California Math Triumphs VOL 4B THE CORE PROCESSES (CALIFORNIA MATH TRIUMPHS VOL 4B)

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Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

Photo Credits Cover Thinkstock/Alamy; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies; v (5 10 13 14) File Photo; ix Digital Vision/PunchStock; vii Ian Grant/Alamy; viii Medioimages/PunchStock; x 1 CORBIS; 2–3 Ray Kachatorian/Getty Images; 17 Martin Harvey/Peter Arnold, Inc.; 40–41 John Giustina/Getty Images; 78–79 Stockbyte/Getty Images; 80 Stockdisc/Getty Images; 85 cre8ive studios/iStock; 91 GABRIEL BOUYS/AFP/Getty Images; 92 Michael Newman/PhotoEdit; 99 James Leynse/Corbis

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. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878210 MHID: 0-07-878210-4 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07

California Math Triumphs Volume 4B

California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii

Authors and Consultants AUTHORS

Frances Basich Whitney

Kathleen M. Brown

Dixie Dawson

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Math Curriculum Leader Long Beach Unified Long Beach, California

Philip Gonsalves

Robyn Silbey

Kathy Vielhaber

Mathematics Coordinator Alameda County Office of Education Hayward, California

Math Specialist Montgomery County Public Schools Gaithersburg, Maryland

Mathematics Consultant St. Louis, Missouri

Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California

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

CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

iv

Instructional Planning and Support

ELL Support and Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant League City, Texas

Author/Educational Consultant Alford, Florida


CONTRIBUTING AUTHORS

California Advisory Board CALIFORNIA ADVISORY BOARD


Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.

Linda Anderson

Cheryl L. Avalos

Bonnie Awes

Kathleen M. Brown

4th/5th Grade Teacher Oliveira Elementary School, Fremont, California

Mathematics Consultant Retired Teacher Hacienda Heights, California

Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Carol Cronk

Audrey M. Day

Jill Fetters

Grant A. Fraser, Ph.D.

Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California

Classroom Teacher Rosa Parks Elementary School San Diego, California

Math Teacher Tevis Jr. High School Bakersfield, California

Professor of Mathematics California State University, Los Angeles Los Angeles, California

Eric Kimmel

Donna M. Kopenski, Ed.D.

Michael A. Pease

Chuck Podhorsky, Ph.D.

Mathematics Department Chair Frontier High School Bakersfield, California

Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

Instructional Math Coach Aspire Public Schools Oakland, California

Math Director City Heights Educational Collaborative San Diego, California

Arthur K. Wayman, Ph.D.

Frances Basich Whitney

Mario Borrayo

Melissa Bray

Professor Emeritus California State University, Long Beach Long Beach, California

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA

Teacher Rosa Parks Elementary San Diego, California

K–8 Math Resource Teacher Modesto City Schools Modesto, California

v

California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire

Math Teacher California College Preparatory Academy Oakland, California

6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California

Eppie Leamy Chung

Monica S. Patterson

Teacher Modesto City Schools Modesto, California

Educator Aspire Public Schools Modesto, California

Judy Descoteaux

Rechelle Pearlman

Mathematics Teacher Thornton Junior High School Fremont, California

4th Grade Teacher Wanda Hirsch Elementary School Tracy, California

Paul J. Fogarty

Armida Picon

Mathematics Lead Aspire Public Schools Modesto, California

5th Grade Teacher Mineral King School Visalia, California

Lisa Majarian

Anthony J. Solina

Classroom Teacher Cottonwood Creek Elementary Visalia, California

Lead Educator Aspire Public Schools Stockton, California

vi


Bobbi Anne Barnowsky

Volume 4A

The Core Processes of Mathematics

Chapter

Operations and Equality

1

1-1 Addition and Subtraction Operations ...........................4. 3AF1.0

1-2 Multiplication and Division Operations .....................11 3AF1.0

Progress Check 1 .............................................................18 1-3 Equality ............................................................................19 4AF2.1, 4AF2.2

1-4 Operations with Unknown Quantities ........................25 4AF1.1

Progress Check 2 .............................................................31 Assessment

Chapters 1 and 2 are contained in Volume 4A. Chapters 3, 4, and 5 are contained in Volume 4B.

Standards Addressed in This Chapter 3AF1.0 Students select appropriate symbols, operations, and properties to represent, describe, simplify, and solve simple number relationships. 4AF1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an understanding and the use of the concept of a variable). 4AF2.1 Know and understand that equals added to equals are equal. 4AF2.2 Know and understand that equals multiplied by equals are equal.

Study Guide .....................................................................32 Chapter Test .....................................................................36 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Standards Practice ...................................................38 Point Lobos State Park

vii

Contents Chapter

Math Fundamentals

2

Standards Addressed in This Chapter 2-1 Commutative Property ..................................................42 2AF1.1, 3AF1.5

2-2 Associative Property .......................................................49 2AF1.1, 3AF1.5

Progress Check 1 .............................................................56 2-3 Distributive Property ......................................................57 5AF1.3

2-4 Order of Operations ....................................................... 63 7AF1.2

Progress Check 2 .............................................................69

2AF1.1 Use the commutative and associative rules to simplify mental calculations and to check results. 3AF1.5 Recognize and use the commutative and associative properties of multiplication (e.g., if 5 × 7 = 35, then what is 7 × 5? and if 5 × 7 × 3 = 105, then what is 7 × 3 × 5?). 5AF1.3 Know and use the distributive property in equations and expressions with variables. 7AF1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2.

Assessment Study Guide .....................................................................70 Chapter Test .....................................................................74 Standards Practice ...................................................76

Mustard plants in Napa Valley Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

viii

Contents Chapter

Math Expressions

3

3-1 Algebraic Expressions .....................................................4 7AF1.1

3-2 Translating Verbal Phrases into Mathematical Symbols ...................................................11 5AF1.2, 7AF1.1

Progress Check 1 .............................................................20 3-3 Simplify Expressions ......................................................21 7AF1.3

3-4 Evaluate Variable Expressions ..................................... 29 5AF1.2, 6AF1.2, 7AF1.3

Progress Check 2 .............................................................35 Assessment Study Guide .....................................................................36


Chapter Test .....................................................................40

Chapters 1 and 2 are contained in Volume 4A. Chapters 3, 4, and 5 are contained in Volume 4B.

Standards Addressed in This Chapter 5AF1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 6AF1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). 7AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used.

Standards Practice ...................................................42

Burney Falls

ix

Contents Chapter

Linear Equations

4

Standards Addressed in This Chapter 4-1 Translate Word Phrases into Equations.......................46 7AF1.1

4-2 Solve Equations Using Addition and Subtraction ...............................................................53 4AF2.1, 7AF4.0

Progress Check 1.............................................................60 4-3 Solve Equations Using Multiplication and Division ....................................................................61 4AF2.2, 7AF4.0

4-4 Multi-Step Equations .................................................... 67 7AF4.0

Progress Check 2.............................................................74 4-5 Symbolic Computation ..................................................75

4AF2.1 Know and understand that equals added to equals are equal. 4AF2.2 Know and understand that equals multiplied by equals are equal. 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). 7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

7NS1.3

Assessment

Chapter Test ....................................................................86 Standards Practice...................................................88

x

Alabama Hills, Owens Valley


Study Guide ....................................................................82

Contents Chapter

Inequalities

5

Standards Addressed in This Chapter 5-1 Translate Phrases into Inequalities ..............................92 7AF1.1

5-2 Solve Inequalities Using Addition and Subtraction ...............................................................99 7AF4.0

Progress Check 1 ...........................................................106 5-3 Solve Inequalities using Multiplication and Division ...................................................................107 7AF4.0

5-4 Solve Multi-Step Inequalities ..................................... 113

7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). 7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

7AF4.0, 7NS1.3

Progress Check 2 ...........................................................120 5-5 Graph Inequalities on a Number Line .......................121 7AF4.0


Assessment Study Guide ...................................................................128 Chapter Test ...................................................................132 Standards Practice .................................................134

California poppies and gazanias

1

Chapter

3

Math Expressions The Transamerica Pyramid in San Francisco is 248 meters lower than the Taipei 101 building in Taiwan. To find the height of the Transamerica Pyramid, let x equal the height of the Taipei 101. The expression x – 248 represents the height of the Transamerica Pyramid.

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 3 Math Expressions

Ray Kachatorian/Getty Images

STEP

STEP

1 Quiz

Are you ready for Chapter 3? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 3. Review these skills and compare them with what you’ll learn in this chapter.

What You Know You know how to translate certain phrases into math symbols. Example: I have 5 more dollars than Teresa. So, the money you have is: Teresa’s amount of money + 5. TRY IT! 1

You have 5 times more songs downloaded than I do. So, the songs that you have downloaded are: 5 × my songs.


2

What You Will Learn Lessons 3-1 and 3-2 Some words usually indicate certain operations, such as: Increased by

Addition

+

Decreased by

Subtraction

-

Times

Multiplication × or ·

Divided by

Division

÷

When you do not know a number, you use a variable, such as x, in place of that number.

Harold ate 2 of my brownies. So, the brownies I have left are: the brownies I had - 2 .

You know how to group items that are alike. Example:

Lesson 3-3 Like terms can be grouped or combined to simplify expressions. Simplify 1a + 1a + 1p + 2p.

= 2 apples + 3 pears

2a

+

3p

3

Lesson

3-1 Algebraic Expressions 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description.

KEY Concept Vocabulary

Examples

variable

a, b, x

constant

-5, 8, 10 8 2x, 7, 5x2, z

VOCABULARY

_

term coefficient algebraic expression

5 is the coefficient in 5y 3z + 4, 7x

2

_x + -2

constant a value that does not change

constant

term each of the quantities connected by plus or minus signs in an algebraic expression

3

variable coefficient

term

coefficient a number that is multiplied by the variable in a term

term

expression

Example 1 Name the variable, constant, and coefficient in 8t + 2. 1. The variable is the letter t. 2. The constant is the number 2. 3. The coefficient is the number 8 because it is multiplied by the variable t.


algebraic expression a combination of numbers, variables, and at least one operation

YOUR TURN! Name the variable, constant, and coefficient in 7x - 5. 1. What is the variable? The variable is the letter 2. What is the constant? The constant is the number

x

.

-5 .

3. What is the coefficient? The coefficient is the number multiplied by the variable. It is the number 7 .


Terms are separated by + or – signs. Constants are terms that do not have a variable. A constant includes the sign that is written before it.

4

variable a letter or symbol used to represent an unknown quantity (Lesson 1-4, p. 25)

Example 2

YOUR TURN!

Write an algebraic expression that has the constant term 9 and the variable h. 1. Write the term with the variable. Follow it with a + or - sign. h + 2. Write the constant term before or after the variable and operation symbol. h + 9 There is more than one expression that has a constant term 9 and a variable h.

Write an algebraic expression that has the constant term 5 and the variable a. 1. Write the term with the variable. Follow or precede it with a + or - sign.

Sample answer: + 4a 2. Write the constant term before or after the variable and operation symbol.

Sample answer: 5 + 4a Write three other expressions that have a constant term 5 and a variable a.

5h + 9 2 +9 h 9 + 2h 9 -h 9 - 5h

_

Sample answers:

a + 5, 3/a + 5, 5 + 2a, 5 - 2a, 5 - a.

Who is Correct?


Write an algebraic expression that has the constant term 2, a variable n, and the coefficient 8.

Hailey

Pablo

Carson

2 + 8n

8 + 2n

8n2 - 2

Circle correct answer(s). Cross out incorrect answer(s). Remember that constant terms do not have a variable. Their values do not change.

Guided Practice 1

Write three different algebraic expressions that have the constant term 1 and the variable h.

_

Answers will vary. Sample answer: h + 1, 5h + 1, 7 + 1, 1 - h. h 2

Write three different algebraic expressions that have the constant term 5 and the variable a.

_

Answers will vary. Sample answer: a + 5, 8a + 5, a + 5, 5 - a. 9

GO ON

Lesson 3-1 Algebraic Expressions

5

Step by Step Practice 3

k. Name the variable, constant, and coefficient in –12 + __ 6 Step 1 Determine the variable. The variable is the letter

k

.

Step 2 Determine the constant. The constant is the number -12 . Step 3 Determine the coefficient. k can be The coefficient is the number mutiplied by the variable. The term __ 6 1 . 1 k. The coefficient is written as __ 6 6

_

Name the variable, constant, and coefficient in each expression. 4

9n + 5

5

variable:

n

variable:

d

constant:

5

constant:

12

9

coefficient: 6

-2x + 9

constant:

9 -2

–2 × 6n

n

constant:

-2

10

3w - 5

constant:

-6

_1

⎛ 1⎞ 7x + ⎪– __ ⎥ ⎝ 7⎠

3

x

_

-1 7 coefficient: 7 constant:

11

4h + 7.9 ___ 5

variable:

w

variable:

constant:

-5

constant: 7.9

coefficient: 6

v

variable:

6

coefficient:

variable:

coefficient: 9

variable:

v -6 + __ 3

3


coefficient:

h

_4 5


x

1

coefficient: 7

variable:

coefficient: 8

d + 12

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram. Look for a pattern. Guess and check. ✓ Use logical reasoning. Solve a simpler problem.

Solve. 12

HEIGHT Doris was 42 inches tall on her birthday. She grew the next year. Write an expression for Doris’s new height in inches. Understand

Read the problem. Write what you know. Doris’s height is expressed by the constant term 42 . Pick a strategy. One strategy is to use logical reasoning.

Plan

Determine the operation that should be used in the expression. addition The phrase “grew” indicates Doris’s height should be greater after one year. Select a variable to represent the number of inches that Doris grew. Sample answer: h

.

Write the expression.

Solve

42 + h Does the expression make sense? Doris’s height will be greater than the height on her birthday.


Check

13

14

MONEY Naveen saved $86. He spent some of the money on comic books. Write an expression for the amount of money Naveen has 86 - m left. Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

PETS Akiko feeds her fish x teaspoons of fish food each day. Write an expression for the amount of food, in teaspoons, Akiko feeds her fish in 7 days.

7x GO ON Lesson 3-1 Algebraic Expressions

7

EATING OUT Dustin took his friend out for lunch. He ordered the soup and salad special. His friend ordered a turkey sandwich. They both ordered regular milkshakes. 15

Dustin could not read the cost of a regular milkshake. Write an expression for the total price of the meal.

'JOFS%JOFS 4PVQBOE4BMBE 4QFDJBM

4BOEXJDIFT 5VSLFZPS)BN

F BL F 4IIBLLF 44IB

.JMLTIBLFT

Sample answer: 6 + 5 + 2m or 6 + 5 + m + m 16

-BSHF 3FHVMBS

Write an expression for the total cost of t sandwiches and two large milkshakes.

Sample answer: 5t + 2(4) List key words that tell you which operation to use when writing an expression.

17

Answers may vary. Sample answer: more than means to add, less than means to subtract, times means to multiply, divided by means to divide

Skills, Concepts, and Problem Solving 18

_

Answers will vary. Sample answer: –15 + d, –15 + 7d, –15 + d , d - 15. 2 19

Write three different expressions that have the constant term 6 and the variable q.

_

Answers will vary. Sample answer: q + 6, 3q + 6, 1 + 6, 6 - q . q Name the variable, constant, and coefficient in each expression. 20

–17 + z variable:

21

z

constant: -17 coefficient:

8

1


16p + 5 variable:

p

constant:

5

coefficient:

16


Write three different expressions that have the constant term –15 and the variable d.

22

3h + 7

23

variable:

h

variable:

s

constant:

7

constant:

–4

3

coefficient: 24

25

3k + __ 2 ___ 4

5

variable:

y

variable:

constant:

-1

constant:

1

k 2 5

_

coefficient:

-b + 4 ___ 5

1

coefficient:

y-1

coefficient: 26

–4 + s

27

_3 4

12 - 7z

variable:

b

variable:

z

constant:

4

constant:

12

_1

coefficient: -

coefficient:

5

-7

Solve.


28

PRODUCE Akshi’s orange tree has 72 oranges. She gave an equal number of oranges to each of her cousins. Write an expression for the number of oranges Akshi gave each cousin.

_

Sample answer: 72 ÷ c or 72 c

29

MOVIES Lamar had 48 DVDs in his collection. He bought more DVDs. Write an expression for the total number of DVDs in Lamar’s collection.

Sample answer: 48 + d 30

BAKING Gavin promised to bring cupcakes for April’s party. April wants to have enough cupcakes so each party guest can eat two. Write an expression for the number of cupcakes Gavin must bring.

Sample answer: 2p 31

MOVIES Carlos and Lucas (both 14 years old) are taking their younger sisters (who are under 13) to a movie. They have x younger sisters. Write an expression for the total cost of admission.

Sample answer: 2 · 8 + 5x or 8 + 8 + 5x

.PWJF5IFBUFS"ENJTTJPO "EVMUT

$IJMESFO VOEFS

GO ON Lesson 3-1 Algebraic Expressions

9

Vocabulary Check sentence.

Write the vocabulary word that completes each

32

variable A(n) have different values.

is a symbol, usually a letter, that can

33

coefficient A(n) variable in a term.

is a number that is multiplied by the

34

A value that does not change is a

35

A(n) algebraic expression is a combination of numbers, variables, and at least one operation.

36

Writing in Math Explain.

constant

.

What is the coefficient in the expression y + 14?

1; the term y can also be written as 1y.

Spiral Review Solve. 37

(Lesson 2-4, p. xx)

SHOPPING Zola bought 4 packs of pencils with 5 pencils each. She gave 4 pencils to Clara. Then Zola bought 2 packs of pencils with 8 pencils each. How many pencils does Zola have now?

32

SHOPPING Janet bought 5 packs of erasers with 10 erasers each. Jorge gave Janet 2 erasers. Then Janet gave 9 erasers to each of 3 friends. How many erasers does Janet have now? 25

Use the Distributive Property and a model to find each product. Show your work. (Lesson 2-3, p. xx) 39

15 · 4 =

60

40

10

13 · 2 =


26


38

Lesson

3-2 Translating Verbal Phrases

5AF1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represent a verbal description.

into Mathematical Symbols KEY Concept You can look for certain words in problems to help you determine which operations to use. Below are the most common of these words.

VOCABULARY

Addition

Subtraction

Multiplication

Division

sum

difference

product

quotient

more than

less than

times

divided by

increased by

decreased by

twice

separate into equal groups

plus

minus

constant a value that does not change (Lesson 3-1, p. 4) term each of the quantities connected by plus or minus signs in an algebraic expression (Lesson 3-1, p. 4)

coefficient a number that is multiplied by the variable in a term (Lesson 3-1, p. 4)

If none of these words are in your problem, the circumstances in the situation will help you decide which operation to use.

algebraic expression a combination of numbers, variables, and at least one operation

Example 1

(Lesson 3-1, p. 4)

1. The words “more than” tell you to use addition (+). 2. The word “seven” indicates that the constant term is 7. 3. The words “a number n” tell you to use the variable n. seven more than a number n ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩

7

+

n

YOUR TURN! Translate “a number x decreased by 5” to an expression. 1. What operation do the words “decreased subtraction by” tell you to use?

x

3. What is the variable?

The expression is 7 + n.

5

2. What is the constant term?

a number x decreased by 5

It can also be written as n + 7 because of the Commutative Property.

⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩


Translate “seven more than a number n” to an expression.

x

-

5


x-5

GO ON

Lesson 3-2 Translating Verbal Phrases into Mathematical Symbols

11

Example 2

YOUR TURN!

Amber has three times as many fish as Arturo. 1. The word “times” tells you to use multiplication (·). 2. The word “three” indicates that the constant term is 3. 3. The “same number of fish as Arturo” is a variable. Choose the letter f as the variable.

Write an expression to represent the following situation. Angelina is separating all of her books into four piles. 1. What operation does the word “separating” tell you to use?

division

2. What is the constant term? four or 4 3. What is the variable?

separating all of her books into four piles ⎫ ⎬ ⎭

b÷

4

⎫ ⎬ ⎭

·

⎧ ⎨ ⎩

⎧ ⎨ ⎩

three times as many fish 3

f

b ⎧ ⎨ ⎩

Write an expression to represent the following situation.


b÷4

The expression is 3 · f or 3f. No operation sign means multiplication.

Who is Correct?

Mitchell

Rico

Marquita

4–x

l+4

n–4

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice For each phrase, name the operation. 1

n decreased by 13

subtraction

2

the sum of a and 8

addition

3

8 divided by w

division

12



Translate “a number minus four” to an expression.

multiplication

4

twice the amount of y

5

7 more than x

addition

6

the quotient of s and 9

division

Translate each phrase to an expression. five times a number

8

multiplication (×)

operation:

none

constant term:

n

variable: ⎧ ⎨ ⎩

n

×

20 5 × n or 5n

10

_

four less than 2 times a number

14

6 ÷ n or 6 n

multiplication, addition n

a number divided by 10

_

n ÷ 10 or n 10

2n - 4

_

two times the sum of a number and 15

expression: 2(n + 15)

12

y 6× 7

n + 20 or 20 + n

variable:

expression: 3n - 11

six times the quotient of y and 7

n

+

operations:

n

6 divided by a number

n

expression:

eleven less than 3 times a number

variable:

none

twenty more than a number

⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ Copyright © by The McGraw-Hill Companies, Inc.

20

variable:

operations: subtraction, multiplication

15

constant term: coefficient:

expression:

13

addition (+)

5

5

11

operation:

coefficient:

five times a number

9

twenty more than a number

⎧ ⎨ ⎩ ⎧ ⎨ ⎩

7

three times the sum of x and 10

3(x + 10) 16

the sum of 12 divided by a number and 8

12 + 8 _ n

GO ON Lesson 3-2 Translating Verbal Phrases into Mathematical Symbols

13

Step by Step Practice Write an expression to represent the following situation. 17

Niles has four fewer magazines than Wilma. Step 1 Decide the operation to use in the expression. The word “fewer” tells you to subtract .

4

Step 2 The constant term is

.

Step 3 Choose a letter for the variable. The number of magazines Wilma has is represented by the variable term Sample answer: m Step 4 Write the expression.

.

m-4

Write an expression to represent each situation. 18

Denise separated her postcards into 3 equal stacks. Which word(s) tell you the operation to use?

separated into equal stacks What is the operation? Choose a letter for the variable. Write the expression. 19

division Sample answer: p p÷3

Joe’s puppy weighed 3 pounds more than it did last month.

20

The blueberry muffin has twice as many Calories as the bran muffin.

2c 21

I have 5 times as many songs on my player as you do.

5s 22

Each box contains the same number of bagels.

This is the total number of bagels. 14


2b


p+3


Problem-Solving Strategies ✓ Draw a diagram.

Solve. 23

GEOMETRY The perimeter of a triangle is the sum of the lengths of its sides. One side length is 12 feet. Another side length is 9 feet. Write an expression for the perimeter of the triangle. Understand

Look for a pattern. Guess and check. Solve a simpler problem. Work backward.

Read the problem. Write what you know. The perimeter of a triangle is the sum of the lengths of 3 sides. One side is

12

feet.

One side is

9

feet.

One side is

unknown

.

Pick a strategy. One strategy is to draw a diagram.

Plan

Sketch the triangle. Choose a variable for the length of the unknown side. The operation to use is

.

Write the expression for the perimeter.

Solve

12 Copyright © by The McGraw-Hill Companies, Inc.

addition

Y

+

9

+

x

Simplify the expression.

21 + x Does the expression make sense? The perimeter of the triangle should be greater than the length of the two known sides combined.

Check

24

FITNESS Tyrus runs 4 miles each day. Write an expression for the number of miles Tyrus runs in d days. Check off each step. 4d

✔

Understand

✔

Plan

✔

Solve

✔

Check


15

25

RIDES Ruben’s dad is taking Ruben and his friends on a hot-air balloon ride. The weight limit is 900 pounds. Ruben’s dad weighs 250 pounds. Ruben weighs 74 pounds. Nathaniel weighs 87 pounds. Belinda weighs b pounds. Write an expression for their combined weights in pounds. 250 + 74 + 87 + b

26

FAMILY Paul is 23 years younger than his mother. Write an a - 23 expression for Paul’s age. Do the expressions 5 – r and r – 5 have the same value? Explain and give an example.

27

No; the order in which you subtract affects the answer. The Commutative Property does not hold true for subtraction. Let r = 1. 5 - 1 = 4 and 1 - 5 = -4

Skills, Concepts, and Problem Solving For each phrase, name the operation. 28

the product of h and 9

29

the quotient of e and 5

multiplication 30

division

a number minus 6

31

a number plus 10

subtraction 13 less than a number

33

the sum of a number and 7

subtraction 34


32

addition addition

24 split into 3 equal groups

35

division

4 groups of 5 each

multiplication

Translate each phrase to an expression.

-

expression: 38

x

37

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

45

minus

x 45 - x

expression: 16

n 14 × n or 14n


+

expression: 39

61 n + 61

b divided by 7 ⎫ ⎬ ⎭ ⎫ ⎬ ⎭ ⎫ ⎬ ⎭

⎫ ⎬ ⎭ ⎫ ⎬ ⎭ ⎫ ⎬ ⎭

×

61

n

14 groups of a number

14

a number plus

⎫ ⎬ ⎭

45

⎫ ⎬ ⎭ ⎫ ⎬ ⎭

36

b

÷

expression:

7

_

b ÷ 7 or b 7

Translate each phrase to an expression. 40

7 divided by a number n

41

_

7 ÷ n or 7 n 42

2 × c or 2c

3 less than m

43

m-3 44

the quotient of 9 and y

45

_

the sum of w and 50

47

20 groups of x each

20x

14 more than h

49

36 split into x equal groups

_

36 ÷ x or 36 x

14 + h 50

12 decreased by w

12 - w

w + 50 48

e increased by 32

e + 32

9 ÷ y or 9 y 46

2 times c

n less than 45

51

45 - n

62 plus a number

62 + n

Write an expression to represent each situation.


52

FOOD A party-sized sandwich feeds 18 students. Write an expression for the number of sandwiches needed for s students.

s ÷ 18 53

ANIMALS Read the photo caption to the right. Write an expression for the speed of the antelope.

c-9 54

GAMES Sabrina scored 63 points less than Aailiyah playing a board game. Aailiyah scored p points. Write an expression for the number of points Sabrina scored.

p - 63 55

ANIMALS A cheetah

MONEY Xavier has $18, Anton has $24, and Len has d dollars. Write an expression for the sum of money Xavier, Anton, and Len have altogether.

18 + 24 + d

can run 9 miles per hour faster than an antelope.


Martin Harvey/Peter Arnold, Inc.

17

56

GEOMETRY The area of a rectangle is 6 times the area of a square. The area of the square is x square units. Write an expression for the area of the rectangle.

6 × x or 6x 57

SCHOOL Last year the number of students at Central High was 425 students. This year the number increased. Write an expression to show the increase in students from last year to this year.

425 + n 58

REAL WORLD Write three real-world examples of variables and constants in the table. Variables

Constants

temperature of water

temperature of water’s boiling point

Answers will vary.

Answers will vary.

Sample answers: age, height, length of a movie

Sample answers: the year you were born, days in a year, length of a wall

59

20 - 5

the quotient of 20 and 5

60

5 · 20

the sum of 5 and 20

61

20 ÷ 5

5 less than 20

62

5 + 20

20 multiplied by 5

63

20 · 5

5 more than 20

64

20 + 5

the product of 5 and 20



Terms

65

are each of the quantities connected by plus or minus signs in an algebraic expression.

18



In Exercises 59–64, match the expressions on the left with the phrases on the right.

change

66

A constant is a value that does not

67

variables A combination of numbers, operation is an algebraic expression.

68

. , and at least one

Writing in Math How do you know when to use a variable when writing an expression?

Answers will vary. Sample answer: Use a variable when one of the terms can have different values (or the value is unknown).

Spiral Review Name the variable, constant, and coefficient in each expression. 69

14 + z

70

(Lesson 3-1, p. 4)

2x + 25

variable:

z

variable:

x

constant:

14

constant:

25

coefficient:

1

coefficient:

2

Solve. Copyright © by The McGraw-Hill Companies, Inc.

71

FOOTBALL A football team scored 4 touchdowns (each worth 6 points) and 2 extra points (each worth 1 point). How many points did the football team score? (Lesson 2-3, p. 57)

26 72

STAMP COLLECTING Hogan has 140 stamps in his stamp collection. He put the stamps in 2 albums. Each album has the same number of stamps. How many stamps are in each album? (Lesson 1-2, p. 11)

70

Find the value of the variable in each equation. 73

x - 5 = 13 x=

18

(Lesson 1-4, p. 25)

74

k + 14 = 22 k=

8

Lesson 3-2 Translating Verbal Phrases into Mathematical Symbols

19

Chapter

Progress Check 1

3 1

(Lessons 3-1 and 3-2)

Write three different expressions that have the constant term 7 and the variable f. 7AF1.1

_

Answers will vary. Sample answer: f + 7, 2f + 7, f + 7, 7 - f. 8

For each phrase, name the operation. 7AF1.1 2

the sum of 4 and y

addition

3

multiplication

5 times p

Name the variable, constant, and coefficient in each expression. 7AF1.1 w 4 __ - 8 5 8k + 6 7 variable:

w

-8 1 coefficient: 7 constant:

_

variable:

k

constant:

6

coefficient:

8

Translate each phrase to an expression. 7AF1.1 6

a number n divided by 9

_

7

13 less than h

n ÷ 9 or n 9

h - 13

8

Lizzie has 6 times as many pencils as Betty.

9

Several stamps – 5 stamps

6p n-5

Solve. 7AF1.1, 5AF1.2 10

BUSINESS Trina made 47 glasses of lemonade. She sold x glasses of lemonade. Write an expression to find how many glasses of lemonade Trina has left.

47 - x 11

CRAFTS Daja made 28 bracelets. She separated the bracelets into p piles. Write an expression to find the number of bracelets in each pile.

_

28 ÷ p or 28 p 20



Write an expression to represent each situation. 5AF1.2

Lesson

3-3 Simplify Expressions 7AF1.3 Simplify numerical expressions by applying properties of rational numbers and justify the process used.

KEY Concept Recall that terms can be a number, a variable, or a combination of numbers and variables. Terms can be positive or negative, and they can have exponents. Examples of Like Terms 6 and 13 -x and 3x

term each of the quantities connected by plus or minus signs in an algebraic expression

Why They Are Like Terms They are both constants. Both contain the variable x and the exponent 1.

2x2 and 5x2

VOCABULARY

Remember that when no exponent is shown, it is really a 1.

(Lesson 3-1, p. 4)

like terms terms that have the same variables to the same powers

Both contain the variable x and the exponent 2.

simplify to combine like terms

Like terms can be combined, or simplified . -x + 3x = -1x + 3x = -2x

Add the coefficients of terms with like variables. -1 + 3 = -2


Unlike terms cannot be combined. For example, in 2y + 4x the terms cannot be combined because the variables are different.

When grouping and simplifying like terms, use the Associative and Commutative Properties of Addition and the Distributive Property of Multiplication.

Example 1

YOUR TURN!

Simplify 3x + 2y + 2x by using a model. 1. Let

represent x and

represent y.

2. Draw a model to represent the expression. Y

represent c and

Y

Z

3. Write the result. 3x + 2y + 2x = 5x + 2y

represent t.

2. Draw a model to represent the expression.

Y

1. Let

Z

Simplify c + 3t + 2c by using a model.

c

+

3. Write the result.

3t

2c 3c + 3t

+

GO ON Lesson 3-3 Simplify Expressions

21

Example 1 Simplify 12 + 9r + 13 + 17r + 61.

Remember that the Commutative Property states that the order in which two numbers are added or multiplied does not change the answer.

1. Name the like terms. Constant terms: 12, 13, 61 r terms: 9r and 17r 2. Rearrange to group the constant terms together and the r terms together. 12 + 9r + 13 + 17r + 61 = 12 + 13 + 9r + 17r + 61

Use the Commutative and Associative Properties.

= 12 + 13 + 9r + 61 + 17r = 12 + 13 + 61 + 9r + 17r 3. Add the like terms. = (12 + 13 + 61) + 9r + 17r

Add the constant terms.

= 86 + 26r

Add the variable terms.

YOUR TURN! Simplify 13p + 13 - 5p + 5. 1. Find the like terms. What are the constant terms? What are the p terms?

13 and 5 13p and -5p

13p – 5p + 13 + 5 3. Add the constant terms. Subtract the variable terms.

8p + 18

Who is Correct? Simplify 14u2 + 6 + 19u + 4 - 3u2 - 8u.

Beatrice

Ronin

Ana

22u + 10

11u + 6 + 11u + 4

11u + 10 + 11u

2

2

2

Circle correct answer(s). Cross out incorrect answer(s). 22



2. Rearrange to group the constant terms together and the p terms together using the Commutative and Associative Properties.

Guided Practice Name the like terms in each expression. 1

3

5t - 8 + 6t + 10

2

9 - c + 4 + 7c

5t and 6t

-c and 7c

-8 and 10

9 and 4

3d2 + 12 – d + 7d2 – 8 + 4d

4

-6n - 2n2 + n + 5 + 8n2 - 3

3d 2 and 7d 2

-2n 2 and 8n 2

-d and 4d

-6n and n

12 and -8

5 and -3


Simplify 23 + 6u2 - 2u - 2 + 18u. Step 1 Combine the constant terms. 23 + 6u2 - 2u - 2 + 18u = 6u2 - 2u + 18u + (

-

2

)

21

= 6u2 - 2u + 18u + Copyright © by The McGraw-Hill Companies, Inc.

23

Step 2 Combine the u terms. = 6u2 + (

-2u

= 6u2 +

16u

+ +

18u

)+

21

21

Step 3 Combine the u2 terms. There is only one term with variable u and exponent 2. So there are no like terms to combine. Step 4 The simplified expression is

6u 2 + 16u + 21

.


23

Simplify each expression. Show your work. 6

7p + 6 - 2p + 5

6

7p + 6 - 2p + 5 = 7p - 2p + (

2p

-

5p

=

7

7p

)+

11

-

15

=( =

25

5x 17x

+ +

12x

)+

3y - 5y2 + 10 + 12y2 - 6y + 9

10

9

3z 2 + 5z + 8 24


8a - 5 + 6a2 - 7a + 10 – 4a2

2a 2 + a + 5 11

7y 2 - 3y + 19 -2 + 11z - z2 + 4z2 - 6z + 10

10

2b3 + b2 + 5b - 4b3 - 4b2 – 8b

-2b 3 - 3b 2 - 3b 13

20c + c2 - 8 + c2 – 10c - 1

2c 2 + 10c - 9


13 + 12k - 2k2 - 6k + 5k2 + 28

)

10

3k 2 + 6k + 41

12

Combine the variable terms.

25 + 5x - 15 + 12x

= 5x + 12x +

10

Combine the constant terms.

11

+

25 + 5x – 15 + 12x = 5x + 12x + (

8

)

11

= 7p - 2p + =(

5

+


Problem-Solving Strategies Act it out. Look for a pattern. Guess and check. ✓ Use a model. Solve a simpler problem. Work backward.

Solve. 14

MONEY Rashin bought 4 hair clips for x dollars each. She also bought shampoo for $5 and a hairbrush for $4. In the checkout lane, she decided to buy 2 ponytail holders that each cost the same as a hair clip. The expression 4x + 5 + 4 + 2x represents the total cost. Simplify the expression to see how much Rashin spent. Understand

Read the problem. Write what you know. The expression that needs to be simplified is

4x + 5 + 4 + 2x Plan

Pick a strategy. One strategy is to use a model.

Solve

You can use a model to find the answer. Let

represent x and

4x hair clips Copyright © by The McGraw-Hill Companies, Inc.

.

represent the constant terms.

+

+

5 shampoo

4

+ 2x =

hairbrush

6x + 9

ponytail holders

Combine the constant terms and the variable terms. 4x + 5 + 4 + 2x = (4x + = Rashin spent Check

6x

2x +

)+(

5

+ 4)

9

6x + 9 dollars.

You can circle one set of like terms and box in another set of like terms to check your answer. 4x + 5 + 4 + 2x


25

15

16

MOWING Over the summer, Zakir, Deltric, and Rafael mowed yards. Each day, Zakir mowed 2 yards, Deltric mowed 3 yards, and Rafael mowed 1 yard. The expression 2d + 3d + d represents the number of yards they mowed in d days. Simplify the expression. 6d Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

AGES Laurita is n years old. Elisa is 2 years older than Laurita. Esteban is 3 years older than Elisa. Represent the sum of their ages with the expression n + (n + 2) + (n + 2 + 3). Simplify the expression.

3n + 7 17

Are 5h2 and -9h2 like terms? Explain.

Yes; both terms have the same variable and same exponent.

Skills, Concepts, and Problem Solving 18

20

22

26

7s - s2 + 5s + 9s2

19

15 + 4f - 2f - 8

-s 2 and 9s 2

4f and -2f

7s and 5s

15 and -8

28x + 17 - 11x + 3

21

y - 2y2 + 6y + 11y2

28x and -11x

-2y 2 and 11y 2

17 and 3

y and 6y

13m - m2 - 9m + 7 - 2m2 + 9

23

5q - 10 + 3q2 + 3q - q2 + 25

-m 2 and -2m 2

3q 2 and -q 2

13m and -9m

5q and 3q

7 and 9

-10 and 25



Name the like terms in each expression.

Simplify each expression. Show your work. 24

4g + 8 + 6g + 18g2 - 10g2 - 13

25

8g 2 + 10g - 5 26

3y - 4y2 + y2 - y + 5 + 7y2

11b 2 - 2b + 10 27

4y 2 + 2y + 5 28

3k + 14 - 8k + 2k2 - 3

5n3 - 2n2 + 10 - n3 - 4n2

16z + 10 - 9z2 - 5z – 3 + z

-9z 2 + 12z + 7 29

2k 2 - 5k + 11 30

9b2 - 6b + 2b2 - 2 + 4b + 12

6p + 17 - 10p - 4 + p2

p 2 - 4p + 13 31

4n 3 - 6n 2+ 10

15 - 7c2 + 22 + 3c - 8 + 5c2

-2c 2 + 3c + 29


Solve. 32

MONEY Mrs. Clark gives her four children an allowance each week. Jodi gets n dollars. Brooke gets twice as much as Jodi. Mark gets $3 less than Brooke. Katrina gets twice as much as Mark. The expression n + 2n + (2n − 3) + 2(2n − 3) represents the total of the four allowances. Simplify the expression. (Hint: Use the Distributive Property.)

33

9n - 9

SCHOOL In Mrs. Garcia’s class, students can earn points for free books. Mrs. Garcia’s bulletin board shows how many points a student can earn. Sergio earned some As and some Bs. He had perfect attendance for the same number of weeks as he earned As. The expression 20a + 15b + 2(20a) represents the number of points that Sergio earned. Simplify the expression.

1OINTS:OU$AN&ARN "POINTS #POINTS $POINTS 1ERFECT"TTENDANCEINAWEEK 5WICEASMANYPOINTSASAN"

60a + 15b GO ON Lesson 3-3 Simplify Expressions

27

Vocabulary Check sentence. 34


Terms that have the same variables to the same powers are called

like terms simplify

.

35

To

means to combine like terms.

36

Writing in Math Explain how to combine the like terms in the expression 7x + 5 + 4x using the Associative Property and the Commutative Property.

Answers will vary. Sample answer: Use the Commutative Property to switch the addends: 7x + 4x + 5. Use the Associative Property to group like terms: (7x + 4x) + 5. Then add the like terms to get 11x + 5.

Spiral Review Name the variable, constant, and coefficient in each expression. 37

6r + 5

38

a-2

variable:

r

variable:

a

constant:

5

constant:

-2

6

Find the value of each expression. 10 · 2 - 6 + 22 - (4 ÷ 2)

16 Solve.

coefficient:

1

(Lesson 2-4, p. 63)

40

7 + 32 · 6 ÷ (1 + 2)

25

(Lesson 2-2, p. 49)

41

FOOD Julio cooked 34 hot dogs. He sold 18 hot dogs. Julio then cooked 6 more hot dogs. How many hot dogs does Julio have 22 ready to sell?

42

SHOPPING Dawn bought 3 packages of paper clips. There were 4 boxes of paper clips in each package. There were 50 paper clips in each box. How many paper clips did Dawn buy? 600

28



coefficient:

39

(Lesson 3-1, p. 4)

Lesson

3-4 Evaluate Variable Expressions KEY Concept To evaluate an algebraic expression , substitute a value for a variable. Then perform the operations.

5AF1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 6AF1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. 7AF1.3 Simplify numerical expressions by applying properties of rational numbers and justify the process used.

VOCABULARY evaluate to find the value of an algebraic expression by replacing variables with numbers algebraic expression

(Lesson

3-1, p. 4)

Remember to use the order of operations after substituting, or replacing, the variables with numbers.

Example 1

a combination of numbers, variables, and at least one operation value the amount of a number order of operations (Lesson 2-4, p. 63)

Evaluate

+ (9 - 2) when

1. Replace

= 3.

with 3 in the expression.


+ (9 - 2) = 3 + (9 - 2) 2. Simplify. Follow the order of operations. 3 + (9 - 2) = 3 + 7 = 10

Simplify grouping symbols. Add.

rules that tell what order to use when evaluating expressions (1) Simplify grouping symbols. (2) Simplify exponents. (3) Multiply and divide in order from left to right. (4) Add and subtract in order from left to right.

YOUR TURN! +4

= 2 and

Evaluate 2

2

1. Replace


when

= 3.

2(2)2 + 4 2. Replace


2(2)2 + 4(3) 3. Simplify. Follow the order of operations.

2(4) + 4(3) 8 + 12 = 20

GO ON Lesson 3-4 Evaluate Variable Expressions

29

Example 2

YOUR TURN! Evaluate 3y2 + x · 3 - 2 when x = 4 and y = 2.

Evaluate 4 ÷ y + x · 3 - 7 when x = 5 and y = 2.

1. Replace y with 2 and x with 4. Write the expression.

1. Replace x with 5 and y with 2 in the expression.

3(2)2 + 4 · 3 - 2

4÷y+x·3-7=4÷2+5·3-7 2. Simplify using the order of operations. 4÷2+5·3-7

2. Simplify using the order of operations. 3(2)2 + 4 · 3 - 2

Divide.

= 3(4) + 4 · 3 - 2

=2+5·3-7

Multiply.

= 12 + 12 - 2

= 2 + 15 - 7

Add.

= 22

= 17 - 7

Subtract.

= 10

Who is Correct? Evaluate the expression 12x - 5 + 2x · 2 when x = 4.

Ines

Sinclair

12x - 5 + 2x · 2 2 = 12(4) - 5 + 2(4) · = 48 - 5 + 8 · 2 = 43 + 8 · 2

12x – 5 + 2x · 2 2 = 12(4) - 5 + 2(4) · = 48 - 5 + 8 · 2 = 48 - 5 + 16

12x - 5 + 2x · 2 = (12x + 2x) - 5 · 2 = 14x - 10 = 14(4) - 10 = 46

= 59


Guided Practice Evaluate each expression when 1

5·

3

4+

5 · 6 = 30 -5·3÷3

4+6-5·3÷3=5 30


= 6. 2

18 ÷

4

2·3+

18 ÷ 6 = 3 +7

2 · 3 + 6 + 7 = 19


= 56 · 2 = 112

Robyn


Evaluate the expression 5y + 2z - x when y = 7, z = 10, and x = 4. Step 1 5y means 5 times y. Replace y with

7

in the expression.

Step 2 2z means 2 times z. Replace z with

10

in the expression.

Step 3 Write the expression with all substitutions made. Simplify using the order of operations. 5 × 7 + 2 × 10 - 4 =

35

+

20

=

55

-

4

=

51

4

-

51 .

The value of the expression is

Evaluate each expression when x = 2, y = 5, and z = 0. 6

5

7y - (5 + 1) ÷ 2 · x2 = 7(

6

=7·5-


= 35 -

3

= 35 -

12

7

4

5 - x ÷ 2 + (3 · 2)2 - 5z = 5 - (

2

) ÷ 2 + (3 · 2)2 - 5(

=5-2÷2+

6

=5-2÷2+

36

= 16 + 42 · x - 5 + (8 - y) - z

46

·4

23

=5+

8

)2

÷ 2 · 22

=7·5-6÷2·

=

2

) – (5 + 1) ÷ 2 · (

1

2

0

)

-5·0

-5·0

0

+ 36 -

40 9

5y2 - 10 ÷ 5 + 3 · 5x

153

GO ON

Lesson 3-4 Evaluate Variable Expressions

31


Draw a diagram.

Solve. 10

Problem-Solving Strategies ✓ Use an equation.

READING It takes Darnita an hour to read 24 pages in a book. The number of pages in the book is p. Use the expression p ÷ 24 to find how many hours it will take her to finish reading a book. How long will it take for Darnita to read a 312-page book? Understand

Guess and check. Act it out. Solve a simpler problem.

Read the problem. Write what you know. Darnita is reading a book that is 312 pages long. She reads

24

pages each hour.

Pick a strategy. One strategy is to use an equation.

Plan

Use h to stand for hours. Write an equation using h and the expression p ÷ 24 h = p ÷ 24 In the equation, replace p with 312 .

Solve

h = 312 ÷ 24 Simplify. h = 312 ÷ 24 h=

13

32

hours to read a

Multiply to check your division.

Check

11

13

ENTERTAINMENT Joven wants to buy an $8 CD and 3 DVDs. Use the variable expression 8 + 3d to find the total cost, where d represents the cost per DVD. Evaluate the expression for DVDs that $62 cost $18 each. Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check



It will take Darnita 312-page book.

12

FOOD The school cafeteria pays $26 per case for hamburger patties. Write an expression for the cost of c cases. Find the cost 26c, $208 of 8 cases. Does the expression 50 ÷ k - 2 have a greater value when k = 5 or k = 10? Explain.

13

k = 5. When k = 5, 50 ÷ k - 2 = 8. When k = 10, 50 ÷ k - 2 = 3.

Skills, Concepts, and Problem Solving Evaluate each expression when 14

4+3-

16

22 - 6 +

4+3-5=2 ·9

= 5. 15

16 · 3 +

17

15 ÷ 3 ·

22 - 6 + 5 · 9 = 43

16 ∙ 3 + 5 = 53 - 11 + 7

15 ÷ 3 · 5 - 11 + 7 = 21


Evaluate each expression when x = 9, y = 3, and w = 1. 18

18 ÷ x · (10 + y - x)

20

x2 ÷ y + 7 · 2 - 6w

22

y2 ÷ y + (x + y) · w

8

19

90 - x2 + 6 ÷ y · 2

13

35

21

8w + 17 · (4y - x)

59

15

23

(2x - w) + x2

98

Solve. 24

RECREATION Dewayne plays a speech game in which nouns are worth 10 points, verbs are worth 15 points, and adjectives are worth 20 points. The total score equals the expression 10n + 15v + 20a, when n represents the number of nouns, v represents the number of verbs, and a represents the number of adjectives. Find Dewayne’s score when n = 7, v = 11, and 535 a = 15.

GO ON

Lesson 3-4 Evaluate Variable Expressions

33

25

GEOMETRY The volume of a box equals the expression lwh, where l represents the length, w represents the width, and h represents the height. Evaluate the expression to find the 480 cubic units volume of the box at the right.


I

Write the vocabulary word that completes each X

value

26

The amount of a number is its

27

Finding the value of an algebraic expression by replacing variables evaluating with numbers is called the expression.

28

Writing in Math

M

.

Explain how to evaluate r - 8 · 2 when r = 30.

Answers may vary. Sample answer: Replace r with 30 in the expression r - 8 · 2, and then simplify the expression using the order of operations.

Spiral Review Solve.

(Lesson 3-2, p. 11)

CRAFTS Eva has 120 beads. She makes bracelets that have b beads. Write an expression to find the number of bracelets 120 ÷ b Eva can make.

30

SOCCER Jake made 9 fewer goals than Leon last soccer season. Leon made g goals. Write an expression to find the number of goals g-9 Jake made.

Use the Commutative and Associative Properties to fill in each blank with the correct value. Check your answer. (Lesson 2-1, p. 42) 31

2

8+2=

10

=

8

+

32

10

63

Name the operation modeled. 33

34

34

9

(Lesson 1-1, p. 4)

subtraction


9

7·9=

addition

=

·

63

7


29

Chapter

Progress Check 2

3


Name the like terms in each expression. 7AF1.3 1

2q + 9 – 5 + 9q2 – 8q + q2

2

9q 2 and q 2

8c 2 and -3c 2

2q and -8q

-5c, -4c, and 7c

9 and -5

7 and 6 = 3. 5AF1.2

Evaluate each expression when 3

8+9·

7 – 5c + 8c2 – 4c – 3c2 + 6 + 7c

-5

4

6÷

8 + 9 · 3 - 5 = 30

+5·3-2

6 ÷ 3 + 5 · 3 - 2 = 15

Simplify each expression. Show your work. 7AF1.3 5

7h - h2 + 5h - 2h + 7h2 - 10

6

-9b + 4b2 - b2 - 5 + 2b2 + 9


6h 2 + 10h - 10

5b 2 - 9b + 4

Evaluate each expression when y = 8 and x = 2. 5AF1.2, 6AF1.2 7

19 - 22 - (6 + 2y) + 3y ÷ 2

5

8

3y + x2 - 32 ÷ 4 + (4 + 2 · 3)

30

Solve. 7AF1.3, 5AF1.2 9

MONEY Cora has x dollars. Danny has twice as much money as Cora. Mary has 5 dollars less than Cora. The expression x + 2x + (x - 5) represents the total amount of money Cora, Danny, and Mary have together. Simplify the expression.

4x - 5 10

UNIFORMS The school band bought uniforms. See the cost of the uniform at right. Write an expression for the cost of u uniforms. Find the cost of 12 uniforms.

80u, $960 Lesson 3-4 Evaluate Variable Expressions

35

Chapter

Study Guide

3

Vocabulary and Concept Check algebraic expression, p. 4 coefficient, p. 4

Write the vocabulary word that completes each sentence. 1

constant, p. 4 2

evaluate, p. 29 like terms, p. 21 simplify, p. 21 term, p. 4

To combine like terms is to simplify

.

term A(n) represents each of the quantities connected by plus or minus signs in an algebraic expression. like terms

3

4x and 3x are examples of

.

4

To find the value of an algebraic expression by replacing evaluate variables with numbers is to the expression.

5

7x + 9 − 3y is an example of a(n)

6

A value that does not change is called a(n) constant .

expression

.

Label each diagram below. Write the correct vocabulary term in each blank.

constant or term

7

8

coefficient

3-1

Algebraic Expressions

Name the variable, constant, and coefficient in each expression. 9

11z - 5 variable:

z

constant:

-5

coefficient:

36

11

Chapter 3 Study Guide

(pp. 4-10)

Example 1 Name the variable, constant, and coefficient in 9y + 4. The variable is the letter y. The constant is the number 4. The coefficient is the number 9 multiplied by the variable y.


Lesson Review

10

f+8

11

-5x = 17

12

2y - 12

variable:

f

variable:

x

variable:

constant:

8

constant:

17

constant: -12

coefficient:

1

coefficient:

-5

y

coefficient:

2

3-2

Translating Verbal Phrases into Mathematical Symbols (pp. 11-19) Translate each phrase to an expression.

Example 2

n + 15

13

n increased by 15

14

the product of q and 4

15

9 less than y

16

g divided by 20

4q

y−9

_g 20

Translate “the number y decreased by eleven” to an expression. The words “decreased by” indicate subtraction (-). The word “eleven” is represented by the constant term, 11. The word “number” is represented by the variable, y. ⎧ ⎨ ⎩


y

⎫ ⎬ ⎭ ⎫ ⎬ ⎭

the number y decreased by eleven -

11

The expression is y − 11. Write an expression to represent each situation. 17

The number of fish in the aquarium is decreased by 2.

Example 3 Write an expression to represent the following situation.

Sample answer: f − 2; answers Mollie has nine more marbles than Megan. could vary based upon letter chosen as The words “more than” indicate addition (+). variable to represent the number of fish. The word “nine” means a constant term, 9.

18

David has three times as many cards as Steven.

Sample answer: 3c; answers could vary based upon letter chosen as variable to represent the number of Steven’s cards.

The number of marbles Megan has is a variable. Choose the letter m as the variable. The expression is 9 + m or m + 9.


37

3-3

Simplify Expressions

(pp. 21–28)

Simplify each expression by using a model. 19

Example 4 Simplify 2x + 3y + x by using a model.

4r + 3v + 2v

Sample answer: Let ♥ represent r

Make a model to represent the expression.

and

Let

represent v.

♥♥♥♥ +

+

represent x and

=

Y

♥♥♥♥ +

Z

represent y.

Y

Y

Z

So, 2x + 3y + x = 3x + 3y.

So, 4r + 3v + 2v = 4r + 5v 20

5s + 2t + s

Sample answer: Let represent s and

represent t.

+

+

= + So, 5s + 2t + s = 6s + 2t

21

17 + 15h - 3h2 - 9h + 7h2 + 10

4h 2 + 6h + 27 22

12b - 3 + 11b2 − 12b + 19 - 7b2

4b 2 + 16 23

21 + 3x - x2 − 7x + x2 - 10

11 - 4x 24

35 + 11z2 − 5z − 4z2 + 5z - 5

7z 2 + 30 38


Example 5 Simplify 15 + 16z + 17 − 12z - 11. Combine like terms using the Associative and Commutative Properties. 15 + 16z + 17 − 12z - 11 = 16z − 12z + (15 + 17 − 11) = 16z − 12z + 21 = 4z + 21


Simplify each expression. Show your work.

3-4

Evaluate Variable Expressions

Evaluate each expression when = 12. 25

144 ÷

144 ÷ 12 = 12 27

Example 6 Evaluate 24 -

4·

4 · 12 = 48 26

(pp. 29–30)

14 +

= 9.

when

1. Replace with 9 in the expression. 24 - = 24 - 9 2. Simplify. Follow the order of operations. 24 - 9 = 15

÷3·5-1

14 + 12 ÷ 3 · 5 − 1 = 33 28

7 · 4 − 23 +

+7

7 · 4 - 23 + 12 + 7 = 39 Evaluate each expression when x = 4, y = 0, and z = 3. 29

15 + z2 · 4 - (x − 1) + 7y

48 Copyright © by The McGraw-Hill Companies, Inc.

30

(2x2 - 2) ÷ 5 + 4xy

6 31

Evaluate 6 ÷ 3 + y · x - 7 when x = 2 and y = 5. Replace x with 2 and y with 5 in the expression. 6÷3+y·x-7=6÷3+5·2-7 Divide.

=6÷3+5·2-7

Multiply.

=2+5·2-7 = 2 + 10 - 7

(5y + 10) ÷ (x - 2) + 4z

17 32

Example 7

Add.

= 12 - 7

Subtract.

=5

(x - 3)2 + (5z - 4y)

16 33

x·y·z+x·z÷6

2 Chapter 3 Study Guide

39

Chapter

Chapter Test

3

Name the variable, constant, and coefficient in each expression. 7AF1.1 1

9k - 15

2

k

variable:

constant: -15 coefficient:

9

t+5 variable:

t

constant:

5

coefficient:

1

Translate each phrase to an expression. 7AF1.1 3

eighteen less than u

u - 18

4

four more than six times x

6x + 4

Write an expression to represent each situation. 5AF1.2, 7AF1.1 5

Sushila has ten fewer hair ribbons than Sarah.

Sample answer: r - 10 6

Joe spent three times as many hours on his research project as Emilio.

Sample answer: 3 × h or 3h Name the like terms in each expression. 7AF1.3 4x + 11 - 2 + 5x

8

4x and 5x; 11 and -2


7

18 - 13d + 6 + 17d

-13d and 17d; 18 and 6

Simplify each expression. Show your work. 7AF1.3 9

15 + 2p + 11p − 18 + 9p

10

22p − 3

7j − 13 + 6j 2 − 9j + 11

6j 2 - 2j - 2

Evaluate each expression when d = 1, b = 4, and f = 2. 5AF1.2, 6AF1.2, 7AF1.3 11

11d · 3 + (b − f )2 − 7

30 40

Chapter 3 Test

12

7d + 4b ÷ f + (f + 3) − b2

4

GO ON

Solve. 5AF1.2, 7AF1.1, 7AF1.3 13

BAKING Angela baked 2 dozen cookies. Her brother ate 3 of them. How many cookies did Angela have left?

(2)12 - 3 = 21 14

MONEY Paulo earns $15 for cutting each lawn. Last week he cut 5 lawns. He also earned $25 for dog-sitting. What was the total amount of money Paulo earned?

(5)$15 + $25 = $100 15

16

NUMBER SENSE The perimeter of a quadrilateral is the sum of the lengths of its sides. Write an expression for the perimeter of the quadrilateral. 12 + 6 + 9 + x

Y

FITNESS Richard bikes 15 miles each day. Write an expression for the number of miles Richard bikes in d days.

15d 17

SHOPPING Anaba bought 2 bottles of perfume. One bottle was $54, and the other was $28. She also bought 2 bars of soap for y dollars each. Write an expression for the total amount she spent.

$54 + $28 + 2y, or $82 + 2y


18

AGES Selma is y years old. Jeremy is 5 years older than Selma. Jack is 2 years younger than Selma. Write an expression that represents the sum of their ages.

y + (y + 5) + (y − 2) = 3y + 3 19

FOOD The school cafeteria pays $54 per case for chicken wings. Each case contains b wings. Write an expression for the cost per chicken wing.w

$54/b,

$54 _ or $54 ÷ b b

Correct the mistakes. 7AF1.1 20

PHOTOGRAPHY Alonso’s photography teacher asked, “If you have photo albums that will each hold 150 pictures, then how many pictures will a photo albums hold?” Alonso’s answer is shown. What mistake did he make?

150 ÷ a

He should have used multiplication instead of division. The number of photos that a albums can hold is actually 150 · a, or 150a. Chapter 3 Test

41

Chapter

3

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

Which expression is “three less than twice a number”? 7AF1.1 A 3n – 2

C 3n + 2

B 3n + 3

D 2n – 3

6

Which property is shown in the sentence below? 5AF1.3 48(13 + 67) = (48 × 13) + (48 × 67) F Associative Property of Addition G Commutative Property of Multiplication

2

Which shows n2 – 9 written in word form? 7AF1.1

H Distributive Property of Multiplication over Addition

F nine less than the square of a number

J Identity Property of Addition

G nine squared more than a number H nine less than a number

7

J a number squared less than nine

Find the perimeter of the rectangle if x = 4 feet. 6AF1.2 Y Y

3

What is the value of the expression?

7AF2.1

20 ÷ 5 + 17 × (7 - 5) C 42

B 38

D 142 8

4

Evaluate (5p - 22) ÷ 4, if p = 8. 5AF1.2 F 9

H 144

G 38

J 361 9

5

C 28 feet

B 24 feet

D 32 feet

Evaluate x2 – 5y, if x = 15 and y = 9. F –30

H –15 6AF1.2

G 180

J 1,980

What is the value of the expression?

7AF2.1

25 × (5 - 2) ÷ 5 - 12

13 × 52 × 6 = 3AF1.5 A 52 × 6 × 15

C 13 × 8 × 52

B 6 × 13 × 50

D 6 × 52 × 13

A –285

C 3

B –3

D 62 GO ON

42

Chapter 3 Standards Practice


A 36

A 12 feet

10

If (5.4 × 2) × 8.1 = 87.48, then what is 5.4 × (2 × 8.1)? 3AF1.5

14

Solomon is 2 years older than 3 times JP’s age. If JP is 5, how old is Solomon?

F 10.8

H 87.48

F 10

H 17

G 16.2

J 88.47

G 11

J 30

ANSWER SHEET 11

Use the Distributive Property to find the perimeter of this rectangle. 7AF1.3 JO

JO


12

13

A 2(5) + 2(11)

C 2(11) + 5

B 2(5) × 2(11)

D 2 + (11 × 5)

Simplify the expression 5x + 9 – 2x - 6 + 4x. 7AF1.3

Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

F 11x + 15

H 10x

10

F

G

H

J

G 7x - 3

J 7x + 3

11

A

B

C

D

12

F

G

H

J

13

A

B

C

D

14

F

G

H

J

Sara sleeps 8.5 hours each night. Which expression represents how many hours she sleeps in z nights? A 8.5 × 7

C 8.5z

B 8.5 + z

D 8.5 ÷ z

Success Strategy Read each problem carefully and look at each answer choice. Eliminate answers you know are wrong. This narrows your choices before solving the problem.


43

Chapter

4

Linear Equations You solve equations all the time in your head. One night you hear a weather forecaster say that an additional 3 inches of snow fell. Now, the total snowfall for the year is 9 inches. You solve the linear equation x + 3 = 9 to find the snowfall before the additional 3 inches fell. The snowfall was 6 inches.


44

Chapter 4 Linear Equations

Royalty Free/Getty/John Giustina

STEP

1 Quiz

2 Preview

STEP

Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out. Get ready for Chapter 4. Review these skills and compare them with what you will learn in this chapter.

What You Know

What You Will Learn

You know how to translate certain phrases into math symbols. Example: I need $4 more to buy a $15 CD. the money you have now + $4 = $15 TRY IT! 1

I made $20 baby-sitting, so now I have $30. the money I had + $ 20

2

= $30

I shared the baseball cards I had equally among three friends. Each friend got six cards.

Lesson 4-1 Key words usually indicate certain operations, such as: Increased by More than

Addition +

Decreased by Difference

Subtraction -

Times Product

Multiplication × or ·

Divided by Separate

Division ÷


the cards I had ÷ 3 friends = 6 cards each Lesson 4-2

You know related addition and subtraction facts.

Addition and subtraction are operations that undo each other.

TRY IT! 3

34 + 11 = 45, so 45 - 11 =

34

4

97 - 19 = 78, so 78 + 19 =

97

You know related multiplication and subtraction facts. TRY IT! 5

128 ÷ 16 = 8, so 8 × 16 = 128

6

14 × 5 = 70, so 70 ÷ 5 =

14

Examples: 5 + 4 = 9, so 9 - 4 = 5 7 - 6 = 1, so 1 + 6 = 7 Lesson 4-3 Multiplication and division are operations that undo each other. Examples: 8 ÷ 2 = 4, so 4 × 2 = 8 5 × 3 = 15, so 15 ÷ 3 = 5

45

Lesson

4-1 Translate Word Phrases into Equations 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represent a verbal description.

KEY Concepts When translating mathematical sentences from words to symbols, you can often change each word or phrase directly to a symbol. For example, “is” means “=.” Addition + sum more than increased by

Subtraction difference less than decreased by

plus

minus

Multiplication × or · product times twice

VOCABULARY algebraic expression a combination of constants, variables, and at least one operation symbol

Division ÷ quotient divided by equal groups

(Lesson 3-1, p. 4)

equation a mathematical sentence that contains an equal sign, =, indicating that the expression on the left side of the equal sign has the same value as the expression on the right side

Group words in a sentence in a way that they can be directly translated.

system of equations a set of two or more equations with the same variables

A system of equations is two or more equations with two different unknowns. The solution of a system of equations is an ordered pair that satisfies each equation. YOUR TURN!

Translate the sentence into an equation. The quotient of 35 and a number is 7.

Translate the sentence into an equation. The product of 12 and a number is 96. 1. What does the word “product” indicate?

1. The word “quotient” means division (÷).

multiplication (× or ·)

2. The word “number” means to use a variable, such as n.

2. What does the word “number” mean?

to use a variable, such as n 3. What does the word “is” mean?

=

=7

35 The equation is 35 ÷ n = 7 or ___ n = 7.

the product of 12 and a number is 96

⎫ ⎬ ⎭

35 ÷ n

{ {

⎫ ⎬ ⎭

the quotient of 35 and a number is 7

12 × n

{ {

3. The word “is” means =.

= 96

Write the equation.

12 × n = 96 or 12 · n = 96 or 12n = 96 46



Example 1

Example 2 Write a system of equations. Rose has a total of 25 fish. There are 6 more goldfish than angelfish. 1. Let g = the number of goldfish, and let a = the number of angelfish. 2.

number of goldfish

plus

number of angelfish

equals

total number of fish

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

g

+

a

=

25

number of goldfish

equals

number of angelfish

plus

seven

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

g

=

a

+

7

3. So, the system of equations is g + a = 25 and g = a + 7. YOUR TURN! Write a system of equations. There are 52 members in the middle school band. There are 8 more girls than boys in the band. 1. Let g = the number of girls, and let b = the number of boys. number of girls

plus

number of boys

equals

total number of band members

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

g

+

b

=

52

number of girls

equals

number of boys

plus

eight

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭


2.

g

=

b

+

8

3. So, the system of equations is

g + b = 52

and

g=b+8

.

Who is Correct? Translate the sentence into an equation. The product of 8 and a number is 32.

Rosana

Brittany

Cid

8 × x = 32

8 ÷ n = 32

8n = 32

GO ON

Circle correct answer(s). Cross out incorrect answer(s). Lesson 4-1 Translate Word Phrases into Equations

47

Guided Practice Identify the operation used in each sentence. 1

Six less than a number is 23.

2

The product of three and a number is 15.

subtraction 3

multiplication

The quotient of a number and 8 is 64.

4

Seven more than a number is 19.

division

addition

Step by Step Practice Translate the sentence into an equation. 5

Eight times a number is 32. Step 1 Determine the operation. The word “times” means

multiplication (×)

.

Step 2 The word “number” means to use a variable. Choose a variable.

Answers may vary. Sample answer: n = is

32.

8×n

=

32

.

{

⎫ ⎬ ⎭

{

Step 3 The word “is” means Step 4 Eight times a number

.

8 × n = 32, 8 · n = 32, or 8n = 32

.

Translate each sentence into an equation. 6

Nine plus a number is 12. operation: variable: equation:

8

+ 9 + n = 12

The product of 5 and a number is 45.


The difference of 14 and a number is 10. operation:

n

5 · n = 45 or 5n = 45 48

7

9

-

variable:

n

equation:

14 - n = 10

A number divided by 3 is 12.

_

n ÷ 3 = 12 or n = 12 3


Step 5 The equation is

Write a system of equations that represents each situation. 10

MONEY Nicholas has a total of 6 dimes and quarters in his pocket. There are 2 more dimes than quarters.

d + q = 6; q + 2 = d 11

AGE The sum of Anthony’s age plus twice Mia’s age is 32. The difference of Anthony’s age minus Mia’s age is 2.

a + 2m = 32; a – m = 2

Step by Step Problem-Solving Practice Solve. 12

SPORTS Demarkus struck out 47 batters this season, which is 12 more than Elias. Write an equation to find b, the number of batters Elias struck out. Understand

Problem-Solving Strategies Draw a diagram. Guess and check. ✓ Write an equation. Solve a simpler problem. Work backward.

Read the problem. Write what you know.

47

batters is 12 more than the number of batters that Elias struck out.


Plan

Pick a strategy. One strategy is to write an equation. List the key words and what they mean. In the sentence you completed above, the key words are:

=

is, which means

more than, which means number, which means Solve

b

Write the equation. 47 =

Check

+ (addition)

12

+

b

Ask yourself, “Who struck out more batters, Demarkus Demarkus or Elias?” The equation 47 = 12 + b makes sense because Demarkus struck out 12 more batters than Elias.

GO ON Lesson 4-1 Translate Word Phrases into Equations

49

13

HOBBIES Martrell divided 90 baseball cards equally among some friends. Each friend got 18 baseball cards. Write an equation to find x, the number of friends. Check off each step.

_

90 ÷ x = 18 or 90 = 18 x ✔ Understand

14

✔

Plan

✔

Solve

✔

Check

PROJECTS Chloe spent 16 hours working on her science fair project. This is 7 fewer hours than Jakira spent on her project. Write an equation to find h, the number of hours Jakira spent on her science fair project.

h - 7 = 16 15

READING To finish a novel, Tadeo needs to spend 4 more hours reading this week than he spent last week. If he needs to spend 12 hours reading this week, write an equation to find n, the number of hours Tadeo spent reading last week.

n + 4 = 12 16

How is writing an equation similar to writing an expression? How is it different?

operation. An equation has an equal sign and an expression does not.

Skills, Concepts, and Problem Solving Translate each sentence into an equation. 17

Five increased by a number is 14.

18

5 + n = 14 19

The product of a number and 7 is 42.

7n = 42

50



_n = 3 6

20

The difference of 16 and a number is 3.

16 - n = 3


Answers may vary. Sample answer: The same key words determine the

21

Seven times a number is 26.

22

Ten divided by a number is 22.

_

10 ÷ n = 22 or 10 = 22 n

7 · n = 26 or 7n = 26 23

A number minus 8 is 38.

24

A number plus 4 is 18.

n - 8 = 38

n + 4 = 18

Write a system of equations that represents each situation. 25

MUFFINS Jorge baked 24 muffins. There are 6 more blueberry muffins than lemon poppyseed muffins.

b + = 24; + 6 = b 26

SURVEY Thirty students were surveyed. There were 8 more who preferred the color red than preferred blue.

r + b = 30; b + 8 = r 27

COWS A farmer has 12 cows. There are twice as many brown cows than white cows.

b + w = 12; 2w = b 28

GRANOLA BARS There are 16 granola bars in a box. There are 4 less chocolate chip bars than peanut butter bars in the box.


c + p = 16; p - 4 = c Solve. 29

FINANCES Refer to Imelda’s baby-sitting rate at the right. Write an equation to find the number of hours, h, Imelda should baby-sit two children in order to earn $106.

8h = 106 30

%0:06/&&%"

#BCZTJUUFS $BMM*NFMEBBU 3BUF QFSIPVS

NUMBER SENSE Langdon’s height is 68 inches, which is 13 inches more than Samantha’s height. Write an equation to find Samantha’s height, h, in inches.

68 = 13 + h 31

MEASUREMENT Chase found a rug shaped like a parallelogram. The area (A) of a parallelogram is the product of the parallelogram’s base (b) and height (h). Write an equation to find the area of the floor that the rug will cover. Use the image shown at the right.

A = 10 · 6

GU

GU

GO ON Lesson 4-1 Translate Word Phrases into Equations

51

Vocabulary Check Write the vocabulary word that completes each sentence. 32

33

34

A mathematical sentence that contains an equal sign, =, indicating that the expression on the left side of the equal sign has the same equation value as the expression on the right side, is a(n)

expression A(n) and operation symbols.

.

is a combination of numbers, variables,

Writing in Math Explain how to write an equation for the sentence, “Five less than a number is ten.”

The words “less than” mean subtraction. The word “number” means to use a variable, n. The word “is” means =. The equation is n - 5 = 10.


(Lesson 3-4, p. 29)

Terry’s class has collected 98 cans of food. They want to divide the total number of cans they collect among three charitable groups. You can use the expression (98 + m) ÷ 3 to find how many cans each group will receive after his class collects m more cans. Evaluate the expression if they collect 22 more cans.

40 cans MONEY Blanca had 38 pennies. She gave 2 away each day. Write an expression for the number of pennies Blanca has left after x days. Then find the number of pennies she has after 7 days.

38 - 2x ; 24 pennies

Name the like terms in each expression. 37

9p2 - 5p + 6p - 3p2 + 8p

(Lesson 3-3, p. 21)

38

5 + 4a - 2a2 + 3a + 3 - 8 + 4a2

9p 2 and -3p 2

-2a 2 and 4a 2

-5p, 6p and 8p

5, 3, and -8 4a and 3a

52



36

Lesson

4-2 Solve Equations Using Addition and Subtraction

4AF2.1 Know and understand that equals added to equals are equal. 7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

KEY Concept Addition and subtraction are inverse operations . Inverse Operations

VOCABULARY Example

Definition

7 + 8 = 15, so addition and 15 - 7 = 8 and subtraction 15 - 8 = 7

Addition Property of Equality adding the same amount to each side of an equation results in a true equation

When a + b = c, then c - a = b and c - b = a

(Lesson 1-3, p. 19)

Solving an equation means to find the value of the variable that makes both sides of an equation equal. To solve an equation, you must isolate the variable on one side of the equal sign. To do this, use inverse operations and the Addition or Subtraction Properties of Equality .

Subtraction Property of Equality subtracting the same amount from each side of an equation results in a true equation linear equation an equation in which the variables appear in separate terms

A linear equation is an equation in which the variables appear in separate terms. Solutions of a linear equation are ordered pairs that make the equation true. One way to find solutions is to make a table.


Example 1

YOUR TURN! Solve x - 3 = 8.

Solve x + 2 = 7.

1. Use the inverse operation of subtraction, which is addition.

1. Use the inverse operation of addition, which is subtraction.

Since 7 - 2 = 5, the value of x must be 5. 2. Check: What number plus 2 equals 7? Use a model to represent the equation.

3

= 8, so 8 +

Since 8 + 3 = be 11 .

=x

11 , the value of x must

2. Check: What number minus 3 equals 8?

11

Draw a model to represent the equation.

3

x-

x + 2 = 7, so 7 - 2 = x

The value of x must be 5. The answer is correct.

What is the value of x?

11

GO ON

Lesson 4-2 Solve Equations Using Addition and Subtraction

53

Example 2

YOUR TURN! Solve a + 3 = -1.

Solve n - 4 = -2. 1. The expression n - 4 can be rewritten as n + (-4). 2. The opposite of -4 is +4, since (-4) + (+4) = 0.

2. Subtract -3 from each side of the equation. a+3=

3. Add 4 to each side of the equation.

-3 , since

1. The opposite of +3 is +3 + (-3) = 0.

-1

-3 -3 a +0 = -4 a = -4

n + (-4) = -2 +4 +4 n+0= 2 n= 2

Example 3 Show four solutions of y = x + 1 in a table of ordered pairs. 1. Choose four values for x.

y

(x, y)

2. Then substitute each value into the equation and solve for y.

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)

y=x+2

y

(x, y)

-1

y = -1 + 2

1

(-1, 1)

0

y=0+2

2

(0, 2)

1

y=1+2

3

(1, 3)

2

y=2+2

4

(2, 4)

-1

3. Four solutions are (-1, 0), (0, 1), (1, 2), and (2, 3). YOUR TURN!

Show four solutions of y = x + 2 in a table of ordered pairs. 1. Choose four values for x.

x

2. Then substitute each value into the equation and solve for y. 3. Four solutions are

(-1, 1) ,

54

(0, 2)


,

(1, 3)

, and

(2, 4)

.


y=x+1

x

Who is Correct? Solve h - 13 = 15.

Toshi

Gabe

Hernan

h - 13 = 15 - 13 = 15 - 13 13 hh=2

h - 13 = 15 + 13 = 15 + 13 13 hh = 28

h - 13 = 15 h = 15 + 13 h = 28


Guided Practice Solve each equation. 1

z - 5 = 10

2

addition

inverse operation: Add

5 5


-3 .

Add -3 to each side of the equation.

5

= 10 +

-3 = 11 + -3 a= 8

a+3+

15

z = 3

The opposite of +3 is

to each side of the equation.

z-5+

a + 3 = 11

13 = 6 + d d = 7

4

8=k-4 k = 12


Solve 17 + h = 8. Step 1 The operation shown is

addition

.

The inverse operation is subtraction . Step 2 Subtract

17

from each side or add (-17) to each side.

Step 3 Solve the equation using both methods. 17 + h = - 17

8

= - 17 h=

-9

17 + h = + (-17)

8

= + (-17) h=

-9

GO ON


55

6

Show four solutions of y = x - 2 in a table of ordered pairs. Four solutions are (-1, -3) ,

(0, -2) , (1, -1) , and (2, 0)

y=x-2

x

.

y = -1 - 2

-3

(-1, -3)

0

y=0-2

-2

(0, -2)

1

y=1-2

-1

(1, -1)

2

y=2-2

0

(2, 0)

Problem-Solving Strategies

Solve. MUSIC Frank’s MP3 player holds 75 songs. He has space for 26 more songs. How many songs are already on Frank’s MP3 player? Understand

Draw a diagram. Guess and check. Act it out. Solve a simpler problem. ✓ Write an equation.

Read the problem. Write what you know. Frank’s player holds 26 more songs.

75

songs. He can add

⎧ ⎨ ⎩

+

26

=

75

⎩

s

equals

⎨

26

plus

⎧

number of songs already on player

⎧ ⎨ ⎩

Let s = the number of songs already on the MP3 player. Write an equation. Solve the equation.

⎧ ⎨ ⎩

Solve

⎧ ⎨ ⎩

Pick a strategy. One strategy is to write an equation. Then solve the equation. You know the total number of songs is 75.

75

- 26 = - 26 s = 49 Frank has

songs on his MP3 player.

Does the answer make sense? The number of songs already on the player plus 26 should equal 75.

49

+ 26 = 75

75

56

49


= 75 ✓


Plan

Check

(x, y)

-1

Step by Step Problem-Solving Practice 7

y

8

BUSINESS Zakir works a paper route. He started out with 166 papers. He has 43 left to deliver. Write and solve an equation to find p, the number of papers Zakir has delivered. Check off each step.

43 + p = 166; 123 papers

9

✔

Understand

✔

Plan

✔

Solve

✔

Check

PARADES Last year’s Labor Day parade had 114 participants, which is 27 less than this year’s parade. Write and solve an equation to find p, the number of participants in this year’s parade.

p - 27 = 114; 141 participants Is 9 the solution to the equation w + 7 = 16? Explain.

10

Yes; substitute 9 for w and then solve: 9 + 7 = 16; 16 = 16.


Skills, Concepts, and Problem Solving Solve each equation. 11

10 + n = 14 n=

13

15

4

w - 7 = 25 w=

12

f = -15 14

32

50 + z = 32

d - 20 = 59 d=

19

16

a=

18

22

25

f + 12 = 88 f=

20

53

c + 12 = 37 c=

79

-5 + a = 17

q + 19 = 72 q=

z = -18 17

f - 3 = -18

76

4 + x = -6 x = -10

GO ON


57

Find four solutions of each equation. Show each solution in a table of ordered pairs. 21

y=x-1 x

23

22

y=x-1

y

(x, y)

y=x+3 y=x+3

y

(x, y)

-1

y = -1 + 3

2

(-1, 2)

x

-1

y = -1 - 1

-2 (-1, -2)

0

y=0-1

-1

(0, -1)

0

y=0+3

3

(0, 3)

1

y=1-1

0

(1, 0)

1

y=1+3

4

(1, 4)

2

y=2-1

1

(2, 1)

2

y=2+3

5

(2, 5)

y=x+5

y

(x, y)

-1

y = -1 + 5

4

(-1, 4)

y=x-3 x

24

y=x-3

y

(x, y)

y=x+5 x

-1

y = -1 - 3

-4 (-1, -4)

0

y=0-3

-3

(0, -3)

0

y=0+5

5

(0, 5)

1

y=1-3

-2

(1, -2)

1

y=1+5

6

(1, 6)

2

y=2-3

-1

(2, -1)

2

y=2+5

7

(2, 7)

Solve. 25

6 + 8 + 12 + x = 36; 10 in. 26

TEMPERATURE On a typical summer day in Death Valley, California, the high temperature is 30ºF higher than the low temperature, which is 85ºF. What is the typical high temperature? Write and solve an equation to find the high temperature, t.

t - 30 = 85; 115°F

58


JO Y

JO

JO


MEASUREMENT The perimeter of a quadrilateral is the sum of the lengths of its sides. The quadrilateral shown has a perimeter of 36 inches. Write and solve an equation to find x, the length of the fourth side, in inches.


A letter or symbol that represents an unknown quantity is a(n)

variable 28

29

30

.

Addition The Property of Equality states that adding the same amount to both sides of an equation keeps the equation balanced. Addition and subtraction are inverse operations because they undo each other. Writing in Math Explain how to solve the equation 9 + t = 4.

Answers will vary. Sample answer: Add -9 to both sides of the equation. 9 + (-9) + t = 4 + (-9); 0 + t = -5; t = -5.

Spiral Review Translate each sentence into an equation. 31

The quotient of 8 and a number is 19.

(Lesson 4-1, p. 46)

32

A number plus 6 is 14.


33

_

8 ÷ n = 19 or 8 = 19 n

n + 6 = 14

GAMES Katie has 45 marbles. Elise has m marbles. Ryuichi has 35 marbles. Write an expression for the total number of marbles Katie, Elise, and Ryuichi have. (Lesson 3-2, p. 11)

45 + m + 35

Use the Distributive Property to find each product. Show your work. (Lesson 2-3, p. 57)

34

9(4 + a)

35

36 + 9a

7(5 - b)

35 - 7b


59

Chapter

Progress Check 1

4


Translate each sentence into an equation. 7AF1.1 1

The quotient of 9 and a number is 14.

_

2

Two times a number is 6.

9 ÷ n = 14 or 9 = 14 n 3

2 × n = 6, 2 · n = 6, or 2n = 6

PIZZA There are 16 slices of pizza. There are 2 more slices of pepperoni pizza than cheese pizza. Write a system of equations that represents the number of slices.

p + c = 16; 2 + c = p

Solve each equation. 4AF2.1, 7AF4.0 4

13 + g = 54 g=

8

5

41

x + 7 = -3

6

x = -10

m-6=5 m=

7

11

-1 = y - 14 y=

13

Show four solutions of y = x + 4 in a table of ordered pairs. y

(x, y)

-1

y = -1 + 4

3

(-1, 3)

0

y=0+4

4

(0, 4)

1

y=1+4

5

(1, 5)

2

y=2+4

6

(2, 6)


y=x+4

x

Solve. 9

ALTITUDE There are 5,280 feet in a mile. A plane flies at an altitude of 30,000 feet. Write an equation to find x, this altitude in miles.

5,280x = 30,000 12

TRAVEL Ethan visited his grandma. He traveled 128 miles by train and still had 17 more miles to travel by bus. The equation m - 128 = 17 represents this situation, where m is the number of miles Ethan lives from his grandma. Solve for m.

m = 145

60


&UIBOT IPVTF

NJ

NJ (SBOENBT IPVTF

Lesson

4-3 Solve Equations Using Multiplication and Division KEY Concept Multiplication and division are inverse operations. Inverse Operations multiplication and division

Example

Definition

4 · 3 = 12, so 12 ÷ 4 = 3 and 12 ÷ 3 = 4

When a · b = c, then c ÷ a = b and c ÷ b = a.

Solving an equation with multiplication or division is similar to solving equations with addition and subtraction. Use inverse operations and the Multiplication or Division Properties of Equality to get the variable alone on one side of the equal sign.

4AF2.2 Know and understand that equals multiplied by equals are equal. 7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

VOCABULARY Multiplication Property of Equality multiplying each side of an equation by the same amount results in a true equation (Lesson 1-3, p. 19) Division Property of Equality dividing each side of an equation by the same amount results in a true equation

Previously you solved linear equations using additiion and subtraction. In this lesson, you will solve linear equations using multiplication and division.


Example 1 Solve 9y = -36. 1. The side of the equation with the variable is 9y. The operation is multiplication. The inverse operation is division. 2. The Division Property of Equality states that dividing each side of the equation by the same amount keeps the equation balanced. Divide each side of the equation by 9. 9 9 ÷ 9, or __ , is an 9 equivalent form of one.

9y -36 _ =_ 9 9 y = -4

3. Check your answer by substituting -4 for y. 9y = -36 9(-4) = -36 -36 = -36 ✓ Both sides of the equation are the same, so the solution is correct.

GO ON

Lesson 4-3 Solve Equations Using Multiplication and Division

61

YOUR TURN! x Solve = 5. -8

_

division

1. What is the operation? What is the inverse operation?

-8

multiplication

x _ =5 -8 x ·_ = -8 -8

·5

x = -40

2. By what number should you multiply each side of the equation? -8

-40 _ =5

3. Check your answer.

-8

5=5✓

Example 2 Show four solutions of y = 2x in a table of ordered pairs. 1. Choose four values for x.

x -1 0 1 2

2. Then substitute each value into the equation and solve for y. 3. Four solutions are (-1, -2), (0, 0), (1, 2), and (2, 4). YOUR TURN!

y y y y y

= = = = =

_x in a table of ordered pairs.

Show four solutions of y =

2

x -1 0

2. Then substitute each value into the equation and solve for y. 3. Four solutions are (-1,-0.5), (0, 0) , (1, 0.5) , and (2, 1)

1 .

2

_x

2 y = -1 ÷ 2

y=0÷2 y=1÷2 y=2÷2

Who is Correct? Solve the equation 2f = 80.

Alanna

Tessa

2f = 80

2f = 80 f = 80 · 2 f = 160

2f 80 _ _ =

2 2 f = 40

Len 2f = 80 2f ÷ 2 = 80 ÷ 2 1f = 40 f = 40

Circle correct answer(s). Cross out incorrect answer(s). 62


y -2 0 2 4

(x, y) (-1, -2) (0, 0) (1, 2) (2, 4)

y

(x, y)

-0.5 (-1, -0.5)

0 0.5 1

(0, 0) (1, 0.5) (2, 1)


1. Choose four values for x.

y=

2x 2(-1) 2(0) 2(1) 2(2)

Guided Practice Solve each equation. 1

4k = 20

2

division inverse operation: Divide each side of the equation 4

by k=

w =7 ___ -2

inverse operation: multiplication Multiply each side of the equation

-2 . w = -14

.

by

5


Show four solutions of y = 3x in a table of ordered pairs.


x -1 0 1 2

4

y y y y y

= = = = =

3x 3(-1) 3(0) 3(1) 3(2)

y -3 0 3 6

(x, y) (-1, -3) (0, 0) (1, 3) (2, 6)

x

Step 1

Choose four values for

.

Step 2

Then substitute each value into the equation and y . solve for

Step 3

Four solutions are (-1, -3) , (1, 3) , and (2, 6) .

(0, 0)

,

x in a table of ordered pairs. Show four solutions of y = __ 4 x y= y (x, y) x 4 -1 y = -1 ÷ 4 -0.25 (-1, -0.25) 0 0 (0, 0) y=0÷4

_

1 2

y=1÷4 y=2÷4

0.25 0.5

(1, 0.25) (2, 0.5)

GO ON Lesson 4-3 Solve Equations Using Multiplication and Division

63


Problem-Solving Strategies Draw a diagram. Look for a pattern. ✓ Write an equation. Act it out. Solve a simpler problem.

Solve. SCRAPBOOKS Ariela has 96 photos from her summer vacation. She wants to make a scrapbook that has 24 pages with the same number of photos on each page. How many photos, p, can Ariela place on each page? Understand


96

24

times

24

equals

⎧ ⎨ ⎩

p

×

24

=

96

⎩

number of photos on page ⎨

Solve

Pick a strategy. One strategy is to write an equation. Then solve an equation.

⎧

Plan

⎧ ⎨ ⎩

pages.

⎧ ⎨ ⎩

photos. She has

⎧ ⎨ ⎩

Ariela has

5

24p _ 24

=

p= Ariela can place

number of photos

24 5

photos on each page.

÷ photos on each page ÷

4

= number of pages =

FINANCE Kendra’s mom bought her a new television. Kendra plans to pay back her mom $27 per month for the next 6 months. Write and solve an equation to find x, the total amount Kendra will pay x = 27; x = 162 her mom.

_ 6

Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check


24


96

64

96 _

Does your answer make sense? Check your arithmetic.

Check

6

4

96

7

FOOD Reya’s father bought 5 smoothies. He paid $20. Write and solve an equation to find c, the cost of one smoothie.

5c = 20; c = $4

8

TRAVEL The Thursdal family had to travel 300 miles to get to their family reunion. Each of the three drivers drove an equal amount. How many miles did each driver drive? Write and solve an equation to find m, the number of miles each driver drove.

3 × m = 300; m = 100 Is 7 a solution to the equation y = x + 6? Explain.

9

No, a solution to a linear equation should be an ordered pair.

Skills, Concepts, and Problem Solving Solve each equation. a 10 __ = -7 8

11

a = -56 y 12 __ = 5 6


y=

-9c = 27 c=

13

30

-3

5b = 45 b=

9


y = 4x x

15

y = 4x

y -4

y = 5x

(x, y)

x

(-1, -4)

-1

y = 5x

y

(x, y)

y = 5(-1)

-5

(-1, -5)

-1

y = 4(-1)

0

y = 4(0)

0

(0, 0)

0

y = 5(0)

0

(0, 0)

1

y = 4(1)

4

(1, 4)

1

y = 5(1)

5

(1, 5)

2

y = 4(2)

8

(2, 8)

2

y = 5(2)

10

(2, 10)

Solve. 16

MEASUREMENT The formula for the area of a parallelogram shown is A = bh. Use the formula to write an equation. Find the height of the parallelogram in centimeters by solving for h.

"DN¤

32 = 8h; 4 cm CDN

Lesson 4-3 Solve Equations Using Multiplication and Division

65

17

BALLOONS Mrs. Collier is bringing balloons to a picnic. She wants to divide the balloons into groups so that each of the 13 students gets 6 balloons. Write and solve an equation for b, the number of balloons Mrs. Collier needs to bring to the picnic.

_b = 6; 78 balloons 13

18

WORK Sylvia makes $6.00 an hour baby-sitting. Make a table that shows Sylvia’s total earnings for working 1, 2, 3, and 4 hours. Number of Hours

Multiply by 6

Total Earnings ($)

1

6×1 6×2

6

2 3 4

12

6×3 6×4

18 24


20

Multiplication The Property of Equality states that multiplying each side of an equation by the same amount keeps the equation balanced. Writing in Math Explain how to solve the equation 5x = 35.

Spiral Review 21

SCHOOL There are 494 students enrolled in Sunnyvale Middle School. Today 58 students were absent. Write and solve an equation to find s, the number of students who attended school today. (Lesson 4-2, p. 53)

58 + s = 494; 436 students

66



_ _

Divide each side of the equation by 5: 5x = 35 ; x = 7. 5 5

Lesson

4-4 Multi-Step Equations 7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

KEY Concept A two-step equation contains two operations on the variable side of the equation. Identify operations and their order.

VOCABULARY

Use inverse operations. Work backward.

First, x is multiplied by 4, . . .

two-step equation an equation that contains two operations on the variable side of the equation

To solve, first subtract 3, . . .

. . . and then 3 is added.

. . . and then divide by 4

You can use algebra tiles to help you solve two-step equations.

Example 1 Solve 4x + 3 = 11.


1. Model the equation using tiles. Y

Y

Y

Y

Y

Y

Y

Y

Y

Y ĕ

Y

Y

Y Y

ĕ

4. Undo multiplication using division.

Y

Y

Y

Y

3. The result is 4x = 8.

Y

2. Undo addition using subtraction.

Yµ Y

5. Check your answer by substituting 2 for x. 4x + 3 = 11 ⇒ 4(2) + 3 = 11 ⇒ 8 + 3 = 11 ⇒ 11 = 11 ✓

µ

GO ON

Lesson 4-4 Multi-Step Equations

67

YOUR TURN! Solve 1 + 2x = 7 using algebra tiles. 1. Model the equation.

1 + 2x = 7

Y Y

addition

2. First, undo the

.

1 + 2x = 7 1

-1 + 2x = 7 -1 0 + 2x = 6

Y Y

2x = 6 3. Next, undo the 2x =

multiplication

_ _ 6 2 2 3

1 + 2(

Y

Y

)=7

6

1+

.

x=3 4. Check your answer.

=7 7=7✓

Solve

_w - 4 = 1. 3

1. Identify the operations and their order.

Identify operations and their order. First, w is divided by 3, . . .

2. Use inverse operations. Work backward to solve for w. w w -4=1 -4=1 3 3 w . . . and then 4 is subtracted. -4+4=1+4 Undo subtraction. 3 w Use inverse operations. Work backward. =5 3 To solve, first add 4, . . . w ·3=5·3 Undo division. 3 w = 15 w -4=1 3 3. Check your answer. 15 - 4 = 1 ⇒ 5 - 4 = 1 ⇒ 1 = 1 ✓ . . . and then multiply by 3. 3

_

_

_

_

_

_

_

68



Example 2

YOUR TURN! y Solve - 2 = 3. 6

_

1. Identify the operations and their order. First, the variable y is

divided

.

_y - 2 = 3

2. Use inverse operations. First, undo the

subtracted

by 6. Then 2 is

subtraction

6

_

y -2 6

.

+2 = 3 +2

_y + 0 = 5 6

_y = 5 Then undo the

division

6

_y

.

·

6

6

6

=5· y=

30

3. Check your answer.

30 - 2 = 3 ⇒ _y - 2 = 3 ⇒ _ 6

6

5

-2=3⇒3=3✓


Who is Correct? Solve the equation 2c - 12 = 12.

Tulio

Ben

2c - 12 = 12 2c - 12 = 12 2 2 6 = 12 c c - 12 + 12 = 6 + 12 c = 18

_

_

2c - 12 = 12 12 2c - 12 - 12 = 12 0 2c = c=0

Pamela 2c – 12 = 12 + 12 = + 12 2c = 24 2c = 24 2 2 c = 12

_ _


GO ON Lesson 4-4 Multi-Step Equations

69

Guided Practice Write the inverse operations in the order that they should be used to solve each equation. k 1 3z + 2 = 38 2 __ - 3 = 13 8

subtraction; division

addition; multiplication


Show four solutions of y = 2x + 1 in a table of ordered pairs. Step 1 Choose four values for x. Step 2 Then substitute each value into the equation and solve for y. Step 3 Four solutions are (-1, -1) , and (2, 5) . x -1

y = 2x + 1 y = 2(-1) + 1

6

(x, y)

-1

(-1, -1)

y = 2(0) + 1

1

(0, 1)

1

y = 2(1) + 1

3

(1, 3)

2

y = 2(2) + 1

5

(2, 5)

4b + 3 = 27

5a - 8 = 22

5

5a - 8


+8 = 22 +8 5a = 30 5a ___

5

= a=

7

6

,

y

·3 =3 ·3 z= 9

b=

70

(1, 3)

y __ -9=2 6

y=

66

30 _ 5 6


3

,

0

Solve each equation. z+5=8 __ 4 3 z + 5 -5 = 8 -5 __ 3 z = 3 __ 3 z __

(0, 1)

Step by Step Problem-Solving Practice Solve. 8

FITNESS Earlier in the week Martin ran 14 miles. He wants to run a total of 30 miles by the end of the week. There are 4 days left this week. How many miles, m, should Martin run each day to meet his goal? Understand

Look for a pattern. Guess and check. Act it out. Solve a simpler problem. ✓ Write an equation.


14

Martin has already run He wants to run

4

He has Plan


30

miles.

total miles.

more days this week.

Pick a strategy. One strategy is to write an equation. Then solve the equation. 30 miles by the end of the week = ⎫ ⎬ ⎭

30 Solve

14 miles he has already run + number of miles in 4 days = 14 + 4m

Solve the equation. Find the value for m, the number of miles Martin should run each day. 30 = 14 + 4m


:PVDBOXSJUFUIF FRVBUJPOIPSJ[POUBMMZ PSWFSUJDBMMZ5IJTNFUIPE JTIPSJ[POUBM

30 -14 = 14 -14 + 4m 16 = ____ 4m

_ 4

4 4

Martin should run Check

4

=m

Undo the addition. Undo the multiplication. Simplify each side.

miles each day.

Substitute for m to check. 30 = 14 + 4( 30 = 14 + 30 =

4

)

16

30

GO ON Lesson 4-4 Multi-Step Equations

71

9

BUSINESS Veronica’s Surf Shop rents snorkel equipment for $5 plus an hourly rate. Esperanza rented equipment for 7 hours. The total cost was $26. Write and solve an equation for r, the hourly rate for renting the equipment. Check off each step.

7r + 5 = 26; r = $3

10

✔

Understand

✔

Plan

✔

Solve

✔

Check

FINANCE Jaunita spent $35 on supplies for necklaces. She wanted to make a profit of $55. Jaunita charged $9 per necklace. Write and solve an equation for n, the number of necklaces Jaunita had to sell to make a profit of $55.

9n - 35 = 55; n = 10 11

COOKING Tomas spent $35 on ingredients to make 5 sweet potato pies. He sold the pies at a bake sale. Write and solve an equation for s, the price Tomas had put on each pie to make a profit of $20.

5s - 35 = 20; s = $11 k + 3 = 5? Explain. Is 8 the solution to the equation __ 4

12

_

Skills, Concepts, and Problem Solving Solve each equation. 13

9d + 6 = 51 d=

16

5

1m - 1 = 3 __ 5

m=

14

4x + 2 = 10 x=

17

20

2

10k + 2 = 62 k=

15

6

g __ -7=5 3

g= 18

4y + 4 = 40 y=

Solve. 19

SPORTS Tickets cost $3 for Friday night’s basketball game. Snack sales were $643. The total sales for the basketball game were $2,029. Write and solve an equation for t, the number of tickets sold.

3t + 643 = 2,029; t = 462 72


36

9


Yes; substitute 8 for k and then solve: 8 + 3 = 5; 2 + 3 = 5; 5 = 5. 4

20

MEASUREMENT The area of the rectangle is 24 square inches. Write and solve an equation for x to determine the width of the rectangle in inches. (Hint: A = lw)

Yo JO

6(x − 5) = 24; x = 9 21

JO

SHOPPING You buy a DVD for $16 and CDs for $12 each. The equation t = 16 + 12c represents the total amount t that you spend if you buy 1 DVD and c CDs. Make a table that shows the total cost of buying a DVD and 1, 2, 3, or 4 CDs. Number of CDs

t = 16 + 12c

Total Cost ($)

1

16 + 12(1) 16 + 12(2)

28

2

40

16 + 12(3) 16 + 12(4)

3 4

52 64


Vocabulary Check Write the vocabulary word that completes each sentence.

subtraction

22

Addition and

are inverse operations.

23

Writing in Math Explain how to solve the equation 3x - 8 = 13.

Add 8 to each side of the equation: 3x - 8 + 8 = 13 + 8; 3x = 21. Divide each side of the equation by 3: 3x = 21 ; x = 7. 3 3

_ _

Spiral Review 24

FASHION Frisco is putting up a fence in his backyard. He bought 16 fence posts. There will be one post for every 3 feet of fence. Write and solve an equation to find p, the perimeter of the fence in feet. (Lesson 4-3, p. 61)

p __ = 16; p = 48 3

Simplify each expression. 25

(Lesson 3-3, p. 21)

13 + 12k - 2k2 - 6k + 5k2

3k 2 + 6k + 13

26

8a - 5 + 6a2 - 7a + 10 - 4a2

2a 2 + a + 5 Lesson 4-4 Multi-Step Equations

73

Chapter

Progress Check 2

4

Solve each equation. 4AF2.2, 7AF4.0 q 1 __ = 6 7 q= 3


2

42

b=

5c + 10 = 45

4

7

c=

-8b = 72

-9

m - 13 = 4 __ 3

m=

51


y = 6x x

7

6

y = 6x

y = -x

y

(x, y)

x

y = -x

y

(x, y)

-1

y = 6(-1)

-6

(-1, -6)

-1

y = -(-1)

1

(-1, 1)

0

y = 6(0)

0

(0, 0)

0

0

(0, 0)

1

y = 6(1)

6

(1, 6)

1

y = -(0) y = -(1)

-1

(1, -1)

2

y = 6(2)

12

(2, 12)

2

y = -(2)

-2

(2, -2)

x+2 y = __ 2 x

y=

8

_x + 2

y

(x, y)

-1 y = -1 ÷ 2 + 2 1.5

(-1, 1.5)

2

(0, 2)

1

y=0÷2+2 y=1÷2+2

2.5

(1, 2.5)

2

y=2÷2+2

3

(2, 3)

0

x

y = 3x - 1

y

(x, y)

-1

y = 3(-1) - 1

-4

(-1, -4)

0

y = 3(0) - 1

-1

(0, -1)

1

y = 3(1) - 1

2

(1, 2)

2

y = 3(2) - 1

5

(2, 5)

Solve. 4AF2.2, 7AF4.0 9

DANCES Lolita bought punch for a school dance. Write and solve an equation to find g, the number of gallons Lolita should buy to make 80 cups of punch. (Hint: 1 gallon = 16 cups)

16g = 80; g = 5 10

NUMBER SENSE The sum of 26 and 88 is the same as a number x multiplied by 6. Write and solve an equation to find x.

26 + 88 = 6x; x = 19 74



2

y = 3x - 1

Lesson

4-5 Symbolic Computation KEY Concept Equations with fractions are solved the same way as equations with whole numbers. Convert a fraction to a decimal. Write an equivalent fraction with a denominator of 100.

⎛

⎞

4 ⎝ 25 ⎠

Write an equivalent fraction with a denominator of 100.

4 ⎝ 25 ⎠

Example 1

_3 as a decimal.


5 1. Divide the numerator by the denominator. 0.6 5 3.0 -30 0

⎞

3 ___ 25 75 __ ⎪ ⎥ = ____

Write as a percent.

Write

100

= 0.75

⎛

VOCABULARY decimal a number that represents whole numbers and fractions; a decimal point separates the whole number from the fraction

3 ___ 25 75 __ ⎪ ⎥ = ____

Write as a decimal to the hundredths place. Convert a fraction to a percent.

7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.

Place the decimal point directly above the decimal point in the dividend.

100

= 75%

YOUR TURN! 1 Write as a decimal. 4

_

1. Divide the numerator by the denominator.

0.25 4 1.00

To divide 3 by 5, place a decimal point after 3. Add as many zeros as necessary to complete the division.

-8 20 -20 0 1 = 0.25 . 2. So, __ 4

3 2. So, __ = 0.6. 5

GO ON Lesson 4-5 Symbolic Computation

75

Example 2 Write

YOUR TURN! 6 Write as a percent. 10

_

_7 as a percent. 5

x _7 = _

1. Set up a proportion.

5 7 × 100 700 700 3. Divide each side by 5. 5 4. Simplify. 140 7 is equivalent to 140%. So, __ 5 2. Cross multiply.

100 1 = 5x = 5x 5x = 5 =x

_ _

x 6 =_ _ 10 100


2. Cross multiply.

6 × 100 = 10x

600 = 10x

3. Divide each side by 10.

10x 600 = _ _ 10 10

4. Simplify.

60 6 So, ___ is equivalent to 10

60 %.

Example 3

YOUR TURN!

1 Last month, Mykia saved _ times as much

A small bottle of water costs

4 as she earned. If she saved $9.75, how much did she earn?

4. Write the new equation. 0.25x = 9.75 0.25x 9.75 5. Solve for x. _____ = ____ 0.25 0.25 x = 39

6. Mykia earned $39. Compare to your estimate for reasonableness.

76


2 as a large bottle of water. If a small bottle of water costs $1.20, how much does a large bottle of water cost? 1 1. Estimate. __ =2 2 1 x = 1.20 2. Write an equation.

2

0.25 4 1.00 -8 20 -20 0

_1 as much

_ 2

3. Convert the fraction to a decimal to make solving the equation easier. 4. Write the new equation.

0.5 2 1.0

-10 0

0.5x = 1.20 0.5x 1.20 5. Solve for x. ____ = ____

0.5

0.5 x = 2.4

6. A large bottle of water costs $2.40 . Compare to your estimate for reasonableness.


1 1. Estimate. __ 40 = 10 4 1 x = 9.75 2. Write an equation. __ 4 3. Convert the fraction to a decimal to make solving the equation easier.

=x

Who is Correct? Write

_4 as a percent. 5

Fidel

Mina

Sherron

x _4 = _

100 5 100x = 5 × 4 20 = 100x 100 100 x 0.2% =

_ _

x _4 = _

_ _

x 4 = 0 10 5 4 × 100 = 5x 400 = 5x 400 400 0.0125% = x

100 5 5x = 0 4 × 10 400 = 5x 5 5 80% = x

_ _

_ _


Guided Practice Write the decimal equivalency for each. 1 = 0.5 __ 1 2


3

37% =

37 = 0.37 _ 100

2

3 __ =

4

80% =

4

0.75 80 = 0.8 _ 100

Write the percent as a fraction with a denominator of 100. Then write the fraction as a decimal.


3 Write __ as a decimal. 8

0.375

Step 1 Divide the numerator by the denominator.

8 3.000

3 Step 2 So, __ = 0.375 8

-24 60 -56 40 -40 0 GO ON Lesson 4-5 Symbolic Computation

77

Write each fraction as a decimal. Round to the nearest hundredth. 1 1 6 __ 7 __ 5 8

0.2

8

4 __

0.13

9

6

1 __ 3

0.67

10

1 __

0.33

11

7

2 __ 9

0.14

13

1 __ 4

225%

14

1 ___

25%

15

10

12 ___ 3

10%

16

2 __

17

5

40% 78

400%


6 __ 4

150%


Write each fraction as a percent. 9 12 __ 4

0.22


Problem-Solving Strategies Look for a pattern. Guess and check. Use logical reasoning. ✓ Write an equation. Work backward.

Solve. 18

SHIPPING Mr. Larson wants a package to weigh 4.75 pounds. He wants to send 4 calculators and a 3 1__ -pound chess set. What is the most that each 4 calculator can weigh? Understand

Read the problem. Write what you know. The total weight needs to be The chess set weighs There will be

4

_

4.75 pounds

1 3 pounds 4

.

.

calculators in the package.

Plan

Pick a strategy. One strategy is to write an equation.

Solve

Let c = weight of a calculator.

_3

1 4c = + 4 total weight = 4 calculators + chess set


4.75

Convert the mixed number to a decimal to make solving the equation easier. 3 1__ = 1.75 4 4.75 = 4c + 1.75 Use inverse operations. Subtract 1.75 from each side.

4.75 - 1.75 = 4c + 1.75 - 1.75

Divide each side. Each calculator can weigh no more than 0.75 pound. Check

3 4

4c 4

____ = ____

0.75 = c

Add the weights to check that the total is 4.75 pounds. 1.75 + 0.75 + 0.75 + 0.75 + 0.75 = 4.75 pounds

GO ON Lesson 4-5 Symbolic Computation

79

19

1 lawns YARD WORK On Saturday Anica mowed 1__ 2 before it rained. How many lawns does she still have to mow? 4 1

_ 2

20

3 1 miles every weekend. If she runs 2__ FITNESS Gina runs 4__ 2 4 miles on Saturday, how many miles does she run on Sunday?

_

13 4 ✔

21

Check off each step. YARD WORK Anica

Understand

✔

Plan

✔

Solve

✔

Check

mows 6 lawns each week.

You have an equation with fractions that have denominators of 4, 10, and 5. By what number could you multiply both sides of the equation to eliminate all the fractions? Why?

20; because 20 is the LCD for fractions having denominators of 4, 10, and 5.

Skills, Concepts, and Problem Solving

0.33 24

5 __

0.8 25

8

2 __ 8

0.625 26

1 __

0.25 27

2

7 __ 4

0.5 28

3 ___

29

15

0.2 80


Getty Images

1.75 27 ___ 3

9


Write each fraction as a decimal. Round to the nearest hundredth. 3 4 22 __ 23 __ 9 5

30

SHOPPING Julia bought the items shown at the right at the grocery store. If Julia bought a total of 10 pounds of vegetables, how much did the 4 potatoes weigh?

LBSPEPPERS LBSTOMATOES LBLETTUCE LBSONIONS POTATOES

_

1 3 lbs 4 31

LUNCH Last week Lise spent 3 times as much on lunch as he did on snacks. If he spent $15.75 on lunch, how much did he spend on snacks?

$5.25 32

1 as SHOPPING Trey bought 2 shirts. The blue shirt costs __ 3 much as the green shirt. If the blue shirt costs $12.00, how much did the green shirt cost?

$36.00 Vocabulary Check Write the vocabulary word that completes each sentence.

decimal

33

A

represents whole numbers and fractions.

34

1 as a decimal. Writing in Math Explain how to write __ 2


Divide the numerator by the denominator.

0.5 1.0 2


(Lesson 4-3, p. 61)

Oliver works three times as many hours each week as Mia works. If Oliver works 15 hours a week, how many hours does Mia work?

5 hours 36

Greg is collecting money for T-shirts. He has collected $105. If each T-shirt costs $7, how many T-shirts has he collected money for?

15 T-shirts Evaluate each expression when s = 0.5, t = 10, and v =

10

37

8v · st =

38

t2 + 100v - 3s = 123.5

_1. 4

(Lesson 3-4, p. 29)

Lesson 4-5 Symbolic Computation

81

Chapter

4

Study Guide

Vocabulary and Concept Check decimal, p. 75

Write the vocabulary word that completes each sentence.

equation, p. 46

1

two-step equation, p. 67

2

equation A(n) is a mathematical sentence that contains an equal sign, =, indicating that the expression on the left side of the equal sign has the same value as the expression on the right side. A(n) two-step equation contains two operations on the variable side of the equation.

Translate the words into an equation. sixteen 3

16

plus 4

+

a number 5

x

equals 6

=

twenty 7

20

Lesson Review

4-1

Translate Word Phrases into Equations

Translate each sentence into an equation. The quotient of a number and 4 is 9.

n÷4=9 9


6n = 42 10

28 decreased by a number is 24.

Example 1 Translate the sentence into an equation. A number increased by 9 is 21. The word “number” means to use a variable, such as n. The words “increased by” mean addition (+). The word “is” means =.

28 − n = 24 11

The sum of 22 and a number is 55.

22 + n = 55

82


The equation is n + 9 = 21.


8

(pp. 46–52)



_n = 8

Translate the sentence into an equation.

4

13


7n = 12 14

Seven times a number is 63. Determine the operation. The word “times” means multiplication (×). The word “number” means to use a variable. Choose a variable.

7 less than a number is 14.

The word “is” means =.

n - 7 = 14 15

Example 2

Twenty two times a number is 88.

22n = 88 7 · n - 5 = 63 or 7n - 5 = 63

4-2

Solve Equations Using Addition and Subtraction

Use a model to solve each equation.


16

17

Example 3 Solve x + 4 = 9.

x + 8 = 15

(pp. 53–59)

What number plus 4 equals 9?

Think: 5 + 4 = 9

x−3=7

Use a model to represent the equation. The value of x must be 5.

Check your answer by using the inverse operations of addition and subtraction. 5 + 4 = 9, so 9 – 4 = 5 The answer is correct.


83

Solve each equation. Check your answers. 18

x - 6 = 12

Solve x - 8 = 5.

x = 18 19

x-8=5 +8=+8 x + 0 = 13 x = 13

y + 9 = 11

y=2 20

w - 8 = 31

BOENBLFB[FSPQBJS

Check your answer by substituting 13 for x.

w = 39 21

Example 4

x-8 =5 13 - 8 = 5 5=5

17 - z = 20

z = -3

4-3

Solve Equations Using Multiplication and Division

Solve. x 22 __ = 3 3

Example 5 Solve

x=9

2. Multiply each side by 2. 3. Solve the equation. x=2·6 2 · __ 2 2x = 12 ___ 2 x = 12

5

y = 30

84

2

1. The operation is division. The inverse operation is multiplication.

y __ =6

_x = 6.



23

(pp. 61–66)

4-4

Multi-Step Equations

(pp. 67–75)

Solve each equation. Check your answers. 24

Example 6 Solve 4x - 3 = 17.

3b + 8 = 29

4x - 3 = 17 +3 +3 4x = 20 20 4x = ___ ___ 4 4 x=5

b=7 25

7x - 1 = 48

x=7 26

BOE NBLF B[FSPQBJS

15y + 9 = 234

y = 15

4-5

Symbolic Computation

Write each fraction as a percent. 5 27 __ 8

62.5%


28

Example 7 Write

10 _ as a percent. 4

10 ____ ___ = x


3 __ 4

75% 29

(pp. 75–79)

2. Cross multiply.

4

10 × 100 = 4x 1,000 ___ _____ = 4x

3. Divide each side by 4.

8 __ 5

160%

100

4

4

250 = x

4. Simplify. 10 5. So, ___ is equivalent to 250%. 4


85

Chapter

Chapter Test

4

Translate each sentence into an equation. 7AF1.1 1

Four more than a number is 17.

2

The product of six and a number is 30.

n + 4 = 17 3

Twenty plus a number is 32.

6n = 30 4

The difference of 25 and a number is 14.

20 + n = 32 5

The product of 9 and n is 39.

25 - n = 14 6


_

n ÷ 3 = 12 or n = 12 3

9 · n = 39 or 9n = 39

Write a system of equations that represents each situation. 7AF4.0 7

FOOTBALL There are 35 players on the football team. There are 5 fewer seventh graders than eighth graders on the team.

s + e = 35; e - 5 = s 8

SURVEY Twenty students were surveyed. There were 2 fewer students who preferred regular milk than chocolate milk. Copyright © by The McGraw-Hill Companies, Inc.

r + c = 20; c - 2 = r

Solve each equation. 4AF2.1, 4AF2.2, 7AF4.0 9

18 + n = 35 n=

11

13

17

q __ =7

42

7 as a decimal. Write ___ 10

0.7 86

Chapter 4 Test

d - 12 = 22

34

d=

12

6

q=

10

-9k = 63 k=

14

-7

9 Write ___ as a percent. 12

75%

GO ON


y = 2x x -1 0 1 2

17

16

y = 2x y = 2(-1)

y 2

(x, y) (-1, -2)

y = 2(0) y = 2(1) y = 2(2)

0 2 4

(0, 0) (1, 2) (2, 4)

y = 2x + 1 y = 2(-1) + 1

y -1

(x, y) (-1, -1)

y = 2(0) + 1 y = 2(1) + 1 y = 2(2) + 1

1 3 5

(0, 1) (1, 3) (2, 5)

y=x-1 x -1 0 1 2

y=x-1 y = -1 - 1

y=0-1 y=1-1 y=2-1

y

(x, y)

-2 (-1, -2) -1 (0, -1) 0 (1, 0) 1 (2, 1)

y = 2x + 1 x -1 0 1 2

Solve. 7AF1.1, 7AF4.0


18

GEOGRAPHY The distance from Loganville to Palmetto is 147 miles. This is 17 more miles than the distance from Loganville to Pendleton. Write and solve an equation to find the distance, d, in miles from Loganville to Pendleton.

d + 17 = 147; d = 130 miles 19

MEASUREMENT The formula for the area of a triangle is 1 bh. Use the formula to write an equation. Find the A = __ 2 height, h, of the triangle shown at the right in inches.

_

21 = 1 6h; 21 = 3h; h = 7 inches 2

I "JO¤

CJO

Correct the mistakes. 20

VOLUNTEERING Emma’s youth group had 32 participants at the senior-center holiday party. That was four times as many participants, p, as last year. Emma wrote an equation to find the number of participants last year. What mistake did she make?

p + 4 = 32

The mistake is that her equation says that the 32 participants are four more than p, rather than four times p. Chapter 4 Test

87

Chapter

4

Standards Practice


2

Which equation represents a number plus 8 equals 17? 7AF1.1 A - 8 = 17

C + 8 = 17

B 8 × = 17

D ÷ 8 = 17

Which expression represents the product of 4 and 15? 7AF1.1

5

6

Jenelle bought 3 CDs for $42. Which expression represents the cost of one CD? 7AF1.1 A 42 × 3

C 3 ÷ 42

B 3 + 42

D 42 ÷ 3

Solve for in the equation. 7AF4.0

F 4 + 15

H 4 × 15

÷ 4 = 128

G 4 - 15

J 4 ÷ 15

F = 32 G = 64

3

H = 124

Solve for in the equation. 7AF4.0

J = 512

24 + = 40 C = 26

B = 16

D = 64

Ella and Jonah collected soda-can tabs for a school fund-raiser. Ella collected 186 tabs. Together they collected 412 tabs. Which equation can be used to find the number of soda-can tabs, n, that Jonah collected? 7AF1.1 F n + 186 = 412 G n - 186 = 412 H n × 186 = 412 J n ÷ 186 = 412

7

Which operation keeps the scale balanced? 4AF2.1

A Add two triangle blocks to the right side. B Add two square blocks to both sides. C Add two square blocks to the left side. D Add two square blocks to the right side.

GO ON 88



4

A =8

8

How is the expression 21 × 18 written in word form? 7AF1.1 F the sum of 21 and 18

9

ANSWER SHEET Directions: Fill in the circle of each correct answer.

G the difference of 21 and 18

1

A

B

C

D

H the product of 21 and 18

2

F

G

H

J

J the quotient of 21 and 18

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

How much does one pair of socks cost?

4AF2.2

QL40$,4


Success Strategy A $0.57 B $2.00 C $1.75

Read the directions carefully. This will help you avoid careless errors. Also, review your answers when you are done. Make sure that you have answered all of the questions and have not mismarked the answer sheet.

D $15.75

10

Find the difference of 18 and n when n = 6. 6AF1.2, 7AF1.1 F 3 G 12 H 24 J 108


89

Chapter

5

Inequalities It is Saturday at noon. If you spend two hours at soccer practice and three hours with your family, what is the most time you can spend with your friends before you have to be home at 6:00 p.m.? You can figure out questions like this by solving inequalities.


90

Chapter 5 Inequalities

Getty/Stockbyte

STEP

1 Quiz

Are you ready for Chapter 5? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

STEP

Get ready for Chapter 5. Review these skills and compare them with what you will learn in this chapter.

What You Know

What You Will Learn

You know how to understand certain phrases. Example: I have less than $5 in my pocket. my money < $5

Lesson 5-1

You know how to add, subtract, multiply, and divide.

Lessons 5-2, 5-3, and 5-4

Examples: 25 + 4 = 29 14 × 2 = 28

29 - 4 = 25 28 ÷ 2 = 14


TRY IT! 1

If 40 + 30 = 70, then 70 - 30 = 40 .

2

If 96 ÷ 12 = 8, then 8 × 12 = 96 .

- subtraction

< less than

> greater than

Addition and subtraction are operations that undo each other. Multiplication and division are operations that undo each other.

8 ÷ 2 = 4, so 4 × 2 = 8

Example: Use the number line to graph the whole numbers less than 6.

+ addition

5 + 4 = 9, so 9 - 4 = 5

You know how to graph numbers on a number line.

Math symbols indicate certain operations.

Lesson 5-5 To graph an inequality such as x > 2, place an open circle at 2. Shade a line to the right to represent all the numbers greater than 2.

TRY IT! Use the number line to graph the whole numbers greater than 6.

3

The open circle means that 2 is not a solution.

91

Lesson

5-1 Translate Phrases into Inequalities 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represent a verbal description.

KEY Concept When translating mathematical sentences from words to symbols, you can often change each word or phrase to a symbol.

VOCABULARY

Writing inequalities is similar to writing equations, but the sentence contains an inequality symbol instead of an equal sign.

expression a combination of constants, variables, and operation symbols (Lesson 3-1, p. 4)

Symbol

Meaning

is greater than

≤

is less than or equal to

≥

is greater than or equal to

≠

is not equal to

inequality a number sentence that compares two unequal expressions and uses < (is less than), > (is greater than), ≤ (is less than or equal to), ≥ (is greater than or equal to), or ≠ (is not equal to)

Remember that:

system of inequalities a set of two or more inequalities with the same variables

• at most translates to is less than or equal to (≤) • at least translates to is greater than or equal to (≥)

Example 1

YOUR TURN!

Translate the sentence to an inequality.


A number is less than five.

A number is more than seven.

1. The words “is less than” translate to .

2. The word “number” means to use a variable, such as n.

92


2. The word “number” means to use a variable, such as n. A number

is more than

seven.

n

>

7


A system of inequalities is two or more inequalities with the same variables.

Example 2 Write a system of inequalities. The number of nickels and dimes is more than 12. There are more than 5 nickels. 1. Let n = the number of nickels, and let d = the number of dimes. 2. number of

nickels

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

n number of nickels

+ is more than

d

>

12

⎫ ⎬ ⎭

⎫ ⎬ ⎭

total number of coins

⎫ ⎬ ⎭

is more than

⎫ ⎬ ⎭

number of dimes

n

>

5

plus

5

3. So, the system of inequalities is n + d > 12 and n > 5. YOUR TURN! Write a system of inequalities. Angel’s Pet Store never has more than a total of 25 cats and dogs and never more than 10 cats. 1. Let c = the number of cats, and let d = the number of dogs.

plus

⎫ ⎬ ⎭

⎫ ⎬ ⎭


total number of cats and dogs

c

+

d

≤

25

⎫ ⎬ ⎭

number of cats

⎫ ⎬ ⎭


⎫ ⎬ ⎭


cats

⎫ ⎬ ⎭

number of dogs

c

≤

10

3. So, the system of inequalities is

⎫ ⎬ ⎭

⎫ ⎬ ⎭

2. number of

10

c + d ≤ 25

and

c ≤ 10

.

Who is Correct? Write an inequality for the sentence. A number increased by ten is less than twenty-six.

Maggie

Kono

Dorotea

10 + n ≤ 26

n + 10 < 26

n + 10 > 26


GO ON

Lesson 5-1 Translate Phrases into Inequalities

93

Guided Practice Translate each sentence to an inequality. 1

Five is greater than a number. variable: symbol:

A number is less than or equal to seven.

2

n >

symbol:

inequality: 5 > n 3

n ≤

variable:

inequality: n ≤ 7

Three times a number is greater than fifty-one.

A number less four is greater than five.

4

3x > 51

x-4>5

Step by Step Practice Translate the sentence to an inequality. 5

Six times a number plus 3 is less than fifty-two. Step 1 Determine the operation. The word “times” means

multiplication (·)

Step 2 Determine the other operation. addition (+) The word “plus” means

. .

Step 3 Choose a variable. The word “number” means to use a variable .

Step 5 Write the inequality.

6 · n + 3 < 52 or 6n + 3 < 52

Write a system of inequalities that represents each situation. 6

PIANO A piano teacher always has at least 30 boys and girls as students. There are always at least 5 more boys than girls.

b + g ≥ 30; b ≥ g + 5 7

FISHING Jerome is making at most 25 small and large fishing lures to sell. He is making at least 4 large lures.

s + ≤ 25; ≥ 4 94


. Copyright © by The McGraw-Hill Companies, Inc.

Step 4 Determine the inequality symbol. < The words “less than” mean

n


Problem-Solving Strategies ✓ Draw a diagram.

Solve. 8

Look for a pattern. Guess and check. Use a model. Solve a simpler problem.

FINANCE Rai charges $10 an hour for tutoring. Write an inequality expressing that the number of hours Rai needs to tutor, h, is at least $85. Understand

Read the problem. Write what you know. Rai wants to earn at least $85 . Rai charges $10 an hour. Pick a strategy. One strategy is to draw a diagram for an equation.

Plan

3BJT IPVSMZ DIBSHF

ˁ

/VNCFS PGIPVST 3BJUVUPST

Write the inequality.

10

·

h

≥ 85

Does the inequality make sense? Think of a number that 10 · h could equal. Is that amount $85 or greater?

Check


5PUBM NPOFZ FBSOFE

Each side of the model is not equal in value, so it is an inequality. Write the inequality. Rai wants to earn at least $85. Change the = sign to ≥ .

Solve

9

BUSINESS Sonja makes and sells candles. She earns $4 in profit for each candle she sells. Write an inequality expressing that c, the number of candles Sonja must sell for her profit, is greater than 4c > 110 $110. Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

¡'

¡$

10

TEMPERATURE The temperature at noon is shown at the right. It dropped 4ºC per hour. Write an inequality expressing that the temperature after h hours was less 30 - 4h < 10 than 10ºC.

o

o

o

o

o

o

GO ON Lesson 5-1 Translate Phrases into Inequalities

95

11

WEATHER During a winter storm, snow fell at a rate of 1.5 inches per hour. Write an inequality expressing that the amount of snow that fell after h hours was less than 12 inches.

1.5h < 12 12

How is writing an inequality similar to and different from writing an equation?

Answers may vary. Sample answer: The same key words are used to determine the operations, but the symbols used for equations and inequalities are different.

Skills, Concepts, and Problem Solving Translate each sentence to an inequality. 13

A number plus eight is not equal to seventy-five.

14

x 30 or 4n > 30 17

12 - n ≤ 15 18

x ≠ 4 + 13 19

A number times nine plus thirteen is greater than thirty-four.

A number divided by two minus five is greater than or equal to thirty-nine.

20

_

22

_

The sum of six and two times a number is less than forty-five.

6 + 2n < 45

96


The quotient of three and a number plus six is less than or equal to twenty-five.

3 ÷ n + 6 ≤ 25 or 3 + 6 ≤ 25 n

n ÷ 2 - 5 ≥ 39 or n - 5 ≥ 39 2 23

Fourteen more than a number is less than or equal to fifty.

14 + x ≤ 50

n · 9 + 13 > 34 or 9n + 13 > 34 21

The difference of twelve and a number is less than or equal to fifteen.

Three times a number minus ten does not equal twenty.

3n - 10 ≠ 20 24

A number times seven plus five is greater than or equal to eighteen.

7x + 5 ≥ 18


A number does not equal four plus thirteen.

The quotient of a number and two is less than five.


BASEBALL CARDS Suki collects New York Yankees and Los Angeles Dodgers baseball cards. She likes to have at least 40 cards. Also, Suki likes to have at least 25 New York Yankees cards.

y + d ≥ 40; y ≥ 25 26

NECKLACES Anna is making at least 12 necklaces to sell. She is making at least 2 more turquoise necklaces than glass beaded necklaces.

t + g ≥ 12; g + 2 ≥ t 27

GARDENING Jake is planting tomatoes and peppers in his garden. He wants at least 40 plants. He does not want any more than 15 pepper plants.

t + p ≥ 40; p ≤ 15

Write an inequality for each situation. 28

WRITING Terron writes at least four sentences in her journal every night. Write an inequality expressing that the number of sentences she wrote after d days was greater than 42.

4d > 42 Copyright © by The McGraw-Hill Companies, Inc.

29

SPORTS Jason scored two goals in each of four soccer games. Write an inequality expressing that the number of Jason’s goals after x games was at least 8.

x≥8 30

MONEY Ivan has $500 in his savings account. He earns $3 per month in interest. Write an inequality expressing that after m months, the amount in Ivan’s account is more than $550.

500 + 3m > 550 31

SCIENCE Lisa started recording a plant’s growth when it was as tall as shown. The plant grew 1 centimeter each day. Write an inequality expressing that after d days of recording the height, the plant was at least 15 centimeters tall.

DN

4 + d ≥ 15 GO ON Lesson 5-1 Translate Phrases into Inequalities cre8ive studios

97


33

34

A number sentence that compares two unequal expressions and uses < (is less than), > (is greater than), ≤ (is less than or equal to), ≥ (is greater than or equal to), or ≠ (is not equal to) is a(n) inequality .

expression A(n) and operation symbols.

is a combination of numbers, variables,

Writing in Math Explain how to write an inequality for the phrase “nine is greater than or equal to a number.”

The words “greater than or equal to” mean to use the greater than or equal to sign (≥). The word “number” means to use a variable, such as n. So, the inequality is 9 ≥ n.


(Lesson 4-4, p. 67)

x = 17; Samuel ate 17 hot dogs, and Shaquille ate 17 - 5 = 12 hot dogs.


(Lesson 4-1, p. 46)

The quotient of seven and a number is twenty.

37

_

7 ÷ n = 20 or 7 = 20 n For each phrase, name the operation. 38

n increased by 5

addition

98


Ten times a number is sixty-four.

10 · n = 64 or 10n = 64

(Lesson 3-2, p. 11)

39

the difference of c and 2

subtraction


NATURE At a hot dog-eating contest, Shaquille ate 5 fewer hot dogs than Samuel. Together they ate 29 hot dogs. This situation is represented by the equation x + (x - 5) = 29, where x is the number of hot dogs Samuel ate. Solve the equation to find the number of hot dogs Shaquille and Samuel ate.

Lesson

5-2

Solve Inequalities Using Addition and Subtraction

KEY Concept Addition and subtraction are inverse operations. When you add or subtract the same number from each side of an inequality, the inequality remains true.

Addition Property of Inequality

Example

Definition

(5 + 1) + 9 < (6 + 1) + 9 6 +9< 7 +9 15 < 16

If a < b, then a + c < b + c.

(9 + 5) + 7 > (8 + 5) + 7 14 + 7 > 13 + 7 21 > 20

If a > b, then a + c > b + c.

Solving an inequality means to find values of the variable that make the inequality true.

These properties are also true for a ≥ b and a ≤ b.

7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

VOCABULARY inverse operations operations that undo each other (Lesson 1-4, p. 4) variable a letter or symbol used to represent an unknown quantity (Lesson 1-4, p. 4) Addition Property of Inequality adding the same amount to each side of an inequality keeps the inequality true


To solve an inequality involving addition or subtraction, get the variable by itself. Use inverse operations.

Example 1

YOUR TURN!

Solve x + 1 < 9.

Solve x - 4 > 12.

1. Locate the side of the inequality with the variable. What operation is used? addition

1. Locate the side of the inequality with the variable. What operation is used?

subtraction

What is the inverse operation? subtraction x+1 16 GO ON Lesson 5-2 Solve Inequalities Using Addition and Subtraction

99

Example 2

YOUR TURN! Solve y + 3.5 ≤ 10.

Solve y - 1.2 ≥ 5.3. 1. Locate the side of the inequality with the variable. What operation is used? subtraction


addition What is the inverse operation?

What is the inverse operation? addition

subtraction

y - 1.2 + 1.2 ≥ 5.3 + 1.2 y ≥ 6.5

y + 3.5

y + 3.5 ≤ 10 ≤ 10 y ≤ 6.5

- 3.5

- 3.5

Who is Correct? Solve q - 26 ≤ 34.

Gail q - 26 ≤ 34 q - 26 ≤ 34 - 26 q≤8

Ajay

Berto

q - 26 ≤ 34 q ≤ 34

q - 26 ≤ 34 26 q - 26 + 26 ≤ 34 + 60 ≤ q

Guided Practice Write the operation that should be used to solve each inequality. 2 1 r - __ ≥ 23 2 n + 7 ≤ 32 5

addition

3

12 ≥ t - 9

addition

100


subtraction

4

13 7 < c + ___ 2

subtraction




5 __ + x ≤ 65. 2

Step 1 Locate the side of the inequality with the variable. What addition operation is used? . Step 2 What is the inverse operation? 5 __ 2

5 __ 2

+ x ≤ 65 x ≤ 62.5

subtraction

.

5 __ 2

Solve each inequality. 6

k - 9 < 12

addition

inverse operation: Add

9

to each side.

k < 21


7

m + 6 > 13 inverse operation: Subtract

6

subtraction

from each side.

m>7 8

7 + p ≤ 1.9

p ≤ -5.1

9

w - 0.8 ≥ 3

w ≥ 3.8

GO ON Lesson 5-2 Solve Inequalities Using Addition and Subtraction

101



Solve. 10

HEALTH A person should eat no more than 2,400 milligrams of sodium per day. Carlos’s breakfast contained 660 milligrams of sodium. His lunch contained 1,300 milligrams of sodium. What is the most sodium, s, Carlos should eat at dinner? Understand

Draw a diagram. Look for a pattern. Guess and check. Solve a simpler problem. ✓ Work backward.

Read the problem. Write what you know. Carlos should have at most 2,400 milligrams of sodium each day. Carlos has already had 660 and 1,300 milligrams today.

Plan

Pick a strategy. One strategy is to work backward. Start with the total sodium for the day. Subtract what Carlos has already had. Use that number to write an inequality.

Solve

2,400 total sodium -

1,740

660 breakfast

-

subtotal

1,300 lunch

= 1,740 = subtotal = =

440 dinner

the most

Write the inequality. s

≤

440

Carlos should eat 440 milligrams of sodium or less at dinner. Check

Add the milligrams of sodium that Carlos had for breakfast (660), lunch (1,300), and dinner (s). Substitute your answer for s to check. 660 + 1,300 + 440 ≤ 2,400

2,400 ≤ 2,400

102



Is the last number the most or the least sodium that Carlos should have at dinner?

11

MUSIC A CD holds up to 80 minutes of music. Raven is making a CD. She has 53 minutes of music on a play list. Write and solve an inequality to find m, the number of minutes of music Raven can add to the CD. Check off each step.

53 + m ≤ 80; m ≤ 27

12

✔

Understand

✔

Plan

✔

Solve

✔

Check

AIRPLANES The ATR-42 can hold a maximum of 66 passengers. There are 22 passengers on board. Write and solve an inequality to find n, the number of additional passengers that can board the plane.

n + 22 ≤ 66; n ≤ 44

AIRPLANES The ATR-42 is a turboprop plane.

Is 44 a solution of the inequality r - 31 < 15? Explain.

13

Yes; substitute 44 for r and then solve: 44 - 31 < 15; 13 < 15.


Skills, Concepts, and Problem Solving Solve each inequality. 14

6 + t > 56 t>

16

18

50

3 + g ≥ 3.1 g≥

15

k> 17

0.1

h + 1 ≤ 19

19

1 94 < f + __ 2

_

93 1 < f 2

10

b - 4.2 ≤ 1.2 b≤

h ≤ 18 20

10 + k > 20

5.4

y - 8 ≥ 54

y ≥ 62 21

2.8 > a - 1.8

4.6 > a GO ON Lesson 5-2 Solve Inequalities Using Addition and Subtraction

Gabriel Bouys/AFP/Getty Images

103

22

j - 21 < 19

40

j
__ __ 3

23

n>2

3

y - 5 ≤ 41.6

b>

25

d-1>0

27

29

31

3.8 + m ≥ 11.4

33

m ≥ 7.6

4 1 _ w≤

c - 4 ≥ 36.9

f - 2 > -2

2 32 5

32

4

f>0

3>4 x + __ 5

_

3 1 + w ≤ __ __

c ≥ 40.9

d>1 30

17

2

y ≤ 46.6 28

b - 5 > 12

9.2 + k < 21.6

k < 12.4 Copyright © by The McGraw-Hill Companies, Inc.

Solve. 34

ENTERTAINMENT Kristen wants to buy tickets to the circus for her family and friends. There are 14 people who are going to the circus. Write and solve an inequality to find n, the number of additional people needed so that Kristen can get the lower price for groups.

14 + n ≥ 20; n ≥ 6 35

NUMBER SENSE The sum of Jessica’s age, x, and Benito’s age, 12, is at least 25. Write and solve an inequality to find Jessica’s age.

x + 12 ≥ 25; x ≥ 13 104


Michael Newman/PhotoEdit

ENTERTAINMENT The circus offers reduced prices for groups of 20 or more people.


37

38

39

A number sentence that compares two unequal expressions and uses < (is less than), > (is greater than), ≤ (is less than or equal to), ≥ (is greater than or equal to), or ≠ (is not equal to) to compare inequality . two unequal expressions is called a(n)

Addition The Property of Inequality states that adding the same amount to both sides of an inequality keeps the inequality balanced. Addition and subtraction are inverse operations because they undo each other. Writing in Math Explain how to solve the inequality 16 + v ≤ 17.

Add -16 to each side of the inequality: 16 + (-16) + v ≤ 17 + (-16); v ≤ 1.

Spiral Review Solve.


40

(Lesson 5-1, p. 90)

BAKING Mrs. Diaz baked 60 cookies for a bake sale. Sebastian ate some cookies. There were less than 50 cookies left for the bake sale. Write an inequality to find c, the number of cookies Sebastian ate.

60 - c < 50

Solve each equation. 41

22 + h = 31 h=

43

(Lesson 4-2, p. 53)

42

9

Show that adding 7 on each side of (19 + 5) = (16 + 8) results in a true equation. (Lesson 1-3, p. 19)

a - 56 = 16 a=

44

72

Show that multiplying by 3 on each side of (4 · 8) = (16 · 2) results in a true equation. (Lesson 1-3, p. 19)

24 + 7 = 24 + 7

32 · 3 = 32 · 3

31 = 31

96 = 96 Lesson 5-2 Solve Inequalities Using Addition and Subtraction

105

Chapter

Progress Check 1

5


Write the symbol that should be used for each inequality. 7AF1.1 1

A number is greater than or equal to 5.

2

≥

Nine divided by a number is less than 27.

23

subtraction

addition

Translate each sentence to an inequality. 7AF1.1 5

The product of twelve and a number is greater than thirty-eight.

6

The quotient of a number and four plus seven is less than or equal to sixty-four.

_

n ÷ 4 + 7 ≤ 64 or n +7 ≤ 64 4

12 · n > 38 or 12n > 38 Solve each inequality. 7AF4.0 7

14 + a ≥ 53

9

39

n - 5 < 110 n < 115

x - 36 < 41 x
b · c.

If a > b, then a · c > b · c.

If a > b, then a · c < b · c.

_ _ _

_ _ _

a b If a < b, then c < __ c a b If a > b, then c > c

a b If a < b, then c > __ c a b If a > b, then c < c

These properties also hold true for a ≥ b and a ≤ b.

VOCABULARY Multiplication Property of Inequality When multiplying each side of an inequality by the same positive number, the inequality remains true. When multiplying each side of an inequality by the same negative number, the inequality symbol reverses and the inequality remains true. Division Property of Inequality When dividing each side of an inequality by the same positive number, the inequality remains true. When dividing each side of an inequality by the same negative number, the inequality symbol reverses and the inequality remains true.

Example 1

YOUR TURN!

Solve 7z ≤ 63.

Solve 8y ≥ 32.

1. Locate the side of the inequality with the variable. What operation is used? multiplication 7z ≤ 63

1. Locate the side of the inequality with the variable. What operation is used? multiplication

2. What is the inverse operation? division

7z _

_

≤ 63 7 7 z≤9

3. Are you dividing by a positive or negative number? positive 4. Will the symbol reverse? no

2. What is the inverse operation? division 3. Are you dividing by a positive or negative number? positive

8y ≥ 32 8y 32 ______ ≥ ______

8

4. Will the symbol reverse? no

8

y≥

4

GO ON

Lesson 5-3 Solve Inequalities Using Multiplication and Division

107

Example 2 Solve

_y > 5.1. 3

1. Locate the side of the inequality with the variable. What operation is used? division What is the inverse operation? multiplication y > 5.1 3 y · 3 > 5.1 · 3 3 y > 15.3

_

YOUR TURN! Solve

_

5


division What is the inverse operation?

multiplication

_

Example 3

_p ≥ 2.1.

_p

_p ≥ 2.1 5

× 5 ≥ 2.1 × 5 5 p ≥ 10.5

YOUR TURN!

Solve - z ≤ 12. 6

Solve -5z ≤ 40.

1. Locate the side of the inequality with the variable. What operation is used? division


3. Are you dividing by a positive or negative number? negative 4. Will the symbol reverse? yes - z ≤ 12 6 z - (-6) ≥ 12(-6) 6

_

_

z ≥ -72

2. What is the inverse operation?

division 3. Are you dividing by a positive or negative negative number? 4. Will the symbol reverse? yes -5z ≤ 40 -5z ≥ 40

_ _ –5 –5 z ≥ -8

108



2. What is the inverse operation? multiplication

multiplication

Who is Correct? Solve 5d ≤ 60.

Ying

Kaylee

Cesar

5d ≤ 60 5d ≤ 60 60 60 d≤1

5d ≤ 60 5d · 5 ≤ 60 · 5 d ≤ 300

5d ≤ 60 5d ≤ 60 5 5 12 d≤

_ _

_ _


Guided Practice Write the operation that should be used to solve each inequality. Will the symbol reverse? p division, yes multiplication, no 1 -9k < 72 2 __ > 3 8 20 m 5 division, no multiplication, yes 3 4b ≤ ___ 4 - __ ≥ __ 3 7 2



n ≤ 2. Solve -__ 6 Step 1 Locate the side of the inequality with the variable. What division operation is used? Step 2 What is the inverse operation? multiplication Step 3 Are you multiplying by a positive or negative number?

negative

n - __ · 6

-6

≥2·

-6

n ≥ -12

Step 4 Will the symbol reverse? yes

GO ON Lesson 5-3 Solve Inequalities Using Multiplication and Division

109


-7c > 21

8

division -7 .

inverse operation: Divide each side by c< 3 8

q __ ≥6

7

-0.3m < 36

9

m > -120

inverse operation: multiplication Multiply each side by 8 . q ≥ 48 0.6h < 54

h < 90

x≥


2

Look for a pattern. Guess and check. Act it out. ✓ Write an inequality. Work backward.

SCHOOL Mrs. Cheeti spent at least 270 minutes grading papers for 90 students. Write and solve an inequality to find how long on average Mrs. Cheeti spent grading each paper. Understand

-45 _


Solve. 11

9 x ≤ __ -__ 5 2

10

Read the problem. Write what you know. Mrs. Cheeti spent at least 270 minutes grading papers. Mrs. Cheeti had 90 papers to grade. Pick a strategy. One strategy is to write an inequality.

Solve

Let n = the number of minutes per paper. Write an inequality that matches the situation. n ≥ ______________________ 90 270 __________________ _· ________________ __

number of minutes · number of papers ≥ total number of minutes

Use the inverse operation to solve.

n · 90 ≥ 270 90n 270 ____ ≥ ____

90 90 n≥

Mrs. Cheeti spent at least Check

Substitute than 3

3

90 (

3


minutes grading each paper.

or any number greater for n to check. ) ≥ 270

270 ≥ 270 110

3

3


Plan

12

MONEY Nadia bought 10 energy bars. She gave the cashier a $20 bill and got change back. Write and solve an inequality to find c, the cost of one energy bar. Check off each step.

10c < 20; c < 2

13

✔

Understand

✔

Plan

✔

Solve

✔

Check

BASEBALL Ginger accidentally broke a window with a baseball. A new window costs at least $128. Write and solve an inequality to find x, the minimum amount Ginger should pay each month for the window to be paid off in 8 months. 8x ≥ 128; x ≥ 16 Is y > 3 the solution of the inequality -4y > -12? Explain.

14

No; when you divide by a negative, the inequalilty symbol must be reversed.


Skills, Concepts, and Problem Solving Solve each inequality. b 15 ___ ≤ 5x 12

16

15w > 7.5

60 ≤ 5 18

x < 14 -___ 13

-6d < 7.2

19

14p > 28

11m < 154

m < 14

s ≤ -50 20

r ≤9 ___ 11

p>2 22

a ≥4 -___ 20

d > -1.2 24

-2s ≥ 100

w > 0.5

x > -182 21

17

r ≤ 99 23

a ≤ -80 25

x ≤ 12 __

x ≤ 48

17

b ≤ -85 26

4

-b ≥ 5 ___

-9c > 72

c < -8 GO ON

Lesson 5-3 Solve Inequalities Using Multiplication and Division

111

Solve. 27

ENTERTAINMENT The total cost of Susan’s purchase was at least $135. Write and solve an inequality to find x, the number of CDs Susan bought. 15x ≥ 135; x ≥ 9

28

TEMPERATURE Mr. Ramirez gave each of his 18 students at least 5 colored pencils. Write and solve an inequality to find n, the number of colored pencils Mr. Ramirez gave out.

_n ≥ 5; n ≥ 90 18

ENTERTAINMENT Susan


bought some CDs for $15 each.

Multiplication The Property of Inequality states that multiplying each side of an inequality by the same positive number keeps the inequality true.

30

division Multiplication and operations because they undo each other.

31

z ≥ 13. Writing in Math Explain how to solve the inequality -__ 4

are inverse

Multiply each side of the inequality by -4 and reverse the symbol since -4 is a negative; z · -4 ≤ 13 · -4; z ≤ -52. 4

_

Solve. 32

(Lesson 5-2, p. 97)

HOBBIES A metro bus can hold a maximum of 90 passengers. There are 53 passengers on the bus. Write and solve an inequality to find n, the number of additional passengers the bus can hold.

n + 53 ≤ 90; n ≤ 37

Solve each equation. 33

(Lesson 4-4, p. 67)

4b + 3 = 27

6

b=

Name the like terms in each expression. 35

5t - 8 + 6t + 10

5t and 6t; -8 and 10 112

34


IT Stock Free/SuperStock

y __ -9=2 6

y=

66

(Lesson 3-3, p. 21)

36

9 - c + 4 + 7c

-c and 7c; 9 and 4


Spiral Review

Lesson

5-4 Solve Multi-Step Inequalities KEY Concept As with multi-step equations, you need to get the variable alone on one side of the inequality sign to solve multi-step linear inequalities. To do this, you use inverse operations more than one time. First undo the addition or subtraction, using the Addition Property of Inequality. Then undo the multiplication or division to get the variable alone using the Multiplication Property of Inequality .

7AF4.0 Students solve simple linear equations and inequalities over the rational numbers. 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.

inverse operations operations that undo each other (Lesson 1-4, p. 25)

Addition Property of Inequality Adding the same amount to each side of an inequality keeps the inequality true. (Lesson 5-2, p. 97)

6TFBOJOWFSTFPQFSBUJPOmSTUUPJTPMBUF UIFUFSNXJUIUIFWBSJBCMF5IFPQQPTJUF PGJT

BOE NBLFB[FSPQBJS

Y @@@@@@@@@ @@@@@ Y


6TFBOJOWFSTFPQFSBUJPOUPDSFBUFBO FRVJWBMFOUGPSNPGPOF5IFPQQPTJUFPG JTµ5IFJOFRVBMJUZTZNCPM SFWFSTFT

Y @@@@ @@@@ Y

To decide which inverse operation to use, it might be easier for you to look at the related equation.

Multiplication Property of Inequality When multiplying each side of an inequality by the same positive number, the inequality remains true. When multiplying each side of an inequality by the same negative number, the inequality symbol reverses and the inequality remains true. (Lesson 5-3, p. 105)

zero pair two numbers when added have a sum of zero; example: +5 and -5 (Lesson 4-2, p. 49)

Example 1 Solve 6a - 3 < 15. 1. Locate the side of the inequality with the variable. What operations are used? subtraction; multiplication 2. What are the inverse operations to be used, in order, to solve the inequality? addition; division 3. Are you dividing by a positive or negative number? positive 4. Will the symbol reverse? no

6a - 3 + 3 < 15 + 3 6a < 18 6a _ 18 _ < 6

6

a a.

Lesson 5-4 Solve Multi-Step Inequalities

113

YOUR TURN! Solve -9y + 4 ≥ 40. 1. Locate the side of the inequality with the variable. What addition; multiplication operations are used? -9y + 4

-4 ≥ 40 -4 -9y ≥ 36

2. What are the inverse operations to be used, in order, subtraction; division to solve the inequality? 3. Are you dividing by a positive or negative number?

negative

9y 36 ______ ≤ ______

–9

–9

y ≤ -4

4. Will the symbol reverse? yes

Example 2 Solve 5 ≥

_c + _1. 4

2

1. Locate the side of the inequality with the variable. What operations are used? addition, division

4.5 · 4 ≥

3. Are you multiplying by a positive or negative number? positive 4. Will the symbol reverse? no

114


_ _

c 1 4 ≥ 2 4

c·4 _

18 ≥ c 3FXSJUFBT

or c ≤ 18

4


2. What are the inverse operations to be used, in order, to solve the inequality? subtraction; multiplication

_ _ _ _

1 5-1 ≥ c + -1 2 2 2 4

YOUR TURN! Solve 10
m ·

3. Are you multiplying by a positive or negative number? negative 4. Will the symbol reverse?

_ _

2 10 < m 5 -22

yes

–22 _ –22

-231 > m or m < -231

Who is Correct?


Solve -2c - 7 ≥ 9.

Jena

Dario

Dorothy

-2c - 7 ≥ 9

-2c - 7 ≥ 9

-2c - 7 ≥ 9

-2c - 7 + 7 ≥ 9 + 7

-2c - 7 + 7 ≥ 9 + 7

-2c - 7 + 7 ≥ 9 + 7

-2c ≥ 17 c ≥ 17 -2 -2 c ≤ -8.5

_ _

-2c ≥ 16 -2c ≥ 16 -2 -2 c ≥ -8

_ _

-2c ≥ 16 -2c ≤ 16 -2 -2 c ≤ -8

_ _


GO ON Lesson 5-4 Solve Multi-Step Inequalities

115

Guided Practice Write the operations in the order that should be used to solve each inequality. Will the symbol reverse? 1

g __ -9≤2 3

addition; multiplication; no

subtraction; yes

-b + 7 ≥ 23

2


w + 5 > 12. Solve ___ -4 Step 1 Locate the side of the inequality with the variable. What operations are used? addition; division

w _ + 5 + (-5) > 12 + (-5) -4

_

Step 2 What are the inverse operations to be used, in order, to solve the inequality? subtraction; multiplication

-w>7 4

Step 3 Are you multiplying by a positive or negative number? negative

-4 · -w > -4 · 7 4

_

w < -28

Step 4 Will the symbol reverse? yes

s≤ 6

116

z+5≥8 __ 9

z ≥ 27


5

-7t + 2 > 16 -7t + 2 -

2

-7t >

14

-7t < ____ 14 ____

-7

-7 t< 2

18 7

2

> 16 -

-6c - 4 < 3.8

c > -1.3

8

x + 6 < 0.8 -__ 8

x > 41.6

9

3f + 5 ≤ 3.2

f < -0.6


Solve each inequality. s 4 __ - 3 ≤ 6 2 s -3+ 3 ≤6+ 3 __ 2 s≤ 9 __ 2 s · 2 ≤9· 2 __ 2


Problem-Solving Strategies Look for a pattern. Guess and check. Act it out. Solve a simpler problem. ✓ Write an inequality.

Solve. 10

FITNESS For an action movie, the lead actor has to lift at least 228 pounds. The actor can only lift 198 pounds. The actor plans to lift an additional 10 pounds each month. Write and solve an inequality to find the number of months it will take the actor to be able to lift enough for the movie. Understand

Read the problem. Write what you know. The actor needs to lift at least 228 pounds. He can lift 198 pounds. He wants to add an additional 10 pounds each month.

Plan

Pick a strategy. One strategy is to write an inequality.

Solve

Let m = the number of months needed to lift the desired weight. Write an inequality that matches the situation.

198

10 · m + 10 pounds · number + each month of months

can lift

≥ ≥

228 wants to lift

Use the inverse operations to solve the inequality. Copyright © by The McGraw-Hill Companies, Inc.

10m + 198 ≥ 228 10m + 198 - 198 ≥ 228 - 198 10m ≥ 10m ____ ≥

10

m≥

30

Simplify each side.

30 ___

10

Divide each side by 10.

3

Simplify each side.

The actor will need at least additional weight. Check

Subtract 198 from each side.

3

months to lift the

Substitute 3 or any number greater than 3 for m to check.

GO ON Lesson 5-4 Solve Multi-Step Inequalities

117

11

FINANCE Meena sells vacuum cleaners. She earns a base salary of $200 per week plus $25 for each vacuum cleaner she sells. Write and solve an inequality to find v, the number of vacuum cleaners Meena needs to sell in order to make at least $500 per week.

200 + 25v ≥ 500; v ≥ 12; Meena needs to sell at least 12 vacuum cleaners per week.

12

✔

Understand

✔

Plan

✔

Solve

✔

Check

TRAVEL Jaime and Brandon are driving 257 miles to a lake. Jaime drove 83 miles. Brandon will drive the rest of the way. Solve 83 + 3s < 257 to find the speed, in miles per hour, Brandon needs to drive to reach the campground in 3 hours. s < 58 q Is q ≤ (-12) the solution to the inequality -__ - 4 ≥ 2? 2 Explain.

13

q -12 ; _ _q ≥ _ ≥ 6; 2 -2 2 -q -q _ + (-4) ≥ 6 + (-4); _ - 4 ≥ 2 Yes; start with q ≤ (-12); 2

2


9k - 3 ≤ 78

15

k≤9 g 16 __ 2

-5≥3

g ≥ 16

17

-9c + 7 ≤ 70

c ≥ -7

p __ + 10 > 16 5

18

p > 30 -3h + 4 > 4

h37 8

5a - 3 < 19

a < 4.4

Solve. 7AF4.0, 7NS1.3 7

SHOPPING Tito has a coupon for $5 off the price of one T-shirt. He bought 7 T-shirts and spent less than $58. Write and solve an inequality to find x, the price of each T-shirt.

7x - 5 < 58; x < 9

TRAVEL The rental agency charges $32 per day to rent a car. They also charge $0.15 per mile driven. You want to take a 5-day trip and have budgeted $250 for the rental car. Write and solve an inequality to find m, the most miles you can drive to stay within your budget.

32 · 5 + 0.15m ≤ 250; m ≤ 600

120



8

Lesson

5-5 Graph Inequalities on a Number Line 7AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

KEY Concept The equation x = 4 has one solution, which is 4. The inequality x > 4 has an infinite number of solutions, such as 1 , 25, and 407. Because you cannot list all of the 4.01, 4.2, 5__ 2 solutions, you can show them on a number line. Example

x>2

x 5. Graph the solution.

Susana 2n + 1 > 5 2n + 1 - 1 > 5 - 1 2n > 4 4 2n > __ ___ 2 2 n>2

0 1 2 3 4 5 6 7 8 9 10

Aisha

Sean

2n + 1 > 5 2n + 1 - 1 > 5 - 1 2n > 4 4 2n > __ ___ 2 2 n>2

2n + 1 > 5 2n + 1 - 1 > 5 - 1 2n > 6 6 2n > __ ___ 2 2 n>3


Guided Practice Write an inequality for each graph. 1

2


x51 2


Graph 7 < c. Step 1 Rewrite the inequality so the variable is on the left side: c > 7.

open circle to graph the inequality, Step 2 Use a(n) because 7 is not a solution. right Step 3 Shade to the greater than 7. Step 4 Graph the inequality.

of 7 because the solution is

GO ON

Lesson 5-5 Graph Inequalities on a Number Line

123

Graph each inequality. 6

z≤3

7

closed

Use a(n)

circle.

8

1 k > -1__ 2

open

Shade to the right –10 –9

10

–8

–7

–6

–6

–5

–4

–3

–2

–1

0

–6

–5

–5


open

–4

–3

–2

–1

–4

–3

–2

–1

0

circle.

–7

–6

–5

–4

–3

–2

–1

0

–8

–7

–6

–5

–4

–3

–2

–1

0

–7

–6

–5

–4

–3

–2

–1

0

0p -5__ 2 –10 –9

–8


circle.

circle.

13

–10 –9

124

11

–10 –9

16

9>t

14

g < 8.2

12

Shade to the right

9

Use a(n)

open

Use a(n)

Shade to the left of 3.

m > 5.5


Problem-Solving Strategies Draw a diagram. Look for a pattern. Guess and check. Solve a simpler problem. ✓ Use logical reasoning.

Solve. 18

FASHION Last year, Odessa had 12 pairs of shoes. This year, she has at least 16 pairs of shoes. Write and solve an inequality to find p, the number of pairs of shoes Odessa bought this year. Graph the solution.

Understand

Read the problem. Write what you know. Odessa had 12 pairs of shoes. Odessa now has at least 16 pairs of shoes.

Plan

Pick a strategy. One strategy is to use logical reasoning. Think first about the total number of shoes that Odessa has now. “At least 16 ” means to use the inequality symbol ≥ so that the right side of ≥16 . the inequality is The expression for the left sign of the inequality is 12 + p . 12 + p ≥ 16 + p ≥ 16 p≥ 4

Solve

12


12 -

12

Graph the inequality. Use a(n) closed circle to graph the inequality, because 4 is a solution. Shade right of 4, because the inequality uses to the the greater than or equal to sign.

Check

Substitute any number greater than or equal to 4 for p to check. Substitute 4 for p. 12 +

4 16

≥ 16 ≥ 16

So, Odessa bought at least shoes this year.

4

or more pairs of GO ON Lesson 5-5 Graph Inequalities on a Number Line

125

19

MUSIC Craig spends 3 times as long practicing the guitar as practicing the piano. He spends more than 80 minutes each day playing both instruments. Write and solve an inequality to find t, the number of minutes Craig spends each day practicing the piano. Graph the solution. 3t + t > 80; t > 20

Check off each step.

20

✔

Understand

✔

Plan

✔

Solve

✔

Check

MONEY Malena has $12 to spend for lunch. She buys a large lemonade for $2 and a sandwich for $6. Write and solve an inequality to find d, the amount of money Malena has to spend on 2 + 6 + d ≤ 12; d ≤ 4 dessert. Graph the solution.

Is 3 a solution of the inequality graphed below? Explain.

21

Skills, Concepts, and Problem Solving Write an inequality for each graph. 22

23

x≤7 24

–10 –9

–8

–7

–6

–5

–4

–3

–2

–1

0

x > -9

_

–8

–7

–6

–5

x ≥ -5 1 4 126

–10 –9


–4

–3

–2

–1

0

25

_

x < 43 4


No; the open circle means that 3 is not a solution.

Graph each inequality. 26

a > -1

27

–10 –9

28

–8

–7

–6

–5

–4

–3

–2

–1

0

z≤6

29

k (is greater than), ≤ (is less than or equal to), ≥ (is greater than or equal to), or ≠ (is not equal to) to compare two unequal expressions. Addition Property of Inequality The that adding the same amount to each side of an inequality keeps the inequality true.

states

4

Is 2 part of the solution? (yes or no)

no

5

yes

Lesson Review Translate Phrases into Inequalities

n - 4 ≤ 12 7

_

15 ÷ n > 5 or 15 > 5 n

128


Eight times a number is greater than or equal to forty. Group the sentence into parts that translate to a number or symbol. Eight multiplied by a number is greater than or equal to 40. ⎫ ⎬ ⎭

Fifteen divided by a number is greater than 5.


8 · n The inequality is 8 · n ≥ 40 or 8n ≥ 40.

≥

{

A number minus four is less than or equal to 12.

Example 1

⎧ ⎨ ⎩

6

(pp. 90–96)

⎧ ⎨ ⎩

Translate each sentence to an inequality.

⎧ ⎨ ⎩

5-1

40.


Is 4 part of the solution? (yes or no)

5-2

Solve Inequalities Using Addition and Subtraction


Locate the side of the inequality with the variable. What operation is used? subtraction What is the inverse operation? addition

28

1 < 11 x - __ 2 x < 11

11

Solve x - 3 > 14.

33

t − 7 < 21 t
56 y>

9

(pp. 97–103)

x - 3 + 3 > 14 + 3 x > 17

_1 2

2 14 + w > __ 3 w > -13 1

_ 3

12

4+m≥8 3__ 5 m≥

_

41 5

Solve each inequality. Copyright © by The McGraw-Hill Companies, Inc.

13

6 14 14

d + 4 < 45

d < 41 15

3.8 + w ≤ 5

Example 3 Solve y + 10 ≥ 19. Locate the side of the inequality with the variable. What operation is used? addition What is the inverse operation? subtraction y + 10 - 10 ≥ 19 - 10 y≥9

w ≤ 1.2 16

8.1 + g ≥ 10

g ≥ 1.9 17

1 < 11 n - 1__ 2 n < 12 1

_ 2


129

5-3

Solve Inequalities Using Multiplication and Division


5r > 55

-2h ≤ 16

h ≥ -8 20

4 -3x > __ 5

4 x < -_ 15

21

2 6x ≤ __ 3

_

x≤1 9


24

25

s ≤ 45

t < 10 ___

What is the inverse operation? division Are you dividing by a positive or negative number? negative Will the symbol reverse? yes -4x ≥ 28 28 -4x ≤ ___ ____ -4 -4 x ≤ -7

Example 5 Solve

_q ≤ 8.1. 4

Locate the side of the inequality with the variable. What operation is used? division

-7 t > -70

What is the inverse operation? multiplication

-__c > 3.2 6 c < -19.2

Will the symbol reverse? no q __ · 4 ≤ 8.1 · 4 4 q ≤ 32.4

p __ ≤ 5.4 2

p ≤ 10.8

130

Locate the side of the inequality with the variable. What operation is used? multiplication


Are you multiplying by a positive or negative number? positive


23

s ≤5 __ 9

Example 4 Solve -4x ≥ 28.

r > 11 19

(pp. 105–110)

5-4

Solve Multi-Step Inequalities

(pp. 111–117)


Example 6

-4t - 1 ≤ 15

Solve -3a + 5 < 17.

t ≥ -4 27

Locate the side of the inequality with the variable. What operations are used? addition, multiplication

6e + 5 ≥ 41

What are the inverse operations to be used, in order, to solve the inequality? subtraction, division

e≥6 f __ -9>0

28

7

Are you dividing by a positive or negative number? negative

f > 63

Will the symbol reverse? yes

Example 1

c 29 2 > ___ - 1 -3 -9 < c or c > -9

-3a + 5 - 5 < 17 - 5 -3a < 12 -3a ___ ____ > 12 -3 -3 a > -4


5-5

Graph Inequalities on a Number Line

Solve and graph each inequality. 30

Example 7

6 14 0

31

6

7

8

9

Use a closed circle to graph the inequality because 7.5 is a solution.

10 11 12 13 14 15

Shade to the left of 7.5 because the solutions are less than or equal to 7.5. Use the symbol ≤.

d + 4 ≤ -2.5

Graph the inequality.

d ≤ -6.5 –10 –9

–8

(pp. 119–125)

–7

–6

–5

–4

–3

–2

–1

0


131

Chapter

Chapter Test

5

Translate the sentence to an inequality. 7AF1.1

n > 11

1

A number is greater than 11.

2

Five times a number is less than or equal to 19.

3

A number increased by six is less than 15.

4

Eight divided by a number is greater than or equal to 27.

5n ≤ 19 n + 6 < 15

_8 ≥ 27 n


SCHOOL SUPPLIES Tasha tries to keep at least 10 pens and pencils. She likes to keep at least 2 more pencils than pens.

p + c ≥ 10; c ≥ p + 2 6

CHICKENS A farmer has red chickens and white chickens. She never has more than 20 chickens at a time. She likes to have at least 5 red chickens.

r + w ≤ 20; r ≥ 5 Solve each inequality. 7AF4.0 m + 9 > 20

8

m > 11 9

11

10

d ≥ 33

s 54

f < -9

s > -36

13

c-45 -5

q < -25

-3x + 9 > 39

x < -10

GO ON

Write an inequality for the graph. 7AF4.0

The inequality is x < 3.

17

Solve. 7AF4.0 18

WEATHER During a thunderstorm, rain fell at a rate of 1.25 inches per hour. Write an inequality to find h, the number of hours the rain fell before there were at least 5 inches total.

1.25h ≥ 5; h ≥ 4 19

AGES The sum of Megan’s age, y, and Paige’s age, 13, is no more than 30. Write and solve an inequality to find Megan’s age.

y + 13 ≤ 30; x ≤ 17


Correct the mistakes. 7AF4.0 20

STICKERS A sticker machine at a factory can make sticker sheets at a rate of 10 per minute. Write an inequality to find m, the number of minutes the machine must operate before there are at least 3,000 finished sticker sheets. To answer this question, Daniel wrote the following:

Daniel 10m < 3,000

What was his mistake? How should the inequality be written?

The inequality should be written as: 10m ≥ 3,000.

Chapter 5 Test

133

Chapter

5

Standards Practice


Which inequality shows sixteen is greater than or equal to nine less than x?

5

9y - 12 > 24

7AF1.1

A 16 ≥ x - 9

A y > 36

B 16 ≤ x - 9

B y > 16

C 16 ≤ 9 + x

C y>4

D 16 ≥ 9 + x

2

D y $50 G $8.50 × n ≤ $50 H n ÷ $8.50 ≥ $50 J $50 - n < $8.50

134


Which inequality is represented by this graph? 7AF4.0 –2

8

–1

0

1

2

3

4

A x+6≤8

C x-6

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