184, 190, 195, 200, 207, 212, 218, 233, 238, 244, 249, 254, 261, 267, 272, 278, 296, 300, 306, 312, 317, 326, 331, 336, 342, 347, 363, 369, 375, 381, 389, 394, 402, 406, 423, 428, 434, 439, 445, 449, 468, 474, 481, 490, 496, 502, 517, 523, 529, 536, 543, 550, 556, 562, 580, 588, 594, 601, 606, 613, 631, 637, 642, 648, 655, 663, 669, 674, 680, 686, 704, 709, 713, 719, 724, 729 How to Use Your Math Book Doing Your Homework, 15 Reading Your Math Book, 14 Studying for a Test, 16 Hypotenuse, 485

I Identity, 426 Identity Property of Addition, 85 Image, 524 Independent events, 682 Independent variable, 365

R66 Index

Indirect measurement, 499 Inductive reasoning, 27, 94 Inequalities, 79, 430, 431, 432, 435 graphing, 432 multi-step, 446-448 solve using addition, 435 using division, 441 using multiplication, 441 using subtraction, 435 symbols, 431 truth, 431 writing, 430, 432 Input, 358, 365. See also functions Integers, 78 absolute value, 80 addition, 84-85, 86-88 comparing, 79 division, 105, 106, 108 multiplication, 99, 100-102 number line, 79 numbers, 234 ordering, 79 subtraction, 92, 93-95 Interdisciplinary connections. See also Applications agriculture, 362, 517 anatomy, 319 archaeology, 494, 517 architecture, 310, 514, 522, 577, 579, 595, 603, 604, 605, 611, 712 art, 305, 412, 535, 542, 580 astronomy, 88, 98 biology, 63, 183, 461, 529 botany, 56, 72, 260, 378, 552 chemistry, 90, 210, 417 civics, 632, 649, 650, 651, 692 computers, 148, 205, 249, 265, 340, 401 driver’s education, 431 earth science, 216, 266, 372, 398, 406 ecology, 702 education, 639 engineering, 586 environment, 324, 425, 511, 669 forestry, 331 genetics, 173 geography, 82, 95, 256, 316, 330, 351, 407, 422, 536, 549, 561, 638, 639 geology, 272, 140, 593 geophysics, 471 government, 632 health, 272, 318, 346 history, 592, 612 life science, 222, 325

marine biology, 438 medicine, 30, 149, 212, 652, 733 meteorology, 156, 271, 735 music, 34, 47, 64, 173, 199, 271, 282, 334, 350, 352, 369, 422, 552, 554, 573, 639, 652, 658, 741 oceanography, 97 performing arts, 330 physical education, 71 physical science, 97, 206, 207, 211, 380, 471, 473, 504 physics, 400, 401, 473, 699 physiology, 699, 703 probability, 674, 680 science, 40, 48, 58, 67, 162, 319, 325, 363, 367, 466, 554, 593, 635, 730 space, 48, 82, 98, 109, 215, 216, 217, 222, 241, 270, 318, 319, 381, 496, 554, 610, 635 statistics, 109, 669, 719 technology, 189, 345, 363, 371, 377, 378, 661, 679 zoology, 67, 325, 382 Interdisciplinary project. See Cross-Curricular Project Interest, 333 Interior angles, 512 polygons, 540 Internet Connections Chapter Test, 73, 119, 173, 223, 285, 353, 413, 455, 507, 569, 619, 695, 735 Concepts in Motion animation, 56, 99, 135, 234, 320, 476, 533, 589 BrainPOP®, 93, 124, 204, 392, 421, 446, 498, 665, 702 Interactive Lab, 42, 112, 143, 197, 257, 344, 365, 426, 485, 552, 599, 651, 710 Cross-Curricular Project, 65, 104, 166, 278, 406, 423, 450, 459, 500, 561, 591, 613, 623, 648, 722, 730 Extra Examples, 27, 33, 39, 45, 51, 56, 63, 80, 87, 93, 100, 107, 113, 125, 130, 137, 143, 149, 153, 159, 163, 181, 186, 193, 197, 205, 209, 214, 229, 236, 240, 246, 250, 257, 265, 269, 275, 292, 303, 309, 314, 323, 327, 333, 340, 344, 360, 367, 377, 385, 391, 398, 404, 420, 424, 430, 437, 443, 447, 465, 471, 478, 486, 493, 498, 513, 519, 526, 533, 540, 547, 552, 560, 577, 585, 591, 597, 599, 603, 610, 627, 635, 640, 646,

Interpolating from data. See Predicting Interquartile range, 634

Inverse operations, 136, 468 Inverse Property of Multiplication, 245 Inverse variation, 722 Investigations. See Graphing calculator investigation; Algebra lab; Spreadsheet lab Online Readiness Quiz, 25, 77, 123, 179, 227, 291, 357, 417, 461, 511, 573, 625, 699 Irrational numbers, 469 graphing, 491 Irregular figures. See Complex figures Isosceles triangle, 478

K Key concept, absolute value, 80 adding like fractions, 250 adding unlike fractions, 263 addition property of equality, 137 addition property of inequalities, 435 additive inverse property, 88 alternate exterior angles, 512 alternate interior angles, 512 angles of a quadrilateral, 533 angles of a triangle, 476 area of a circle, 553 area of a parallelogram, 545 area of a rectangle, 163 area of a trapezoid, 547 area of a triangle, 546 associative property of addition, 44 associative property of multiplication, 44 circumference of a circle, 551 classify triangles by angles acute, 478 obtuse, 478 right, 478 classify triangles by sides equilateral, 478 isosceles, 478 scalene, 478 commutative property of addition, 43 commutative property of multiplication, 43 corresponding angles, 512 corresponding parts of congruent triangles, 518 corresponding parts of similar figures, 498

direct variation, 378 distance formula, 492 distributive property, 124 dividing fractions, 246 division property of equality, 141 division property of inequalities, 441, 443 exterior angle, 512 Fundamental Counting Principle, 671 inequalities, 431 integer addition with different sign, 87 integer addition with same sign, 86 integer division with different signs, 107 integer division with same sign, 106 interior angles, 512 interior angles of a ploygon, 540 interquartile range, 634 inverse property of multiplication, 245 irrational numbers, 469 lateral area of cylinders, 599 lateral area of rectangular prisms, 597 measures of central tendency, 274 multiplication property of equality, 143 multiplication property of inequalities, 441, 443 multiply integers with different sign, 100 multiply integers with same sign, 101 multiplying fractions, 239 mutually exclusive, 684 negative exponents, 209 negative relationship, 62 parallel lines cut by a transversal, 513 percent proportion, 322 percents and decimals, 314 perimeter of a rectangle, 163 positive relationship, 62 probability, 665 probability of two dependent events, 683 probability of mutually exclusive events, 684 product of powers, 203 properties of numbers, 44 property of proportions, 303 proportion, 302 Pythagorean Theorem, 485 quotient of powers, 204

Index

R67

Index

652, 660, 666, 671, 678, 684, 701, 706, 710, 716, 726 Online Readiness Quiz, 25, 77, 123, 179, 227, 291, 357, 417, 461, 511, 573, 625, 699 Other Calculator Keystrokes, 67, 280, 643, 650, 681, 731 Personal Tutor, 27, 33, 39, 45, 50, 55, 63, 79, 87, 94, 101, 108, 113, 125, 131, 138, 142, 149, 154, 159, 164, 182, 187, 192, 198, 205, 210, 215, 231, 235, 241, 247, 257, 259, 265, 270, 277, 293, 298, 303, 310, 315, 324, 328, 334, 340, 344, 359, 366, 371, 378, 386, 393, 399, 404, 421, 425, 431, 437, 442, 448, 466, 471, 477, 487, 494, 499, 514, 520, 525, 533, 541, 548, 552, 559, 578, 585, 589, 598, 603, 611, 628, 634, 639, 645, 653, 661, 667, 670, 677, 683, 702, 707, 711, 717, 722, 727 Practice Test, 73, 119, 173, 223, 285, 353, 413, 455, 507, 569, 619, 659, 735 Reading in the Content Area, 37, 86, 136, 187, 236, 297, 430, 464, 512, 583, 633, 701 Real-World Career, 63, 156, 199, 253, 310, 392, 449, 499, 535, 605, 652, 702 Self-Check Quiz, 10, 30, 35, 40, 47, 52, 58, 65, 82, 90, 96, 103, 109, 114, 127, 132, 140, 144, 150, 156, 160, 166, 183, 189, 194, 199, 206, 212, 217, 232, 237, 243, 248, 253, 260, 266, 271, 278, 295, 299, 305, 312, 317, 326, 330, 336, 341, 346, 362, 369, 374, 380, 388, 393, 401, 406, 422, 427, 433, 438, 444, 449, 467, 473, 480, 489, 495, 501, 517, 522, 529, 535, 542, 549, 555, 562, 580, 587, 593, 601, 605, 612, 630, 636, 641, 648, 655, 663, 669, 673, 679, 685, 703, 709, 712, 718, 724, 729 Standardized Test Practice, 74, 120, 174, 224, 286, 354, 414, 456, 508, 570, 620, 696, 736 Vocabulary Review, 69, 116, 169, 219, 281, 348, 408, 451, 503, 564, 615, 690, 732

Index

ratios of similar solids, 610 scatter plots, 62 scientific notation, 214 slope, 385 square roots, 464 subtracting integers, 93 subtracting like fractions, 251 subtracting unlike fractions, 264 subtraction property of equality, 136 subtraction property of inequalities, 435 surface area of a cone, 604 surface area of cylinders, 598 surface area of rectangular prisms, 597 types of angles acute, 477 obtuse, 477 right, 477 straight, 477 types of relationships, 62 volume of a cone, 590 volume of a cylinder, 585 volume of a prism, 583 volume of a pyramid, 589 volume of a sphere, 591

L Labs. See Algebra Labs, Geometry Labs, Graphing Calculator Labs, Spreadsheet Labs Lateral area, 597 cylinders, 596 rectangular prism, 596 Lateral faces, 597 Lateral surface area, 596, 597, 598, 599 LCD. See Least common denominator LCM. See Least common multiple Least common denominator (LCD), 258, 263 Least common multiple (LCM), 257 monomials, 258 Leaves, 626 Legs, 485 Less than (), 79 inequalities, 435 less than (

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