Algebra: Concepts and Skills

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Author: Laurie Boswell | Holt Mcdougal

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Copyright © 2004 by McDougal Littell, a division of Houghton Mifflin Company. Warning. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without prior written permission of McDougal Littell, a division of Houghton Mifflin Company, unless such copying is expressly permitted by federal copyright law. With the exception of not-for-profit transcription in Braille, McDougal Littell, a division of Houghton Mifflin Company, is not authorized to grant permission for further uses of copyrighted selections reprinted in this text without the permission of their owners. Permission must be obtained from the individual copyright owners as identified herein. Address inquiries to Manager, Rights and Permissions, McDougal Littell, a division of Houghton Mifflin Company, P.O. Box 1667, Evanston, IL 60204. ISBN: 0-618-37420-5 123456789–DWO–07 06 05 04 03 Internet Web Site: http://www.classzone.com ii

About the Authors 䉴

RON LARSON is a professor of mathematics at Penn State University at Erie, where he has taught since receiving his Ph.D. in mathematics from the University of Colorado in 1970. He is the author of a broad range of instructional materials for middle school, high school, and college. Dr. Larson has been an innovative writer of multimedia approaches to mathematics, and his Calculus and Precalculus texts are both available in interactive form on the Internet.

䉴

LAURIE BOSWELL is a mathematics teacher at Profile JuniorSenior High School in Bethlehem, New Hampshire. A recipient of the 1986 Presidential Award for Excellence in Mathematics Teaching, she is also the 1992 Tandy Technology Scholar and the 1991 recipient of the Richard Balomenos Mathematics Education Service Award presented by the New Hampshire Association of Teachers of Mathematics.

䉴

TIMOTHY D. KANOLD is Director of Mathematics and a mathematics teacher at Adlai E. Stevenson High School in Lincolnshire, Illinois. In 1995 he received the Award of Excellence from the Illinois State Board of Education for outstanding contributions to education. A 1986 recipient of the Presidential Award for Excellence in Mathematics Teaching, he served as President of the Council of Presidential Awardees of Mathematics.

䉴

LEE STIFF is a professor of mathematics education in the College of Education and Psychology of North Carolina State University at Raleigh and has taught mathematics at the high school and middle school levels. He is the 1992 recipient of the W. W. Rankin Award for Excellence in Mathematics Education presented by the North Carolina Council of Teachers of Mathematics, and a 1995–96 Fulbright Scholar to the Department of Mathematics of the University of Ghana. All authors contributed to planning the content, organization, and instructional design of the program, and to reviewing and writing the manuscript. Ron Larson played a major role in writing the textbook and in establishing the program philosophy. iii

y 5

(8, 4)

3

rise ⴝ 4ⴚ2ⴝ2

(3, 2) run ⴝ 8 ⴚ 3 ⴝ 5 1

3

5

7

9

11 x

䉴 REVIEWERS Pauline Embree Mathematics Department Chair Rancho San Joaquin Middle School Irvine, CA

Diego Gutierrez Mathematics Teacher Crawford High School San Diego, CA

The reviewers read and commented on textbook chapters in pre-publication format, particularly with regard to classroom needs.

䉴 TEACHER PANEL Courteney Dawe Mathematics Teacher Placerita Junior High School Valencia, CA

Diego Gutierrez Mathematics Teacher Crawford High School San Diego, CA

Dave Dempster Mathematics Teacher Temecula Valley High School Temecula, CA

Roger Hitchcock Mathematics Teacher Buchanan High School Clovis, CA

Pauline Embree Mathematics Department Chair Rancho San Joaquin Middle School Irvine, CA

Louise McComas Mathematics Teacher Fremont High School Sunnyvale, CA

Tom Griffith Mathematics Teacher Scripps Ranch High School San Diego, CA

Viola Okoro Mathematics Teacher Laguna Creek High School Elk Grove, CA

The Teacher Panel helped plan the content, organization, and instructional design of the program.

CALIFORNIA CONSULTING MATHEMATICIANS Kurt Kreith Don Chakerian Professor of Mathematics Professor of Mathematics University of California, Davis University of California, Davis

The California Consulting Mathematicians prepared the Mathematical Background Notes preceding each chapter in the Teacher’s Edition of this textbook.

iv

1

Connections to Algebra 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 2, 8, 16, 26, 36, 37, 42, 44, 49 Skills Review 7, 16, 32, 43 Reading Algebra 9, 11, 30, 31, 32 Writing Algebra 3 Vocabulary Tip 5, 24, 31, 43 Keystroke Help 13 Test Tip 14, 41, 60

APPLICATION HIGHLIGHTS

INT

Scuba Diving 1, 53 Race Cars 4 Basketball 17 Veterinarians 26 Mach Number 28 Northwest Territory 34 Plant Growth 40 Braking Distance 46 Hot-air Ballooning 49 Chisholm Trail 52 NE ER T

INTERNET

1, 7, 10, 19, 25, 34, 38, 40, 42, 44, 46, 50, 52, 61

2

1.1 Variables in Algebra

3

1.2 Exponents and Powers

9

1.3 Order of Operations

15

QUIZ 1, 21

1.4 Equations and Inequalities 䉴 DEVELOPING CONCEPTS: Finding Patterns, 22

24

1.5 Translating Words into Mathematical Symbols

30

QUIZ 2, 35

1.6 A Problem Solving Plan Using Models

36

1.7 Tables and Graphs

42

1.8 An Introduction to Functions

48

QUIZ 3, 54

ASSESSMENT Chapter Readiness Quiz, 2 Quizzes, 21, 35, 54 Standardized Test Practice, 8, 14, 20, 29, 35, 41, 47, 53 Chapter Summary and Review, 55 Chapter Test, 59 Chapter Standardized Test, 60 Maintaining Skills, 61

Getting Ready

Pre-Course Test

xviii

A diagnostic test on key skills from earlier courses, referenced to the Skills Review (pp. 759–782)

Pre-Course Practice

xx

Additional practice on the skills in the Pre-Course Test, also referenced to the Skills Review

Contents

v

2

Properties of Real Numbers 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 66, 86, 93, 94, 101, 102, 108, 109, 113, 114 Skills Review 66, 111 Reading Algebra 65, 71, 107 Writing Algebra 94 Vocabulary Tip 100 Look Back 72, 79, 84, 87, 97, 99, 119, 120 Keystroke Help 80 Test Tip 126


INT

Helicopters 63, 75 Nome, Alaska 67 Stars 69 Space Shuttle 73 Planets 75 Golf Scores 82 Stock Market 88 Water Cycle 90 Flying Squirrels 95 Rappelling 97

64

2.1 The Real Number Line

65

2.2 Absolute Value

71

2.3 Adding Real Numbers 䉴 DEVELOPING CONCEPTS: Addition of Integers, 77

78

QUIZ 1, 83

2.4 Subtracting Real Numbers 䉴 DEVELOPING CONCEPTS: Subtraction of Integers, 84

86

2.5 Multiplying Real Numbers 䉴 DEVELOPING CONCEPTS: Multiplication of Integers, 92

93

2.6 The Distributive Property 䉴 DEVELOPING CONCEPTS: The Distributive Property, 99

100

QUIZ 2, 106

2.7 Combining Like Terms

107

2.8 Dividing Real Numbers

113

QUIZ 3, 118 Extension: Inductive and Deductive Reasoning

119

NE ER T

INTERNET

63, 67, 69, 73, 75, 80, 90, 91, 95, 97, 101, 104, 108, 115, 117, 127

vi

Contents

ASSESSMENT Chapter Readiness Quiz, 64 Quizzes, 83, 106, 118 Standardized Test Practice, 70, 76, 82, 91, 98, 105, 112, 117

Chapter Summary and Review, 121 Chapter Test, 125 Chapter Standardized Test, 126 Maintaining Skills, 127

3

Solving Linear Equations 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 132, 133, 138, 146, 151, 157, 158, 178, 184 Skills Review 165 Writing Algebra 177 Vocabulary Tip 144 Look Back 131, 152, 171 Test Tip 143, 194


INT

Bald Eagles 129, 181 City Parks 136 Newspaper Recycling 142 Thunderstorms 142 Earth’s Temperature 145 Steamboats 155 Cheetahs 155 Moons of Jupiter 162 Cocoa Consumption 167 Bottle-nosed Whales 175 NE ER T

INTERNET

129, 133, 136, 139, 142, 145, 148, 153, 155, 159, 161, 162, 164, 167, 170, 172, 173, 179, 181, 182, 185, 187, 195

130

3.1 Solving Equations Using Addition and Subtraction 䉴 DEVELOPING CONCEPTS: One-Step Equations, 131

132

3.2 Solving Equations Using Multiplication and Division

138

3.3 Solving Multi-Step Equations

144

QUIZ 1, 149

3.4 Solving Equations with Variables on Both Sides 䉴 DEVELOPING CONCEPTS: Variables on Both Sides, 150

151

3.5 More on Linear Equations

157

3.6 Solving Decimal Equations

163

QUIZ 2, 169

䉴

GRAPHING CALCULATOR: Solving Multi-Step Equations, 170

3.7 Formulas

171

3.8 Ratios and Rates

177

3.9 Percents

183

QUIZ 3, 188 Project: Planning a Car Wash

198

ASSESSMENT Chapter Readiness Quiz, 130 Quizzes, 149, 169, 188 Standardized Test Practice, 137, 143, 148, 156, 162, 168, 176, 182, 188 Chapter Summary and Review, 189

Chapter Test, 193 Chapter Standardized Test, 194 Maintaining Skills, 195 Cumulative Practice, Chapters 1–3, 196

Contents

vii

4

Graphing Linear Equations and Functions 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 203, 211, 212, 216, 223, 231, 238, 250, 251, 253, 254 Reading Algebra 230, 236, 254 Look Back 218, 252 Keystroke Help 250 Test Tip 264


INT

Cable Cars 201, 247 Wing Length 207 Triathlon 214 Boiling Point 214 Mount St. Helens 218 Zoo Fundraising 226 U.S.S. Constitution 234 Gold Bullion 237 Violin Family 240 Monarch Butterflies 257

4.1 The Coordinate Plane

203

4.2 Graphing Linear Equations 䉴 DEVELOPING CONCEPTS: Linear Equations, 209

210

4.3 Graphing Horizontal and Vertical Lines

216

QUIZ 1, 221

4.4 Graphing Lines Using Intercepts

222

4.5 The Slope of a Line 䉴 DEVELOPING CONCEPTS: Investigating Slope, 228

229

4.6 Direct Variation

236

QUIZ 2, 241

4.7 Graphing Lines Using Slope-Intercept Form 䉴 DEVELOPING CONCEPTS: Slope-Intercept Form, 242 䉴 GRAPHING CALCULATOR: Graphing a Linear Equation, 250

243

4.8 Functions and Relations

252

QUIZ 3, 258

NE ER T

INTERNET

201, 204, 207, 214, 215, 217, 224, 226, 227, 232, 237, 245, 248, 250, 256, 265

viii

Contents

202


5

Writing Linear Equations 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 270, 271, 273, 278, 279, 285, 292, 293, 295, 298, 299, 307, 308 Vocabulary Tip 274, 291 Look Back 300 Test Tip 318


INT

Archaeology 267, 303 Hurdling 273 Old Faithful 274 Water Pressure 282 Airplane Descent 289 The Chunnel 289 Speed of Sound 290 Movie Theaters 298, 299 Car Costs 302 City Street Plan 311 NE ER T

INTERNET

267, 271, 273, 274, 280, 282, 287, 289, 295, 298, 303, 306, 310, 319

268

5.1 Slope-Intercept Form

269

5.2 Point-Slope Form 䉴 DEVELOPING CONCEPTS: Point-Slope Form, 276

278

QUIZ 1, 284

5.3 Writing Linear Equations Given Two Points

285

5.4 Standard Form

291

QUIZ 2, 297

5.5 Modeling with Linear Equations

298

5.6 Perpendicular Lines 䉴 DEVELOPING CONCEPTS: Perpendicular Lines, 305

306

QUIZ 3, 312

ASSESSMENT Chapter Readiness Quiz, 268 Quizzes, 284, 297, 312 Standardized Test Practice, 275, 283, 290, 296, 304, 311 Chapter Summary and Review, 313 Chapter Test, 317 Chapter Standardized Test, 318 Maintaining Skills, 319

Contents

ix

6

Solving and Graphing Linear Inequalities 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 324, 330, 337, 338, 342, 343, 344, 348, 349, 350, 356, 362, 363, 368, 369 Skills Review 340 Reading Algebra 323 Writing Algebra 325, 331 Vocabulary Tip 367 Keystroke Help 374 Test Tip 380

APPLICATION HIGHLIGHTS Music 321 Astronomy 325, 359 Mercury 327 Fly-fishing 338 Mountain Plants 343 Steel Arch Bridge 346 Water Temperature 352 Poodles 357 Fireworks 365 Nutrition 371

6.1 Solving Inequalities Using Addition or Subtraction

323

6.2 Solving Inequalities Using Multiplication or Division 䉴 DEVELOPING CONCEPTS: Investigating Inequalities, 329

330

6.3 Solving Multi-Step Inequalities

336

QUIZ 1, 341

6.4 Solving Compound Inequalities Involving “And”

342

6.5 Solving Compound Inequalities Involving “Or”

348

6.6 Solving Absolute-Value Equations 䉴 DEVELOPING CONCEPTS: Absolute-Value Equations, 354

355

QUIZ 2, 360

6.7 Solving Absolute-Value Inequalities

361

6.8 Graphing Linear Inequalities in Two Variables

367

QUIZ 3, 373

䉴

GRAPHING CALCULATOR: Graphing Inequalities, 374

Project: Investigating Springs INT

322

384

NE ER T

INTERNET

321, 327, 332, 337, 340, 346, 347, 349, 359, 365, 369, 371, 372, 374, 381

x

Contents

ASSESSMENT Chapter Readiness Quiz, 322 Quizzes, 341, 360, 373 Standardized Test Practice, 328, 335, 341, 347, 353, 360, 366, 372 Chapter Summary and Review, 375


7

Systems of Linear Equations and Inequalities 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 397, 409, 424 Reading Algebra 391 Look Back 390, 415, 417, 426 Keystroke Help 395 Test Tip 436


INT

Housing 387, 413 Web Sites 391 Softball 400 Volume and Mass 406 Beehive 407 Chemistry 410 Salary Plan 411 Gardening 413 Jewelry 421 Carpentry 421 NE ER T

INTERNET

387, 389, 391, 395, 398, 400, 404, 411, 413, 421, 428, 429, 437

388

7.1 Graphing Linear Systems 䉴 GRAPHING CALCULATOR: Graphing Linear Systems, 395

389

7.2 Solving Linear Systems by Substitution

396

7.3 Solving Linear Systems by Linear Combinations

402

QUIZ 1, 408

7.4 Linear Systems and Problem Solving

409

7.5 Special Types of Linear Systems 䉴 DEVELOPING CONCEPTS: Special Types of Systems, 415

417

7.6 Systems of Linear Inequalities 䉴 DEVELOPING CONCEPTS: Systems of Inequalities, 423

424

QUIZ 2, 430

ASSESSMENT Chapter Readiness Quiz, 388 Quizzes, 408, 430 Standardized Test Practice, 394, 401, 407, 414, 422, 429 Chapter Summary and Review, 431 Chapter Test, 435 Chapter Standardized Test, 436 Maintaining Skills, 437

Contents

xi

8

Exponents and Exponential Functions 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 444, 445, 456, 462, 463, 469, 476, 477, 478 Reading Algebra 441 Writing Algebra 449, 477 Look Back 443, 444, 471 Keystroke Help 451, 461, 471 Test Tip 494


INT

Bicycle Racing 439, 480 Irrigation Circles 445 Alternative Energy 447 U.S. History 453, 471, 473 Shipwrecks 457 World Wide Web 459 Baseball Salaries 467 Compound Interest 477 Car Depreciation 482 Pharmacists 486 NE ER T

INTERNET

439, 447, 450, 457, 461, 464, 466, 470, 473, 480, 486, 487, 495

xii

Contents

440

8.1 Multiplication Properties of Exponents 䉴 DEVELOPING CONCEPTS: Investigating Powers, 441

443

8.2 Zero and Negative Exponents

449

8.3 Graphs of Exponential Functions

455

QUIZ 1, 460

䉴

GRAPHING CALCULATOR: Exponential Functions, 461

8.4 Division Properties of Exponents

462

8.5 Scientific Notation

469

QUIZ 2, 474

8.6 Exponential Growth Functions 䉴 DEVELOPING CONCEPTS: Exponential Functions, 475

476

8.7 Exponential Decay Functions

482

QUIZ 3, 488

ASSESSMENT Chapter Readiness Quiz, 440 Quizzes, 460, 474, 488 Standardized Test Practice, 448, 454, 459, 468, 474, 481, 487 Chapter Summary and Review, 489 Chapter Test, 493 Chapter Standardized Test, 494 Maintaining Skills, 495

9

Quadratic Equations and Functions 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 500, 505, 507, 511, 512, 521, 522, 528, 534, 535, 537, 540, 548 Skills Review 516 Reading Algebra 499, 500, 533 Look Back 542, 546 Keystroke Help 501 Test Tip 504, 558


INT

Baseball 497, 538 Chess 503 Minerals 509 Sailing 513 Tsunamis 515 Dolphins 524 Golden Gate Bridge 528 Microgravity 530 Red-tailed Hawk 537 Financial Analysis 544

498

9.1 Square Roots

499

9.2 Solving Quadratic Equations by Finding Square Roots

505

9.3 Simplifying Radicals

511

QUIZ 1, 517

9.4 Graphing Quadratic Functions 䉴 DEVELOPING CONCEPTS: Graphing Quadratic Functions, 518

520

9.5 Solving Quadratic Equations by Graphing 䉴 GRAPHING CALCULATOR: Approximating Solutions, 532

526

9.6 Solving Quadratic Equations by the Quadratic Formula

533

QUIZ 2, 539

9.7 Using the Discriminant

540

9.8 Graphing Quadratic Inequalities 䉴 DEVELOPING CONCEPTS: Graphing Quadratic Inequalities, 546

547

QUIZ 3, 552 Project: Designing a Stairway

NE ER T

562

INTERNET

497, 503, 506, 509, 512, 513, 524, 525, 527, 532, 534, 537, 538, 542, 544, 549, 559

ASSESSMENT Chapter Readiness Quiz, 498 Quizzes, 517, 539, 552 Standardized Test Practice, 504, 510, 516, 525, 531, 538, 545, 551 Chapter Summary and Review, 553


Contents

xiii

10

Polynomials and Factoring 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 570, 582, 588, 596, 597, 604, 610 Skills Review 590, 616 Reading Algebra 568 Look Back 567, 569, 575, 583, 605 Test Tip 628

566

10.1 Adding and Subtracting Polynomials 䉴 DEVELOPING CONCEPTS: Addition of Polynomials, 567

568

10.2 Multiplying Polynomials 䉴 DEVELOPING CONCEPTS: Multiplying Polynomials, 574

575

10.3 Special Products of Polynomials

581

QUIZ 1, 587

10.4 Solving Quadratic Equations in Factored Form

588

Radio Telescope 565, 592 Picture Framing 579 Genetics 584, 586 Meteorite Crater 592 Landscape Design 598 Taj Mahal 600 Cliff Diver 605 Block and Tackle 612 Pole-vaulting 614 Terrarium 619

10.5 Factoring x 2 + bx + c 䉴 DEVELOPING CONCEPTS: Factoring x 2 + bx + c, 594

595

10.6 Factoring ax 2 + bx + c 䉴 DEVELOPING CONCEPTS: Factoring ax 2 + bx + c, 602

603

INT


NE ER T

INTERNET

565, 569, 576, 579, 583, 586, 589, 592, 598, 600, 605, 607, 612, 618, 629

QUIZ 2, 608

10.7 Factoring Special Products

609

10.8 Factoring Cubic Polynomials

616

QUIZ 3, 622


xiv

Contents

11

Rational Expressions and Equations 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 634, 639, 640, 647, 648, 652, 653, 659, 664, 666, 670, 671, 672, 673, 679 Skills Review 663 Reading Algebra 633, 678 Vocabulary Tip 633, 640 Keystroke Help 645 Test Tip 686


INT

Scale Models 631, 637 Clay Warriors 635 Fence Mural 637 Bicycling 641 Snowshoes 643 Ocean Temperatures 643 Air Pressure 650 Car Trip 666 Batting Average 675 NE ER T

INTERNET

631, 635, 641, 643, 645, 647, 654, 656, 659, 661, 665, 672, 675, 687

632

11.1 Proportions

633

11.2 Direct and Inverse Variation 䉴 GRAPHING CALCULATOR: Modeling Inverse Variation, 645

639

11.3 Simplifying Rational Expressions

646

QUIZ 1, 651

11.4 Multiplying and Dividing Rational Expressions

652

11.5 Adding and Subtracting with Like Denominators

658

11.6 Adding and Subtracting with Unlike Denominators

663

11.7 Rational Equations

670

QUIZ 2, 677 Extension: Rational Functions

678

ASSESSMENT Chapter Readiness Quiz, 632 Quizzes, 651, 677 Standardized Test Practice, 638, 644, 650, 657, 662, 669, 676 Chapter Summary and Review, 681 Chapter Test, 685 Chapter Standardized Test, 686 Maintaining Skills, 687

Contents

xv

12

Radicals and More Connections to Geometry 䉴 CHAPTER STUDY GUIDE

STUDENT HELP Study Tip 692, 693, 699, 717, 724, 726, 736, 741 Reading Algebra 710 Vocabulary Tip 722, 731 Look Back 698, 715, 726, 742 Test Tip 752


INT

Centripetal Force 689, 706 Dinosaurs 696 Sailing 700 De-icing Planes 708 Penguins 718 Diving 720 Staircase Design 728 Soccer 732 Maps 734 Pony Express 739 NE ER T

INTERNET

689, 693, 696, 702, 705, 708, 712, 720, 725, 728, 732, 734, 737, 739, 742, 743, 744

690

12.1 Functions Involving Square Roots 䉴 DEVELOPING CONCEPTS: Functions with Radicals, 691

692

12.2 Operations with Radical Expressions

698

12.3 Solving Radical Equations

704

QUIZ 1, 709

12.4 Rational Exponents

710

12.5 Completing the Square 䉴 DEVELOPING CONCEPTS: Completing the Square, 715

716

12.6 The Pythagorean Theorem and Its Converse 䉴 DEVELOPING CONCEPTS: The Pythagorean Theorem , 722

724

QUIZ 2, 729

12.7 The Distance Formula

730

12.8 The Midpoint Formula

736

12.9 Logical Reasoning: Proof

740

QUIZ 3, 746 Project: Investigating the Golden Ratio

756

ASSESSMENT Chapter Readiness Quiz, 690 Quizzes, 709, 729, 746 Standardized Test Practice, 697, 703, 709, 714, 721, 729, 735, 739, 745 Chapter Summary and Review, 747 xvi

Contents

Chapter Test, 751 Chapter Standardized Test, 752 Cumulative Practice, Chapters 1–12, 754

Contents of Student Resources Skills Review Handbook

pages 759–782

• Decimals

759–760

• Factors and Multiples

761–762

• Fractions

763–766

• Writing Fractions and Decimals

767–769

• Comparing and Ordering Numbers

770–771

• Perimeter, Area, and Volume

772–773

• Estimation

774–776

• Data Displays

777–779

• Measures of Central Tendency

780

• Problem Solving

781–782

Extra Practice for Chapters 1–12

pages 783–794

End-of-Course Test

pages 795–796

Tables

pages 797–802

• Symbols

797

• Squares and Square Roots

• Formulas

798

• Measures

• Properties

799

800–801 802

Appendices

pages 803–816

Glossary

pages 817–823

English-to-Spanish Glossary

pages 824–831

Index

pages 832–844

Selected Answers Contents

xvii

Getting Ready

Pre-Course Test Skills Review pp. 759–760

DECIMALS Find the sum, difference, product, or quotient. 1. 3.4 6.005

2. 27.77 18.09

3. 23.7 13.67

4. 9.744 0.87 Skills Review pp. 761–762

FACTORS AND MULTIPLES Find the greatest common factor of the pair of numbers. 5. 8, 28

6. 36, 42

7. 54, 81

8. 50, 150

Find the least common multiple of the pair of numbers. 9. 6, 7

10. 10, 15

11. 24, 38

12. 12, 36

Find the least common denominator of the pair of fractions. 1 7 13. , 2 10

5 6 14. , 8 7

5 7 15. , 9 12

11 15 16. , 20 32 Skills Review pp. 763–766

FRACTIONS Find the reciprocal of the number. 3 17. 12 18. 16

9 19. 5

1 20. 2 3

Add, subtract, multiply, or divide. Write the answer in simplest form. 3 1 21. 4 4

1 1 22. 2 8

6 5 23. 7 9

5 1 24. 11 2 8 4

6 1 25. 11 2

7 3 26. 11 5

4 8 27. 15 3

2 1 28. 4 3 8 Skills Review pp. 768–769

FRACTIONS, DECIMALS, AND PERCENTS Write the percent as a decimal and as a fraction in simplest form. 29. 7%

30. 26%

31. 48%

32. 84%

Write the decimal as a percent and as a fraction in simplest form. 33. 0.08

34. 0.15

35. 0.47

36. 0.027

7 39. 8

11 40. 20

Write the fraction as a decimal and as a percent. 9 37. 10 xviii

4 38. 5 Pre-Course Test

Skills Review pp. 770–771

COMPARING AND ORDERING NUMBERS Compare the two numbers. Write the answer using , or . 41. 138 and 198

42. 781 and 718

43. 8.4 and 8.2

44. 7.88 and 4.88

5 3 45. and 12 4

3 4 46. and 6 8

5 1 47. and 1 3 2

7 2 48. 16 and 16 8 3

Write the numbers in order from least to greatest. 49. 47, 74, 44, 77

50. 80, 808, 88, 8

51. 0.19, 0.9, 0.49, 0.4

52. 6.5, 5.4, 6.4, 6

5 4 3 1 53. , , , 8 7 5 2

9 6 5 6 54. , , , 7 4 4 13

5 3 13 7 55. 1, 1, , 9 4 11 5

2 1 1 1 56. 16, 15, 16, 15 3 4 9 8 Skills Review pp. 772–773

PERIMETER, AREA, AND VOLUME Find the perimeter. 57. a triangle with sides of length 18 feet, 27 feet, and 32 feet 58. a square with sides of length 4.7 centimeters Find the area. 59. a square with sides of length 13 yards 60. a rectangle with length 7.7 kilometers and width 4.5 kilometers Find the volume. 61. a cube with sides of length 19 meters

62. a rectangular prism with length 5.9 inches, width 8.6 inches, and height 1.2 inches Skills Review pp. 777–779

DATA DISPLAYS

63. The list below shows the distribution of gold medals for the 1998 Winter Olympics.

Choose an appropriate graph to display the data. Germany 12 United States 6 South Korea 3 Switzerland 2

Norway 10 Japan 5 Finland 2 Bulgaria 1

Source: International Olympic Committee

Russia 9 Netherlands 5 France 2 Czech Republic 1

Canada 6 Austria 3 Italy 2

Skills Review p. 780

MEASURES OF CENTRAL TENDENCY Find the mean, median, and mode(s) of the data set. 64. 1, 3, 3, 3, 4, 5, 6, 7, 7, 9

65. 17, 22, 36, 47, 51, 58, 65, 80, 85, 89

66. 5, 23, 12, 5, 9, 18, 12, 4, 10, 21

67. 101, 423, 564, 198, 387, 291, 402, 572, 222, 357 Pre-Course Test

xix

Getting Ready

Pre-Course Practice Skills Review pp. 759–760

DECIMALS Find the sum or difference. 1. 14 7.1

2. 11 0.003

3. 19.76 48.19

4. 73.8 6.93

5. 10.2 3.805 1.1

6. 7.2 3.56

8. 6.08 3.15

9. 15.2 5.02

Find the product or quotient. 7. 17 3.9 10. 0.019 0.27

11. 45.28 16.1

12. 26.01 5.1

13. 7.03 1.9

14. 21.84 0.84

15. 0.0196 0.056 Skills Review pp. 761–762

FACTORS AND MULTIPLES List all the factors of the number. 1. 12

2. 41

3. 54

4. 126

Write the prime factorization of the number if it is not a prime number. If a number is prime, write prime. 5. 54

6. 60

7. 35

8. 47

List all the common factors of the pair of numbers. 9. 16, 20

10. 24, 36

11. 28, 42

12. 60, 72

Find the greatest common factor of the pair of numbers. 13. 8, 12

14. 10, 25

15. 15, 24

16. 24, 30

17. 36, 42

18. 54, 81

19. 68, 82

20. 102, 214

Find the least common multiple of the pair of numbers. 21. 9, 12

22. 8, 5

23. 14, 21

24. 24, 8

25. 12, 16

26. 70, 14

27. 36, 50

28. 22, 30

Find the least common denominator of the pair of fractions. 5 5 29. , 8 6

5 7 30. , 12 8

7 9 31. , 12 20

5 8 32. , 6 15

3 15 33. , 4 28

9 8 34. , 11 13

5 20 35. , 6 27

17 27 36. , 40 52

xx

Pre-Course Practice


FRACTIONS Find the reciprocal of the number. 1 1. 8 2. 16

9 3. 5

Add or subtract. Write the answer as a fraction or a mixed number in simplest form. 7 1 9 3 1 3 5. 6. 7. 12 12 10 10 8 8 1 2 9. 3 9

17 3 10. 20 5

1 5 11. 6 8

4 4. 3 7

5 2 8. 15 15 8 2 12. 1 9 3

Multiply or divide. Write the answer as a fraction or a mixed number in simplest form. 3 1 2 3 3 1 13. 14. 15. 1 5 2 3 8 5 2

3 2 16. 2 3 8 3

2 4 17. 5 5

1 4 20. 4 1 3 5

2 8 18. 3 9

1 7 19. 5 4 8

Add, subtract, multiply, or divide. Write the answer as a fraction or a mixed number in simplest form. 2 5 3 1 2 3 21. 22. 9 5 23. 8 5 3 6 8 4 5 8

1 4 24. 4 5

2 1 25. 1 1 3 4

3 2 28. 1 4 3

3 3 26. 5 3 8 4

1 1 27. 4 5


FRACTIONS, DECIMALS, AND PERCENTS Write the percent as a decimal and as a fraction or a mixed number in simplest form. 1. 8%

2. 25%

3. 38%

4. 73%

5. 135%

6. 350%

7. 6.4%

8. 0.15%

Write the decimal as a percent and as a fraction or a mixed number in simplest form. 9. 0.44 13. 1.6

10. 0.09

11. 0.13

12. 0.008

14. 3.04

15. 6.6

16. 4.75

Write the fraction or mixed number as a decimal and as a percent. Round decimals to the nearest thousandth. Round percents to the nearest tenth of a percent. 11 3 5 17 17. 18. 19. 20. 12 5 8 25 1 21. 5 5

1 22. 2 4

1 23. 3 16

3 24. 8 7 Pre-Course Practice

xxi

continued from page xxi Skills Review pp. 770–771

COMPARING AND ORDERING NUMBERS Compare the two numbers. Write the answer using , or ⴝ. 1. 13,458 and 14,455

2. 907 and 971

3. ⫺8344 and ⫺8434

4. ⫺49.5 and ⫺49.05

5. 0.58 and 0.578

6. 0.0394 and 0.394

9 15 7. ᎏᎏ and ᎏᎏ 10 16

13 1 8. ᎏᎏ and ᎏᎏ 20 4

9 3 9. ᎏᎏ and ᎏᎏ 24 8

1 1 10. 7ᎏᎏ and 7ᎏᎏ 4 5

11 2 11. ⫺2ᎏᎏ and ⫺3ᎏᎏ 16 9

5 2 12. 18ᎏᎏ and 18ᎏᎏ 8 3

Write the numbers in order from least to greatest. 13. 1507, 1705, 1775, 1075

14. 38,381, 30,831, 38,831, 30,138

15. ⫺0.019, ⫺0.013, ⫺0.205, ⫺0.035

16. 6.034, 6.30, 6.33, 6.34

1 2 5 5 17. ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ 2 7 11 8

4 3 3 4 18. ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ 5 4 7 9

3 4 2 4 19. ⫺ᎏᎏ, ⫺ᎏᎏ, ⫺ᎏᎏ, ⫺ᎏᎏ 2 2 3 3

3 5 7 4 20. ᎏᎏ, ᎏᎏ, ᎏᎏ, 1ᎏᎏ 8 4 9 7

3 7 5 4 21. 1ᎏᎏ, ᎏᎏ, ᎏᎏ, 1ᎏᎏ 5 5 3 5

5 5 2 5 22. 15ᎏᎏ, 14ᎏᎏ, 14ᎏᎏ, 15ᎏᎏ 8 9 3 7 Skills Review pp. 772–773

PERIMETER, AREA, AND VOLUME Find the perimeter. 1.

2.

2.5 m 2.5 m

2.5 m

3. 13 in.

2.5 m

13 in.

4.5 km 6.8 km

9 in.

4.5 km

4. a rectangle with length of 12.5 centimeters and width of 11.6 centimeters 5. a regular pentagon with sides of length 19 feet Find the area. 6. a square with sides of length 1.67 yards 7. a rectangle with length 1.4 inches and width 2.8 inches 8. a triangle with base 15 centimeters and height 10 centimeters Find the volume. 9. a cube with sides of length 34 feet 10. a rectangular prism with length 18 meters, width 6 meters, and height 3 meters 11. a rectangular prism with length 6.5 millimeters, width 5.5 millimeters, and

height 2.2 millimeters xxii

Pre-Course Practice

6.8 km


DATA DISPLAYS In Exercises 1 and 2, use the table shown below. Hurricane categories are determined by wind speed, with Category 5 the most severe. U.S. Mainland Hurricane Strikes by Category from 1900 –1996 Category

One

Two

Three

Four

Five

Number

57

37

47

15

2

1. The data range from 2 to 57. The scale must start at 0. Choose a reasonable

scale for a bar graph. 2. Draw a bar graph to display the number of hurricane strikes by category. In Exercises 3 and 4, use the table shown below. U.S. Mainland Hurricane Strikes by Decade from 1900–1989 Decade

1900– 1909

1910– 1919

1920– 1929

1930– 1939

1940– 1949

1950– 1959

1960– 1969

1970– 1979

1980– 1989

Number

16

19

15

17

23

18

15

12

16

Source: National Hurricane Center

3. The data range from 12 to 23. The scale must start at 0. Choose a reasonable

scale for a histogram. 4. Draw a histogram to display the number of hurricane strikes by decade. Choose an appropriate graph to display the data. Draw the graph. 5.

Reported House Plant Sales for One Week Type Number

6.

Violets

Begonias

Coleus

Orchids

Cacti

90

46

39

70

60

Republicans in the Senate by Congress Number Congress Republicans

100th

101st

102nd

103rd

104th

105th

106th

45

45

44

43

52

55

55

Source: Statistical Abstract of the United States: 1999

Skills Review p. 780

MEASURES OF CENTRAL TENDENCY Find the mean, median, and mode(s) of the data set. 1. 1, 3, 7, 2, 6, 3, 7, 9, 4, 7

2. 16, 19, 15, 17, 23, 18, 15, 12, 16, 7

3. 10, 48, 86, 32, 58, 73, 89, 39, 59, 27

4. 53, 54, 53, 45, 45, 44, 43, 52, 55, 55 Pre-Course Practice

xxiii

Getting Ready

A Guide to Student Help 䉴 Each chapter begins with a Study Guide

CHAPTER PREVIEW gives an overview of what you will be learning. KEY WORDS lists important new words in the chapter. READINESS QUIZ checks your understanding of words and skills that you will use in the chapter, and tells you where to go for review.

STUDY TIP suggests ways to make your studying and learning easier.

xxiv

䉴 Student Help notes

throughout the book STUDY TIPS help you understand and apply concepts and avoid common errors.

MORE EXAMPLES indicates that there are more worked-out examples on the Internet.

READING ALGEBRA guides you in reading and understanding your textbook.

SKILLS REVIEW refers you to the pages where you can go for review and practice of topics from earlier courses.

Student Help READING ALGEBRA

Order is important for subtraction. “4 less than a number” means y ⫺ 4, not 4 ⫺ y.

Student Help SKILLS REVIEW

For help with writing fractions in lowest terms, see p. 763.

HOMEWORK HELP tells you which textbook examples may help you with homework exercises, and lets you know when there is extra help on the Internet.

Student Help HOMEWORK HELP

Extra help with problem solving in Exs. 34–39 is available at www.mcdougallittell.com

Other notes included are: • WRITING ALGEBRA

• TEST TIP

• KEYSTROKE HELP

• LOOK BACK

VOCABULARY TIPS explain the meaning and origin of words.

Student Help VOCABULARY TIP

Equation comes from a Latin word that means “to be equal”.

xxv

1

Connections to to Connections Algebra Algebra How much does it cost to rent scuba diving equipment?

APPLICATION: Scuba Diving Scuba divers must be certified divers and must use scuba equipment in order to breathe underwater. Equipment such as wet suits, tanks, buoyancy compensator devices, and regulators can be rented at a cost per day at many sporting good stores.

Think & Discuss The table shows the cost per day for renting a regulator. Number of days

Rental charge

1

$12.00 1

2

$12.00 2

3

$12.00 3

4

$12.00 4

1. You decide to rent a regulator for 4 days. What is

the rental charge? 2. Use the pattern in the table to predict the rental cost if you rent a regulator for 10 days.

Learn More About It

INT

You will learn more about the price of renting scuba equipment in Exercise 24 on page 53. NE ER T

APPLICATION LINK More about the prices of renting scuba

diving equipment is available at www.mcdougallittell.com

1

Chapter

1

PREVIEW

Study Guide What’s the chapter about? • Writing and evaluating variable expressions • Checking solutions to equations and inequalities • Using verbal and algebraic models • Organizing data and representing functions KEY WORDS • base, p. 9 • order of operations, p. 15 • equation, p. 24 • solution, p. 24

• variable, p. 3 • variable expression, p. 3 • numerical expression, p. 3 • power, p. 9 • exponent, p. 9

PREPARE

• • • • •

inequality, p. 26 modeling, p. 36 function, p. 48 domain, p. 49 range, p. 49

Chapter Readiness Quiz Take this quick quiz. If you are unsure of an answer, look at the reference pages for help.

VOCABULARY CHECK (refer to pp. 3, 9) 1. Which of the following is not a variable expression? A 

Student Help

9 4y

B 

10 4(2)

C 

2x 3

D 

2m 3n

base

D 

variable

2. Which term describes the expression 73?

STUDY TIP

“Student Help” boxes throughout the chapter give you study tips and tell you where to look for extra help in this book and on the Internet.

A 

power

B 

exponent

C 

SKILL CHECK (refer to pp. 772, 770) 3. Find the perimeter of the figure. A  C 

B  D 

30 feet 120 feet

20 ft

60 feet

10 ft

10 ft

200 feet

20 ft

䡵

4. Complete the statement 0.5 > ? . A 

STUDY TIP

1 4

B 

Keep a Math Notebook Keeping a notebook will help you remember new concepts and skills.

2

Chapter 1

Connections to Algebra

1 2

C 

3 4

D 

3 2

Keeping a Math Notebook • Keep a notebook of math not es about each chapter separate from your hom ework exercises. • Review your notes each day before you start your next homework assignment.

1.1

Variables in Algebra How many miles has a race car traveled?

Goal Evaluate variable expressions.

A race car zooms around the Indianapolis Motor Speedway at 180 miles per hour. In Example 3 you will find how many miles the car will travel in 2 hours.

Key Words • • • • •

variable value variable expression numerical expression evaluate

In algebra, you can use letters to represent one or more numbers. When a letter is used to represent a range of numbers, it is called a variable. The numbers are called values of the variable. For example, the distance traveled by the race car in the picture above can be expressed as the variable expression 180t, where t represents the number of hours the car has traveled. ASSIGNING VARIABLES

A variable expression consists of constants, variables, and operations. An expression that represents a particular number is called a numerical expression. For example, the distance traveled by the race car in two hours is given by the numerical expression 180 2.

Student Help WRITING ALGEBRA

The multiplication symbol is usually not used in algebra because of its possible confusion with the variable x.

EXAMPLE

1

Describe the Variable Expression

Here are some variable expressions, their meanings, and their operations. VARIABLE EXPRESSION

MEANING

OPERATION

8y, 8 p y, (8)(y)

8 times y

Multiplication

16 , 16 b b

16 divided by b

Division

4s

4 plus s

Addition

9x

9 minus x

Subtraction

Describe the Variable Expression State the meaning of the variable expression and name the operation. x 1. 10 x 2. 13 x 3. 4. 24x 16

1.1

Variables in Algebra

3

To evaluate a variable expression, you write the expression, substitute a number for each variable, and simplify. The resulting number is the value of the expression. EVALUATING EXPRESSIONS

Write expression.

EXAMPLE

2

Substitute numbers.

Simplify.

Evaluate the Variable Expression

Evaluate the variable expression when y 2. Solution EXPRESSION

SUBSTITUTE

SIMPLIFY

a. 5y

5(2)

10

10 b. y

10 2

5

c. y 6

26

8

d. 14 y

14 2

12

Evaluate the Variable Expression Evaluate the variable expression when x 3. 6. 5 x

5. 7x

12 7. x

8. x 2

The variable expression r times t can be written as rt, r p t, or (r)(t).

Sports

EXAMPLE

3

Evaluate rt to Find Distance

RACE CARS Find the distance d traveled in 2 hours by a race car going an average speed of 180 miles per hour. Use the formula: distance equals rate r multiplied by time t.

Solution d rt RACE CARS The fastest average speed in the Indianapolis 500 is 185.981 miles per hour set by Arie Luyendyk, shown above.

Write formula.

180(2)

Substitute 180 for r and 2 for t.

360

Simplify.

ANSWER 䊳

The distance traveled by the race car was 360 miles.

Evaluate rt to Find Distance 9. Using a variable expression, find the distance traveled by a car moving at an

average speed of 60 miles per hour for 3 hours.

4

Chapter 1



The word perimeter comes from perimeaning around and -meter meaning measure. Perimeter is a measure of the distance around a geometric figure.

EXAMPLE

4

Find the Perimeter

GEOMETRY LINK The perimeter P of a triangle is equal to the sum of the lengths of its sides:

c 17 a8 b 15

Pabc Find the perimeter of the triangle in feet. Solution

Pabc

Write the formula. Substitute the side lengths of 8, 15 and 17.

8 15 17

Add the side lengths.

40

ANSWER 䊳

The triangle has a perimeter of 40 feet.

Find the Perimeter 10. Find the perimeter of a square with each side 12 inches long.

EXAMPLE

5

Estimate the Area

GEOGRAPHY LINK

The area A of a triangle is equal to half the base b times

1 2

the height h: A bh. Use this formula to estimate the area (in square miles) of Virginia. height h right angle

h 200 mi b 410 mi

base b

Solution Write the formula.

1 2

A bh 1 2

Substitute 410 for b and 200 for h.

(410)(200)

Simplify the formula.

41,000

ANSWER 䊳

The area of Virginia is about 41,000 square miles.

h7

Find the Area 11. Find the area of the triangle in square meters.

1.1

b 12 Variables in Algebra

5

1.1

Exercises

Guided Practice Vocabulary Check

Identify the variable or variables. 1. y 15

2. 20 s

b 3. 10

4. rt

5. Complete: You ____ ? an expression by substituting numbers for variables

and simplifying. The resulting number is called the ____ ? of the expression.

Skill Check

State the meaning of the variable expression and name the operation. 5 6. c

7. p 4

8. 5 n

9. (8)(x)

Evaluate the variable expression when k 3. 10. 11 k

11. k 2

12. 7k

k 13. 33

18 14. k

15. 18 p k

Find the perimeter of each triangle.

16. a.

b. 8 cm

6 cm

9 in.

10 cm

15 in.

12 in.

Practice and Applications DESCRIBING EXPRESSIONS Match the variable expression with its

meaning. 17. y 8

18. y 8

A. 8 times y

B. y divided by 8

y 19. 8 C. y plus 8

20. 8y D. y minus 8

EVALUATING EXPRESSIONS Evaluate the expression for the given value

of the variable.


Example 1: Exs. 17–20 Example 2: Exs. 21–32 Example 3: Exs. 33–39 Example 4: Exs. 40–42 Example 5: Exs. 43–46

6

Chapter 1

21. 9 p when p 11

1 22. t when t 2 2

b 23. when b 14 7

d 24. when d 36 12

25. (4)(n) when n 5

26. 8a when a 6

27. 12 x when x 3

28. 9 y when y 8

29. 10r when r 7

30. 13c when c 3

18 31. when x 3 x

63 32. when k 9 k


Student Help

33. DRIVING DISTANCE You are driving across the country at an average speed

of 65 miles per hour. Using an appropriate formula, find the distance you travel in 4 hours.

HOMEWORK HELP NE ER T

INT


FINDING DISTANCE Find the distance traveled using d rt.

34. A train travels at a rate of 75 miles per hour for 2 hours. 35. An athlete runs at a rate of 8 feet per second for 5 seconds. 36. A horse trots at 8 kilometers per hour for 30 minutes. 37. A racecar driver goes at a speed of 170 miles per hour for 2 hours. 38. A plane travels at a speed of 450 miles per hour for 3 hours. 39. A person walks at a rate of 4 feet per second for 1 minute.

Student Help SKILLS REVIEW

The perimeter of any geometric figure is the sum of all its side lengths. To review perimeter and area formulas, see p. 772.

Find the perimeter of the geometric figure. 40.

41.

13 ft

42.

22 m 10 m

8 ft

10 ft

10 m

12 in.

12 in.

12 in.

12 in.

22 m 12 in.

Find the area of each triangle. 43.

h4m

44.

h 6 mi

45. h 3 yd

46.

b 4 yd

b 10 mi

b5m

To find the area A of a rectangle, you multiply the length times the width: Alpw Use the formula to estimate the area of Wyoming.

275 mi

362 mi

47. CHALLENGE A tsunami is a huge fast-moving series of water waves that

can be caused by disturbances such as underwater earthquakes or volcanic explosions. If a tsunami is traveling at a speed of 500 miles per hour across the Pacific Ocean, how far has it gone in 15 minutes? HINT: Convert 15 minutes to hours. 1.1

Variables in Algebra

7

EXAMPLE

Unit Analysis

Writing the units of measure helps you determine the units for the answer. This is called unit analysis. When the same units occur in the numerator and the denominator, you can cancel them. Use unit analysis to evaluate the expression. The letter h is an abbreviation for hours, while mi stands for miles.

冢

冣

冢

25 mi a. (3 h) 1h

冣

45 mi b. (90 mi) 1h

Solution

冢

Cancel hours. 冣 1h 45 mi b. (90 mi) 冢冣 (90 mi)冢冣 2 h 45 mi 1h

25 mi a. (3 h) 75 mi 1h

Student Help STUDY TIP

When you divide by a fraction, you multiply by the reciprocal. See Skills Review, p. 765.

Cancel miles.

冢

冣 20 mi 49. Evaluate the expression (80 mi) 冢冣. 1h 60 mi 48. Evaluate the expression (4 h) . 1h

Standardized Test Practice

50. MULTIPLE CHOICE How many miles does Joyce travel if she drives for

6 hours at an average speed of 60 miles per hour? A 

66 miles

B 

180 miles

C 

360 miles

D 

420 miles

51. MULTIPLE CHOICE The lengths of the sides of a triangle are 4 centimeters,

8 centimeters, and 7 centimeters. What is the perimeter of the triangle? F 

Mixed Review

7 cm

G 

H 

16 cm

19 cm

J 

28 cm

OPERATIONS WITH DECIMALS Find the value of the expression.

(Skills Review p. 759) 52. 32.8 4

53. 3.98 5.50

54. 0.1(50)

SIMPLIFYING EXPRESSIONS Simplify the expression without using a calculator. (Skills Review p. 765)

冢冣

30 55. (5) 10

Maintaining Skills

8

Chapter 1

冢冣

2 56. (60) 12

冢冣冢冣

3 5 57. 15 6

ADDING DECIMALS Add. (Skills Review p. 759)

58. 2.3 4.5

59. 16.8 7.1

60. 0.09 0.05

61. 1.0008 10.15

62. 123.8 0.03

63. 46 7.55

64. 0.32 0.094

65. 6.105 7.3

66. 2.008 1.10199


1.2

Exponents and Powers How much water does the tank hold?

Goal Evaluate a power.

How much water do you need to fill a fish tank? You will use a power to find the answer in Example 5.

Key Words • • • •

power exponent base grouping symbols

An expression like 23 is called a power. The exponent 3 represents the number of times the base 2 is used as a factor. base

exponent

23 ⫽ 2 p 2 p 2 power 3 factors of 2

The expression 2 3 means “multiply 2 by itself 3 times.” The numbers you multiply are factors. In general, a n ⫽ a p a p a p ... p a. n times

EXAMPLE

1

Read and Write Powers

Express the power in words. Then write the meaning. EXPONENTIAL FORM


Note that x 1 is customarily written as x with the exponent omitted.

WORDS

MEANING

a. 42

four to the second power or four squared

4p4

b. 53

five to the third power or five cubed

5p5p5

c. x6

x to the sixth power

xpxpxpxpxpx

Write the Power Write the expression in exponential form. 1. 3 squared

2. x to the fourth power 1.2

3. s cubed Exponents and Powers

9

Student Help

EXAMPLE

More examples are available at www.mcdougallittell.com

INT

Evaluate the Power

Evaluate x 4 when x 2.

MORE EXAMPLES NE ER T

2

Solution x4 24

Substitute 2 for x. Write out the factors.

2p2p2p2

Multiply the factors.

16

ANSWER 䊳

The value of the power is 16.

Parentheses ( ) and brackets [ ] are grouping symbols. They tell you the order in which to do the operations. You must do the operations within the innermost set of grouping symbols first:

GROUPING SYMBOLS

First multiply. Then add.

First add. Then multiply.

(3 p 4) 7 12 7 19

3 p (4 7) 3 p 11 33

You will learn more about the order of operations in the next lesson.

EXAMPLE

3

Evaluate Exponential Expressions

Evaluate the variable expression when a 1 and b 2. a. (a2) (b2)

b. (a b)2

Solution a. (a2) (b2) (12) (22)

Substitute 1 for a and 2 for b.

(1 p 1) (2 p 2)

Write factors.

14

Multiply.

5

Add.

b. (a b)2 (1 2)2

Substitute 1 for a and 2 for b.

(3)2

Add within parentheses.

3p3

Write factors.

9

Multiply.

Evaluate Exponential Expressions Evaluate the variable expression when s 2 and t 4.

10

Chapter 1

4. (t s)3

5. (s 2) (t 2)

6. (t s)2

7. (t 2) (s 2)

8. (s 2) t

9. (t 2) s


In Lesson 1.3 you will learn several rules for order of operations. One of those tells us that 2x3 is to be interpreted as 2(x3).


Notice that in part (a) of Example 4, the exponent applies to x, while in part (b), the exponent applies to 2x.

4

EXAMPLE

Exponents and Grouping Symbols

Evaluate the variable expression when x 4. a. 2x3

b. (2x)3

Solution a. 2x3 2(43)

Substitute 4 for x.

2(64)

Evaluate power.

128

Multiply.

b. (2x) (2 p 4) 3

3

Substitute 4 for x.

8

Multiply within parentheses.

512

Evaluate power.

3

Exponents can be used to find the area of a square and the volume of a cube. s s s s Area of square: A s 2

s Volume of cube: V s 3

Units of area, such as square feet, ft2, can be written using a second power. Units of volume, such as cubic feet, ft3, can be written using a third power.

EXAMPLE

5

Find the Volume of the Tank

The fish tank has the shape of a cube. Each inner edge s is 2 feet long. Find the volume in cubic feet.

FISH TANKS

2 ft

Solution V s3

Write formula for volume of a cube.

23

Substitute 2 for s.

8

Evaluate power.

ANSWER 䊳

2 ft 2 ft

The volume of the tank is 8 cubic feet.

Find Area and Volume 10. Use the formula for the area of a square to find the area of each side of the

fish tank in Example 5. Express your answer in square feet.

1.2

Exponents and Powers

11

1.2 Exercises Guided Practice Vocabulary Check

Complete the sentence. 1. In the expression 37, the 3 is the ____ ? . 2. In the expression 54, the 4 is the ____ ? . 3. The expression 912 is called a ____ ? . 4. Two kinds of grouping symbols are ____ ? and ____ ? .

Skill Check

Match the power with the words that describe it. 5. 37

A. four to the sixth power

6. 73

B. three to the seventh power

7. 46

C. seven to the third power

4

8. 6

D. six to the fourth power

Evaluate the variable expression when t 3. 9. t 2

10. 1 t 3

11. 4t 2

12. (4t)2

Practice and Applications WRITING POWERS Write the expression in exponential form.

13. two cubed

14. p squared

15. nine to the fifth

power 16. b to the eighth power 19.

17. 3 p 3 p 3 p 3

18. 4x p 4x p 4x

A square painting measures 5 feet by 5 feet. Write the power that gives the area of the painting. Then evaluate the power. 5 ft

5 ft EVALUATING POWERS Evaluate the power.

Student Help

20. 92

21. 24

22. 73

23. 26

24. 54

25. 18

26. 103

27. 06

HOMEWORK HELP


12

Chapter 1

EVALUATING POWERS WITH VARIABLES Evaluate the expression for the

given value of the variable. 28. w2 when w 12

29. b3 when b 9

30. c4 when c 3

31. h5 when h 2

32. n2 when n 11

33. x3 when x 5


Student Help KEYSTROKE HELP

Your calculator may have a key or a key that you can use to evaluate powers.

EVALUATING POWERS Use a calculator to evaluate the power.

34. 86

35. 135

36. 59

37. 127

38. 66

39. 312

EVALUATING EXPRESSIONS Evaluate the variable expression when

c 4 and d 5. 40. (c d)2

41. (d 2) c

42. (c3) d

43. (d 2) (c2)

44. (d c)7

45. (d 2) d

EXPONENTIAL EXPRESSIONS Evaluate the expression for the given

value of the variable. 46. 2x 2 when x 7

47. 6t 4 when t 1

48. 7b2 when b 3

49. (5w)3 when w 5

50. (4x)3 when x 1

51. (5y)5 when y 2

52. INTERIOR DESIGN The floor of a room is 14 feet long by 14 feet wide.

How many square feet of carpet are needed to cover the floor? 53. VOLUME OF A SAFE A fireproof safe is designed in the shape of a cube.

The length of each edge of the cube is 2 meters. What is the volume of the fireproof safe?

Art

54. ARTISTS In 1997, the artist Jon Kuhn of North Carolina created a cubic

sculpture called Crystal Victory, shown at the left. Each edge of the solid glass cube is 9.5 inches in length. What is the volume of the cubic structure? CRITICAL THINKING Count the number of cubic units along the edges of

the cube. Write and evaluate the power that gives the volume of the cube in cubic units. 55.

ARTIST Jon Kuhn used mathematics when creating the cubic sculpture Crystal Victory. The solid glass cube is made of lead crystal and colored glass powders.

56.

57.

CHALLENGE You are making candles. You melt paraffin wax in the cubic container shown below. Each edge of the container is 6 inches in length. The container is half full.

58. What is the volume of the wax in this container? 6 in.

59. Each edge of a second cubic container is 4 inches in

length. Can this second candle mold hold the same amount of melted wax that is in the candle mold shown at the right? Explain your answer.

6 in. 6 in.

60. Design a third cubic candle mold different from the one given above that will

hold all the melted wax. Draw a diagram of the mold including the measurements. Explain why your mold will hold all the melted wax. 1.2

Exponents and Powers

13


61. MULTIPLE CHOICE Evaluate the expression 2x 2 when x 5. A 

B 

20

C 

40

D 

50

100

62. MULTIPLE CHOICE One kiloliter is equal to 103 liters. How many liters are

in one kiloliter? F 

G 

10

H 

100

J 

1000

10,000

63. MULTIPLE CHOICE Sondra bought

Student Help

this trunk to store clothes. What is the volume of the trunk?

TEST TIP

Jotting down the formula for the volume of a cube will help you answer Exercise 63.

A  C 

B  D 

9 ft 9 ft3

3 ft

9 ft2 None of these 3 ft 3 ft

Find the perimeter of the geometric figure when x 3.

Mixed Review (Lesson 1.1) x

64.

x

65. x

x

x

66. x

x

2x x

x

2x

x x


of the variable. (Lesson 1.1)

Maintaining Skills

67. 9j when j 5

68. 6 t when t 21

b 69. when b 18 2

70. 25 n when n 3

71. c 4 when c 24

72. (7)(r) when r 11

24 73. when s 8 s

74. 3m when m 7

75. d 13 when d 22

SIMPLIFYING FRACTIONS Simplify. (Skills Review p. 763)

4 76. 8

10 77. 2

15 78. 10

6 79. 20

8 80. 14

18 81. 21

8 82. 6

27 83. 3

25 84. 15

21 85. 7

3 86. 24

28 87. 4

ESTIMATING Estimate the answer. Then evaluate the expression.

(Skills Review p. 774)

14

Chapter 1

88. 2.5 0.5

89. 0.3 0.03

90. 10.35 5.301

91. 3.71 1.054

92. 2.1 0.2

93. 5.175 1.15


1.3 Goal

Order of Operations How many points ahead are you?

Use the established order of operations.

You are playing basketball. You make 8 field goals and 2 free throws. Your friend makes half as many field goals as you and no free throws. You will find how many points ahead you are in Example 5.

Key Words • order of operations • left-to-right rule

In arithmetic and algebra there is an order of operations to evaluate an expression involving more than one operation. ORDER OF OPERATIONS

ORDER OF OPERATIONS STEP

First do operations that occur within grouping symbols.

STEP

Then evaluate powers.

STEP

Then do multiplications and divisions from left to right.

STEP

Finally, do additions and subtractions from left to right.

EXAMPLE

1

Evaluate Without Grouping Symbols

Evaluate the expression 3x 2 1 when x 4. Use the order of operations. Solution 3x 2 1 3 p 42 1

Substitute 4 for x.

3 p 16 1

Evaluate power.

48 1

Multiply 3 times 16.

49

Add.

Evaluate Expressions Without Grouping Symbols Evaluate the variable expression when x 2. Use the order of operations. 1. 2x 2 5

2. 8 x 2

3. 6 3x 3 1.3

4. 20 4x 2 Order of Operations

15

Some expressions have operations that have the same priority, such as multiplication and division or addition and subtraction. The left-to-right rule states that when operations have the same priority, you perform them in order from left to right.

LEFT-TO-RIGHT RULE

Use the Left-to-Right Rule

2

EXAMPLE

Evaluate the expression using the left-to-right rule. a. 24 8 6 (24 8) 6

Work from left to right.

16 6

Subtract 8 from 24.

10

Subtract 6 from 16.

b. 15 p 2 6 (15 p 2) 6


You multiply first in part (c) of Example 2, because multiplication has a higher priority than addition and subtraction.


30 6

Multiply 15 times 2.

5

Divide 30 by 6.

c. 16 4 p 2 3 16 (4 p 2) 3

Do multiplication first.

16 8 3

Multiply 4 times 2.

(16 8) 3


24 3

Add 16 and 8.

21

Subtract 3 from 24.

12 41

A fraction bar can act as a grouping symbol: (1 2) (4 1) .

Expressions with Fraction Bars

3

EXAMPLE

Evaluate the expression. Then simplify the answer. 7p4 7p4 8 72 1 8 49 1 28 8 49 1

Simplify the numerator.

28 57 1


28 56

Subtract.

1 2

Simplify.

Student Help

Evaluate power.

SKILLS REVIEW

For help with writing fractions in simplest form, see p. 763.

Use the Order of Operations and Left-to-Right Rule Evaluate the variable expression when x 1. 5. 4x 2 5 3

16

Chapter 1


6. 5 x 3 1

2x 7. x2 1 5

You need to know if your calculator uses the order of operations or not. If it does not, you must input the operations in the proper order yourself. USING A CALCULATOR

EXAMPLE

4

Use a Calculator

Enter the following in your calculator. Does the calculator display 6 or 1? 10

6

2

1

Solution a. If your calculator uses the order of operations, it will display 6.

10 6 2 1 10 (6 2) 1 (10 3) 1 6 b. If your calculator does not use the order of operations and performs the

operations as they are entered, it will display 1. [(10 6) 2] 1 (4 2) 1 21 1

Sports

EXAMPLE

5

Evaluate a Real-Life Expression

You are playing basketball. You make 8 field goals and 2 free throws. Your friend makes half as many field goals as you and no free throws. How many points ahead of your friend are you?

BASKETBALL SCORES

Solution 8p2 2

16 2

8 p 2 2 p 1 16 2

BASKETBALL SCORES

A field goal is worth 2 points. A free throw is worth 1 point.

ANSWER 䊳

Multiply from left to right.

16 2 8

Divide.

18 8

Add.

10

Subtract.

You are 10 points ahead of your friend.

Evaluate a Real-Life Expression 8. Your friend makes 4 field goals. You make three times as many field goals as

your friend plus one field goal. How many points do you have? Explain the order of operations you followed. 9. Your friend makes 6 field goals and 2 free throws. You make twice as many

field goals as your friend and half the number of free throws. How many points do you have? Explain the order of operations you followed. 1.3

Order of Operations

17


1. Place the operations in the order in which you should do them. a. Multiply and divide from left to right. b. Do operations within grouping symbols. c. Add and subtract from left to right. d. Evaluate powers. 2. What rule must be applied when evaluating an expression in which the

operations have the same priority?

Skill Check

Evaluate the expression. 3. 5 p 6 p 2

4. 16 4 2

5. 4 9 1

6. 2 p 82

7. 15 6 3

8. 9 3 p 2

9. 2 p 32 5

10. 23 p 32

Evaluate the variable expression when x 3. 11. x2 5

12. x3 5x

13. x 3x4

27 14. 2 16 x

15 15. 23 10 x

24 16. p 5 x

Practice and Applications NUMERICAL EXPRESSIONS Evaluate the expression.

17. 13 3 p 7

18. 7 8 2

19. 24 5 p 3

20. 62 4

21. 43 9 p 2

5 22. 3 p 2 9

VARIABLE EXPRESSIONS Evaluate the expression for the given value of

the variable.



18

Chapter 1

23. 6 p 2p2 when p 5

24. 2g p 5 when g 4

25. 14(n 1) when n 2

x 26. 16 when x 14 7

NUMERICAL EXPRESSIONS Evaluate the expression.

27. 23 5 2

28. 4 p 2 15 3

29. 6 3 2 p 7

30. 5 8 p 2 4

31. 16 8 p 22

32. 2 p 32 7

33. 10 3 (2 5)

34. 7 18 (6 3)

35. [(7 p 4) 3] 15


Student Help HOMEWORK HELP NE ER T

INT


EXPRESSIONS WITH FRACTION BARS Evaluate the expression. Then

simplify the answer. 6p4 36. 4 32 1

13 4 37. 18 42 1

52 p 2 38. 2 1 6 12

21 9 39. 52 40 5

33 8 7 40. 2p7

4 p 25 41. 16 42 1

42. LOGICAL REASONING Which is correct? 92 3 A. 92 3 5 5

92 3 B. [92 3] 5 5

CRITICAL THINKING In Exercises 43–46, two calculators were used

to evaluate the expression. Determine which calculator performed the correct order of operations. 43. 15

6

3

4

Calculator A: 12 44. 15

9

Calculator B: 7

3

7


10

Calculator B: 9 5


3

4 Calculator B: 9

6

Calculator A: 9

2 Calculator B: 15

FOOTBALL UNIFORMS

In Exercises 47 and 48, use the table showing the costs of parts of a football player’s uniform. 47. A sporting goods company

offers a $2000 discount for orders of 30 or more complete uniforms. Your school orders 35 complete uniforms. Write an expression for the total cost. 48. Evaluate the expression you

wrote in Exercise 47. Part of uniform

Jersey and pants

Shoulder pads

Lower body pads

Knee pads

Cleats

Helmet

Cost

$230

$300

$40

$15

$100

$200

1.3

Order of Operations

19

In Exercises 49 and 50, refer to the squares shown at the right. 49. Write an expression that represents the area of the

1 x 1 2 x 2

shaded region. HINT: Subtract the area of the inner square from the area of the outer square. 50. If x 8, what is the area of the shaded region?

x

x

ADMISSION PRICES In Exercises 51 and 52, use the table below. It

State Fairs

shows the admission prices for the California State Fair. California State Fair Admission Prices Age

ADMISSION PRICES Every

year nearly 1,000,000 people attend the California State Fair.

Admission price

General Admission (13–61 years of age)

$7.00

Seniors (62 years and above)

$5.00

Children (5–12 years)

$4.00

Children (4 years and under)

Free

䊳 Source: Sacramento Bee

51. Write an expression that represents the admission price for a group

consisting of 2 adults, 1 senior, and 3 children. The children’s ages are 12 years, 10 years, and 18 months. 52. Evaluate the expression you wrote in Exercise 51. Then find the total cost

of admission for the group. 53. CHALLENGE At a concert you buy a hat for $10.00, a hot dog for $2.75, and

nachos for $3.50. There is a 6% sales tax on the hat. Your calculator follows the established order of operations. Write a keystroke sequence for the amount you owe. Then find the amount you owe. HINT: 6% 0.06


54. MULTIPLE CHOICE Evaluate the expression 42 10 2. A 

3

B 

11

C 

13

D 

21

55. MULTIPLE CHOICE Evaluate the expression 32 x2 9 when x 2. F 

19

G 

21

H 

37

J 

39

56. MULTIPLE CHOICE Which expression has a value of 12? A  B  C  D 

3352

F 

1

18 6 3 3 7 14 7 4 22 p 3 6 p 2

32 6 5 57. MULTIPLE CHOICE Evaluate the expression . 2p5

20

Chapter 1


G 

5

H 

7

J 

10

Mixed Review


of the variable. (Lesson 1.1) 58. (8)(a) when a 4

24 59. when x 3 x

60. c 15 when c 12

x 61. p x when x 18 2

62. 9t when t 7

63. 25 y when y 14

WRITING POWERS Write the expression in exponential form. (Lesson 1.2)

64. twelve squared

65. z to the sixth power

66. 2b p 2b p 2b

EXPONENTIAL EXPRESSIONS Evaluate the expression for the given

value of the variable. (Lesson 1.2) 67. 9t 2 when t 3

Maintaining Skills

68. (7h)3 when h 1

69. (6w)2 when w 5

FACTORS Determine whether the number is prime or composite. If it is composite, list all of its factors. (Skills Review p. 761)

70. 15

71. 9

72. 13

73. 38

74. 46

75. 50

76. 64

77. 29

Quiz 1 Evaluate the variable expression when x 3. (Lessons 1.1, 1.2) 1. 6x

2. 42 x

3. x 29

4. 12 x

5. 5x 10

6. 10 2x

7. x2 3

8. 2x3

9. (2x)3

Find the distance traveled using d rt. (Lesson 1.1) 10. A car travels at an average speed of 50 miles per hour for 4 hours. 11. A plane flies at 500 miles per hour for 4 hours. 12. A marathon runner keeps a steady pace of 10 miles per hour for 2 hours. Write the expression in exponential form. (Lesson 1.2) 13. six cubed

14. 4 p 4 p 4 p 4 p 4

15. 5y p 5y p 5y

16. 3 p 3 p 3

17. 2x p 2x p 2x p 2x

18. eight squared

19. PACKING BOXES A cubic packing box has dimensions of 4 feet on each edge. What is the volume of the box? (Lesson 1.2) Evaluate the expression. Then simplify the answer. (Lesson 1.3) 7 p 22 20. 7 (23 1)

62 11 22. 2(17 2 p 4)

(32 3) 21. 2p9 1.3

Order of Operations

21

DEVELOPING CONCEPTS

1.4

Finding Patterns Question

GOAL

For use with Lesson 1.4

How can you use algebra to describe a pattern?

Use algebraic expressions to describe patterns.

MATERIALS • graph paper • toothpicks

Copy the first four figures on graph paper. Then draw the fifth and sixth figures of the sequence.

Figure 1

Figure 2

Figure 3

Figure 4

The table shows the mathematical pattern for the perimeters of the first four figures. Copy and complete the table. Figure

1

2

3

4

5

6

Perimeter

4

8

12

16

?

?

4p1

4p2

4p3

4p4

4p?

4p?

Pattern

Observe that 4(1) 4, 4(2), 8, 4(3) 12, and so on. This suggests that the perimeter of the nth figure is 4n, where n 1, 2, 3, 4, . . . . Find the perimeter of the 10th figure.

Think About It 1. Copy the four figures below. Then draw the fifth and sixth figures.

Figure 1

Figure 2

Figure 3

Figure 4

2. Calculate the perimeters of all six figures. Organize your results in a table. 3. What is the perimeter of the 10th figure? Can you guess a formula for the

nth figure? 22

Chapter 1


Use toothpicks to model the perimeter of all six figures in Explore on page 22. Notice that the perimeter of each figure is equal to the number of toothpicks used to form the figure. Change the shape of Figures 2–6 by moving toothpicks until the figures consist of n unit squares. Figures 2 and 3 in the sequence are shown below. Complete Figures 4, 5, and 6 on a separate sheet of paper. Figure 2

Figure 3

You should be able to conclude that if one square unit has a perimeter of 4 p 1, then n squares must have a perimeter of 4n. This conclusion verifies the pattern you found on page 22.

Think About It 1. Use toothpicks to model the perimeter of all six figures in Exercise 1 on

page 22. 2. Change the shape of the figures modeled above in Exercise 1 until they

consist of n unit squares. 3. Do the number of unit squares verify the pattern found in Exercise 3 on

page 22? Explain your reasoning. Developing Concepts

23

1.4

Equations and Inequalities How much do the ingredients cost?

Goal Check solutions of equations and inequalities.

You can use an equation to solve a real-life problem. In Example 3 you will use an equation to estimate the cost of ingredients for nachos.

Key Words • equation • solution • inequality

Student Help

An equation is a statement formed by placing an equal sign () between two expressions. An equation has a left side and a right side.

VOCABULARY TIP

Equation comes from a Latin word that means “to be equal”.

Left side

Right side

4x 1 9 Equation

When the variable in an equation is replaced by a number, the resulting statement is either true or false. If the statement is true, the number is a solution of the equation.

EXAMPLE

1

Check Possible Solutions

Check to see if 2 and 3 are solutions of the equation 4x 1 9. Solution Substitute the x values 2 and 3 into the equation. If both sides of the equation are equal in value, then the number is a solution. X VALUE

SUBSTITUTE

SIMPLIFY

CONCLUSION

2

4(2) 1 ⱨ 9

99

True, 2 is a solution.

3

4(3) 1 ⱨ 9

13 9

False, 3 is not a solution.

is not equal to ANSWER 䊳

24

Chapter 1


The number 2 is a solution of the equation 4x 1 9, because the statement is true. The number 3 is not a solution, because the statement is false.

Finding all the solutions of an equation is called solving the equation. Some equations are simple enough to be solved with mental math. Later in the book you will learn how to systematically solve more complex equations.

SOLVING EQUATIONS

Student Help

EXAMPLE

MORE EXAMPLES

Solve Equations with Mental Math

To solve equations with mental math, think of the equation as a question.

NE ER T


INT

2

EQUATION

QUESTION

SOLUTION

2x 10

2 times what number gives 10?

2 p 5 10, so x 5

4x3

4 is equal to what number minus 3?

4 7 3, so x 7

2x6

2 plus what number gives 6?

2 4 6, so x 4

x 1 3

What number divided by 3 gives 1?

3 1, so x 3 3

Then check each solution by substituting the number in the original equation. If the statement is true, the number is a solution.

Solve Equations and Check Solutions Use mental math to solve the equation. Then check your solution. 1. 2 6 x

EXAMPLE

Nachos

ips 20 tortilla ch s n ea b s p 1 1/2 cu ed ic 1 cup d tomatoes

ted 1/2 cup gra e chees d 2 1/ cup slice olives

Add ns on chips. ves. Spread bea n eese a d oli ch en th , es tomato inutes. °F for 5 m Bake at 400

2. x 3 11

3

x 3. 5 4

4. 14 2x

Use Mental Math to Solve a Real-Life Equation

You are buying ingredients for nachos. At the market you find that tortilla chips cost $2.99, beans cost $.99, cheese costs $3.99, two tomatoes cost $1.00, and olives cost $1.49. There is no tax. You have $10. About how much more money do you need? Solution Ask: The total cost equals 10 plus what number of dollars? Let x

represent the extra money you need. Use rounding to estimate the total cost. 3 1 4 1 1.5 10 x 10.5 10 x ANSWER 䊳

The total cost is about 10.5 or $10.50, so you need about $.50 more to purchase all the ingredients.

Use Mental Math to Solve a Real-Life Equation 5. Solve the equation in Example 3 if a large bag of chips costs $3.99. About

how much more money would you need to buy the nacho ingredients? 1.4

Equations and Inequalities

25

Student Help

An inequality is a statement formed by placing an inequality symbol, such as 35

20. 8 ≥ 64 b

21. 3b 24 > 0

22. 16 ≤ b2

23. 60 > 7b 3

24. 18 b < 10

25. 37 ≥ 4b

Practice and Applications CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value of the variable is or is not a solution of the equation.

26. 3b 1 13; b 4

27. 5r 10 11; r 5

28. 4c 2 10; c 2

29. 6d 5 31; d 6

30. 5 x2 17; x 3

31. 2y3 3 5; y 1

32. 9 2t 15; t 12

33. n2 5 20; n 5

SOLVING WITH MENTAL MATH Use mental math to solve the equation.

Student Help

34. x 3 8

35. n 6 11

36. p 13 20

37. r 1 7

38. 3y 12

39. 4p 36

40. z 4 5

x 41. 3 7

42. 2b 28

43. 11t 22

44. 29 d 10

45. 3 y 8

46. r 30 70

42 47. 7 x

48. 7m 49

HOMEWORK HELP

Example 1: Exs. 26–33 Example 2: Exs. 34–48 Example 3: Exs. 49, 50 Example 4: Exs. 51–56 Example 5: Exs. 57, 58

1.4


27

49. TIME MANAGEMENT You have a hair appointment in 60 minutes. It takes

20 minutes to get to the gas station and fill your tank. It takes 15 minutes to go from the gas station to the hair stylist. You wait x minutes before leaving your house and arrive on time for your appointment. Use the diagram to help decide which equation best models the situation. GAS

Hair Salon

2 3 0 0 6 1

x

20

15

A. 20 15 x 60

B. 60 20 15 x

C. 60 20 15 x 60

D. x 20 15 60

50. MENTAL MATH Solve the equation you chose in Exercise 49. CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value

of the variable is or is not a solution of the inequality. 51. n 2 < 6; n 3

52. a 7 ≥ 15; a 22

53. 6 y ≤ 8; y 3

54. s 5 > 8; s 4

55. 7g ≥ 47; g 7

56. 72 t > 6; t 12

57. SELLING CARDS Your community center is selling cards. Your goal is to

sell $100 worth of cards. Each box sells for $3. Using mental math, solve the inequality 3b ≥ 100 to determine at least how many boxes you must sell to meet your goal. 58. BUYING A GUITAR You are budgeting money to buy a guitar that costs

$150 including tax. If you save $20 per month, will you have enough money in 6 months? Use the inequality 20n ≥ 150 to model the situation, where n represents the number of months.

Science 59.

Mach number is the maximum speed at which a plane can fly divided by the speed of sound. Copy and complete the table. Use the v 660

equation m , where m is the Mach number and v is the speed (in miles per hour) of the aircraft, to find the Mach number for each type of aircraft. Airplane type Speed v Mach number m

Test aircraft

Supersonic

Jet

4620

1320

660

?

?

?

CHUCK YEAGER in 1947

became the first person to fly faster than the speed of sound (Mach 1) or about 660 miles per hour. 28

Chapter 1


Test aircraft

Supersonic aircraft

Jet aircraft

60.

Use mental math to fill in the missing number so that all the equations have the number 6 as a solution.

䡵

䡵

a. ? x 18


? c. 6 x

b. ? x 30

61. MULTIPLE CHOICE Which is a solution of the equation 5(8 x) 25? A 

2

B 

3

C 

D 

4

5

62. MULTIPLE CHOICE For which inequality is x 238 a solution? F  H 

G  J 

250 ≥ x 12 250 > x 12

250 < x 12 250 ≤ x 1

63. MULTIPLE CHOICE The width of a soccer field cannot be greater than

100 yards. The area cannot be greater than 13,000 square yards. Which of the following would you use to find the possible length x of a soccer field? A  C 

Mixed Review

B  D 

100x ≥ 13,000 100 x ≤ 13,000

100x ≤ 13,000 100x 13,000


of the variable. (Lesson 1.1) 64. b 12 when b 43

65. 12 x when x 4

66. 12n when n 4

y 67. when y 30 15

WRITING POWERS Write the expression in exponential form.

(Lesson 1.2) 68. 3 p 3 p 3 p 3 p 3

69. seven squared

70. y p y p y p y

71. 9 p 9 p 9 p 9 p 9 p 9

72. twelve cubed

73. 8d p 8d p 8d

NUMERICAL EXPRESSIONS Evaluate the expression. Then simplify the

answer. (Lesson 1.3)

Maintaining Skills

74. 9 12 4

75. 7 56 8 2

76. 63 3 p 3

77. 4 p 2 5

78. 3 13 6

79. 49 7 2

80. (28 4) 32

42 2 81. 2

82. 2[(2 3)2 10]

ROUNDING Round the number to the underlined place value.

(Skills Review p. 774) 83. 5.64

84. 0.2625

85. 0.45695

86. 15.295

87. 758.949

88. 32.6582

89. 0.325

90. 26.96

91. 4.0965 1.4


29

1.5

Translating Words into Mathematical Symbols

Goal Translate words into mathematical symbols.

How long were you on the phone? In Example 6 you will translate words into an algebraic equation to find the length of a long distance phone call.

Key Words • translate • phrase • sentence

To solve real-life problems, you often need to translate words into mathematical symbols. To do this, look for words, such as sum or difference, that indicate mathematical operations.

y Summar Account ges New Char 36.00 ice..........$ rv Basic Se 6.00 nce ........$ Long Dista 42.00 rges .......$ Total Cha 42.00 t Due $ un

o Total Am

EXAMPLE

1

Translate Addition Phrases

Write the phrase as a variable expression. Let x represent the number. PHRASE

TRANSLATION

The sum of 6 and a number

6x

8 more than a number

x8

A number plus 5

x5

A number increased by 7

x7

EXAMPLE

2

Translate Subtraction Phrases

Write the phrase as a variable expression. Let y represent the number. PHRASE


Order is important for subtraction. “4 less than a number” means y 4, not 4 y.

TRANSLATION

The difference between 5 and a number

5y

7 minus a number

7y

A number decreased by 9

y9

4 less than a number

y4

Translate Addition and Subtraction Phrases Write the phrase as a variable expression. Let x represent the number. 1. 11 more than a number 30

Chapter 1


2. A number decreased by 10


Quotient comes from a word meaning “how many times.” When you divide you are finding how many times one quantity goes into another.

Notice that order does not matter for addition and multiplication. “The sum of 6 and a number” can be written as either 6 x or x 6. Order is important for n 4

4 n

subtraction and division. “The quotient of a number and 4” means , not .

EXAMPLE

3

Translate Multiplication and Division Phrases

Write the phrase as a variable expression. Let n represent the number. PHRASE

TRANSLATION

The product of 9 and a number

9n

10 times a number

10n

A number multiplied by 3

3n

One fourth of a number

1 n 4

The quotient of a number and 6

n 6

7 divided by a number

7 n

Translate Multiplication and Division Phrases Write each phrase as a variable expression. Let x represent the number. 3. The quotient of 8 and a number

4. The product of 2 and a number

In English there is a difference between a phrase and a sentence. Phrases are translated into variable expressions. Sentences are translated into equations or inequalities.

TRANSLATING SENTENCES

EXAMPLE Student Help

PHRASE

EXPRESSION

SENTENCE

EQUATION OR INEQUALITY

4

Translate Sentences

Write the sentence as an equation or an inequality.

READING ALGEBRA

SENTENCE

TRANSLATION

The word is by itself means “.”

The sum of a number x and 12 is 16.

x 12 16

The words is less than mean “m2

Find the volume of a cube when each side x is 10 feet.

42.

(Lesson 1.2) CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value

of the variable is or is not a solution of the equation. (Lesson 1.4)

Maintaining Skills

43. 8k 2 30; k 4

44. 15 2c 5c; c 5

r2 45. 40; r 9 2

46. 50 3w; w 15

PERCENTS AND DECIMALS Write the percent as a decimal.

(Skills Review p. 768) 47. 28%

48. 25%

49. 40%

50. 22%

51. 45%

52. 90%

53. 17.4%

54. 6.51%

Quiz 2 Check to see if x 4 is or is not a solution of the equation. (Lesson 1.4)

x 2. 0 4

1. 10x 5 35

3. x2 5 21

Check to see if a 20 is or is not a solution of the inequality. (Lesson 1.4) 4. 3a > 50

5. 10 a < 30

6. 40 3a ≥ 50

a 7. ≤ 5 5

80 8. ≥ 5 a

a 9. 2 ≤ 5 5

10.

The rectangle shown at the right has an area of 32 square units. Write an equation to find the width x. Use mental math to solve the equation. HINT: The area of a rectangle equals length times width. (Lesson 1.4)

x 8

Write the sentence as an equation or an inequality. (Lesson 1.5) 11. A number x divided by 9 is less than 17. 12. The product of 10 and a number x is 50. 13. A number y plus 10 is greater than or equal to 57. 14. A number y minus 6 is 15. 1.5

Translating Words into Mathematical Symbols

35

1.6

A Problem Solving Plan Using Models

Goal Model and solve real-life problems.

How much food did you order? In Chinese restaurants the bill is sometimes totaled by counting the number of plates ordered. In Example 1 you will use an algebraic model to find out how many plates of food you ordered.

Key Words • modeling • verbal model • algebraic model

Writing algebraic expressions, equations, or inequalities that represent real-life situations is called modeling. First you write a verbal model using words. Then you translate the verbal model into an algebraic model. Write a verbal model.

EXAMPLE

1

Assign labels.

Write an algebraic model.

Write an Algebraic Model

You and some friends are at a Chinese restaurant. You order several $2 plates of wontons, egg rolls, and dumplings. Your bill is $25.20, which includes tax of $1.20. Use modeling to find how many plates you ordered.

Student Help

Solution

STUDY TIP

Be sure you understand the problem before you write a model. For example, notice that the tax is added after the cost of the plates is figured.

VERBAL MODEL

LABELS

ALGEBRAIC MODEL

Cost per p Number Amount of bill Tax of plates plate Cost per plate 2

(dollars)

Number of plates p

(plates)

Amount of bill 25.20

(dollars)

Tax 1.20

(dollars)

2 p 25.20 1.20 2 p 24 p 12

ANSWER 䊳

36

Chapter 1


Write algebraic model. Subtract. Solve using mental math.

Your group ordered 12 plates of food costing $24.

A PROBLEM SOLVING PLAN USING MODELS VERBAL MODEL

Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know.

LABELS

Assign labels to each part of your verbal model. Use the labels to write an algebraic model based on your verbal model.

ALGEBRAIC MODEL SOLVE

Solve the algebraic model and answer the original question.

CHECK

Check that your answer is reasonable.

EXAMPLE

2


A football field is about 53 yards wide and 120 yards long. A soccer field has the same area, but is 60 yards wide. How long is the soccer field?

STUDY TIP

Sometimes a diagram can help you see what you know and what you need to find to solve the problem.

Solution Write a verbal model showing that the area (width length) of the soccer field equals the area (width length) of the football field. 10 20 30 40 50 40 30 20 10

Student Help

53 yd

60 yd 10 20 30 40 50 40 30 20 10

120 yd

x yd VERBAL MODEL

Width of p Length of Width of p Length of soccer field soccer field football field football field

LABELS

Width of soccer field 60

(yards)

Length of soccer field x

(yards)

Width of football field 53

(yards)

Length of football field 120

(yards)

ALGEBRAIC MODEL

60 x 53 120 60 x 6360 x 106

ANSWER 䊳

Write algebraic model. Simplify. Solve with mental math. (60 p 106 6360)

The soccer field is 106 yards long.

Write an Algebraic Model 1. You want two rectangular gardens to have equal areas. The first garden is

5 meters by 16 meters. The second garden is 8 meters wide. How long should the second garden be? Apply the problem solving plan to find the answer. 1.6


37

EXAMPLE

Careers

3


A jet pilot is flying from Los Angeles to Chicago at a speed of 500 miles per hour. When the plane is 600 miles from Chicago, an air traffic controller tells the pilot that it will be 2 hours before the plane can get clearance to land.

JET PILOTS

a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? b. The pilot knows that at her present altitude, the speed of the jet must be

greater than 322 miles per hour or the plane could stall. Is it reasonable for the pilot to fly directly to Chicago at the reduced speed from part (a) or should the pilot take some other action?

INT

JET PILOTS select a route, an altitude, and a speed that will provide the fastest and safest flight. NE ER T More about jet pilots available at www.mcdougallittell.com

Solution a. You can use the formula (rate)(time) (distance) to write a verbal model. VERBAL MODEL LABELS

ALGEBRAIC MODEL

ANSWER 䊳

Speed of jet p Time Distance to travel Speed of jet x

(miles per hour)

Time 2

(hours)

Distance to travel 600

(miles)

2 x 600 x 300

Write algebraic model. Solve using mental math.

To arrive in 2 hours, the pilot would have to slow the jet down to a speed of 300 miles per hour.

b. It is not reasonable for the pilot to fly at 300 miles per hour, because the

plane could stall. The pilot should take some other action, such as circling in a holding pattern, to use some of the time.

You can ignore information that is not needed to solve a problem. To solve Example 3, you do not need to know that the plane is traveling at 500 miles per hour.

Write an Algebraic Model Use the following information to write and solve an algebraic model.

You are running in a marathon. During the first 20 miles, your average speed is 8 miles per hour. During the last 6.2 miles, you increase your average speed by 2 miles per hour. 2. How long will it take you to run the last 6.2 miles of the marathon? Use the

problem solving plan with models to answer the question. 3. A friend of yours completed the marathon in 3.2 hours. Did you finish ahead

of your friend or behind your friend? Explain. 38

Chapter 1



In Exercises 1 and 2, complete the sentence. 1. Writing expressions, equations, or inequalities to represent real-life situations

is called ____ ? . 2. A ____ ? model with labels is used to form an algebraic model. 3. Write the steps of the problem solving plan.

Skill Check

4. ADMISSION PRICES Your family and friends are going to an amusement

park. Adults pay $25 per ticket and children pay $15 per ticket. Your group has 13 children and your total bill for tickets is $370. How many adults are in your group? Choose the verbal model that represents this situation. A.

Cost per Number Cost per Number of Total adult p of adults child p children cost

B.

Cost per Number Cost per Number of Total adult p of adults child p children cost

C.

Cost per Number Cost per Number of Total adult of adults p child children cost

Practice and Applications WALK OR TAKE THE SUBWAY? In Exercises 5–10, use the following

information.

You are one mile from your home. You can walk at a speed of 4 miles per hour. The subway comes by every 15 minutes, and you heard one come by 3 minutes ago. The subway ride takes 8 minutes. 5. How many minutes will it take to get home by subway if you take the next

train? 6. Write a verbal model that relates the time it would take to walk home, your

walking speed, and the distance to your home. ____ ? ____ ? ____ ?

Student Help

7. Assign labels to your verbal model. Use t to represent the unknown value.

HOMEWORK HELP

8. Use the labels to translate your verbal model into an equation.

Example 1: Exs. 5–20 Example 2: Exs. 5–20 Example 3: Exs. 5–20

9. Use mental math to solve the equation. 10. Which will get you home faster, walking or taking the subway? Explain. 1.6


39

In Exercises 11–15, use the following information.

Kudzu is a type of Japanese vine that grows at a rate of 1 foot per day during the summer. On August 1, the length of one vine was 50 feet. What was the length on July 1? HINT: July has 31 days.

Science

11. Use the verbal phrases to complete the verbal model.

Total length Original length Number of days

____ ? ____ ? ____ ? ____ ?

Growth rate 12. Assign labels to the verbal model. Use x to represent the unknown value. PLANT GROWTH Kudzu

was introduced to the United States in 1876. Today kudzu covers over 7 million acres of the southeastern United

13. Choose the algebraic model that best represents the verbal model. A. (x 1) p 31 50

B. x (1 p 31) 50

C. x 50 1

D. x (1 31) 50

14. Use mental math to solve the algebraic model you chose in Exercise 13. 15. Check that your answer is reasonable.

Student Help

BUYING A STEREO In Exercises 16–20, use the following information.

HOMEWORK HELP NE ER T

INT


An appliance store sells two stereo models. The model without a CD player is $350. The model with a CD player is $480. Your summer job allows you to save $50 a week for 8 weeks. At the end of the summer, you have enough to buy the stereo without the CD player. How much would you have needed to save each week to buy the other model? 16. Write a verbal model that relates the number of weeks worked, the amount

you would have needed to save each week, and the price of the stereo with the CD player. ____ ? ____ ? ____ ? 17. Assign labels to your verbal model. Use m to represent the unknown value. 18. Use the labels to translate your verbal model into an equation. 19. Use mental math to solve the equation. 20. Check that your answer is reasonable. 21. CHALLENGE You are running for class president. By two o’clock on

election day you have 95 votes and your opponent has 120 votes. Forty-five more students will be voting. Let x represent the number of students (of the 45) who vote for you. a. Write an inequality that shows the values of x that will allow you to win

the election. b. What is the smallest value of x that is a solution of the inequality? 40

Chapter 1



Student Help TEST TIP

In Exercise 22, you can use unit analysis (p. 8) to see which answer choice is correct.

22. MULTIPLE CHOICE A jet

Baltimore

is flying from Baltimore to Orlando at a speed r of 500 miles per hour. The distance d between the two cities is about 793 miles. Which equation can be used to find the time t it takes to make the trip? A 

793 500t

B 

t

C  D 

793t 500

500 793

Orlando

t 793(500)

23. MULTIPLE CHOICE Jim lives in a state in which speeders are fined $25 for a

speeding ticket plus $10 for each mile per hour over the speed limit. Jim was given a ticket for $175 for speeding in a 45 mile per hour zone. Which equation can be used to find how fast Jim was driving? F  H 

Mixed Review

45 (x 45) p 10 175 25 (x 45) p 10 175 25

G  J 

x 45 p 10 175 25 (x 45) p 10 175 25


of the variable. (Lesson 1.2) 24. x2 2 when x 7 25. (2x)3 when x 5

26. (10 x)2 when x 6

NUMERICAL EXPRESSIONS Evaluate the expression. (Lesson 1.3)

27. 22 42 2

28. 4 8 p 4 1

29. 2 p 4 (7 3)

CHECKING SOLUTIONS Check to see if the given value of the variable is

or is not a solution of the equation or the inequality. (Lesson 1.4) 30. 2x 3 < 15; x 9 31. 3x 4 ≤ 16; x 4

32. 16 x2 4 17; x 2

33. FUNDRAISING Your fundraising group earns 25¢ for each lemonade and

50¢ for each taco sold. One hundred tacos are sold. Your total profit is $100. How many lemonades are sold? Write an equation that models this situation. Solve the equation using mental math. (Lesson 1.5)

Maintaining Skills

FRACTIONS AND MIXED NUMBERS Write the improper fraction as a

mixed number. (Skills Review p. 763) 3 34. 2

7 35. 4

11 36. 3

13 37. 6

16 38. 5

21 39. 9

18 40. 4

15 41. 7

30 42. 8

54 43. 12

84 44. 36

20 45. 3

1.6


41

1.7

Tables and Graphs How much does it cost to make a movie?

Goal Organize data using a table or graph.

Almost every day you have the chance to interpret data that describe real-life situations. In Example 3 you will interpret data about the average cost of making a movie.

Key Words • data • bar graph • line graph

Data are information, facts, or numbers that describe something. It is easier to see patterns when you organize data in a table.

EXAMPLE

Organize Data in a Table

1

The table shows the top three categories of food eaten by Americans.

Student Help

Top Categories of Food Consumed by Americans (lb per person per year)

STUDY TIP

INT

To find how much dairy was consumed in 1980, you go across the row labeled Dairy and stop at the column for 1980.

Year

1970

1975

1980

1985

1990

1995

2000

Dairy

563.8

539.1

543.2

593.7

568.4

584.4

590.0

Vegetables

335.4

337.0

336.4

358.1

382.8

405.0

410.0

Fruit

237.7

252.1

262.4

269.4

273.5

285.4

290.0

NE ER T

DATA UPDATE of U.S. Department of Agriculture at www.mcdougallittell.com; 2000 data are estimated by authors.

Make a table showing total dairy and vegetables consumed (pounds per person) per year. In which year did Americans consume the least dairy and vegetables? In which year did Americans consume the most dairy and vegetables? Solution To make the table, add the data for dairy and vegetables for the given year. Year

1970

1975

1980

1985

1990

1995

2000

Total

899.2

876.1

879.6

951.8

951.2

989.4

1000.0

ANSWER

䊳 The least consumption was in 1975 and the greatest in 2000.

Organize Data in a Table 1. Make a table showing the total dairy products, vegetables, and fruit

consumed (pounds per person) per year. Which year had the least consumption? Which had the greatest consumption? 42

Chapter 1



Horizontal bars go across parallel with the horizon. Vertical bars go straight up and down.

One way to represent the data in a table is with a bar graph. The bars can be either vertical or horizontal. Example 2 shows a vertical bar graph of the data from Example 1. BAR GRAPHS

EXAMPLE

2

Interpret a Bar Graph

The bar graph shows the total amount of dairy products, vegetables, and fruit consumed by the average American in a given year. It appears that Americans ate about five times the amount of dairy products, vegetables, and fruit in 1995 as compared with 1970. If you study the data in Example 1, you can see that the bar graph could be misinterpreted. Explain why the graph could be misinterpreted.

00

95

20

19

90

80

85

19

19

19

19

19

75

1300 1275 1250 1225 1200 1175 1150 1125 0

70

Consumption (lb/person)

Consumption of Dairy, Vegetables, and Fruit

Year

Solution The bar graph could be misinterpreted because the vertical scale is not consistent. The zigzag line shows a break where part of the scale is not shown. Because of the break, the first tick mark on the vertical scale represents 1125 pounds of food consumed per person. The other tick marks on the vertical scale represent 25 pounds of food consumed per person. To make a bar graph that could not be misinterpreted, you must evenly space the tick marks and make sure that each tick mark represents the same amount.

Make and Interpret a Bar Graph 2. The bar graph at the

00

95

20

90

19

85

19

80

19

75

19

19

For help with drawing bar graphs, see p. 777.

70

SKILLS REVIEW

Copy and complete the bar graph using the data from Example 2. Describe the pattern from 1970 through 2000.

1400 1200 1000 800 600 400 200 0

19

Student Help

Total of Dairy, Vegetables, and Fruit Consumed Consumption (lb/person)

right is set up so that it is not misleading. The first two bars are drawn for you.

Year

1.7

Tables and Graphs

43

As an alternative to a vertical bar graph, data is sometimes represented by a line graph. Here the vertical bars are replaced by a single point located at the top of the bar. These points are then connected by line segments. Line graphs are especially useful for showing changes in data over time. LINE GRAPHS

Careers

EXAMPLE

3

Make and Interpret a Line Graph

MOVIE MAKING From 1983 to 1996, the average cost (in millions of dollars) of making a movie is given in the table. Draw a line graph of the data. Then determine in which three years did the cost decrease from the prior year.

Average Cost of Making a Movie 1983

1984

1985

1986

1987

1988

1989

Cost (millions)

$11.8

$14.0

$16.7

$17.5

$20.0

$18.1

$23.3

DIRECTORS OF PHOTOGRAPHY decide the

Year

1990

1991

1992

1993

1994

1995

1996

type of film and equipment used and the composition of the movie. NE ER T More about movie making is available at www.mcdougallittell.com

Cost (millions)

$26.8

$26.1

$28.9

$29.9

$34.3

$36.4

$33.6

INT

Year

䊳 Source: International Motion Picture Almanac

Solution Draw the vertical scale from 0 to 40 million dollars. Mark the number of years on the horizontal axis starting with 1983. For each average cost in the table, draw a point on the graph. Then draw a line from each point to the next point.

Cost (millions of dollars)

Average Cost of Making a Movie


19

83 19 84 19 85 19 86 19 87 19 88 19 89 19 90 19 91 19 92 19 93 19 94 19 95 19 96

This point represents the year 1983 and the cost $11.8 million.

40 35 30 25 20 15 10 5 0

In 1988, 1991, and 1996 the average cost of making a movie decreased from the prior year.

Make and Interpret a Line Graph 3. Make a line graph of the data above changing the tick marks on the vertical

scale to 0, 10, 20, 30, and 40. Which graph is easier to interpret? Why? 44

Chapter 1



1. Explain what data are. Give an example. 2. Name two ways to display organized data.

Skill Check

WEATHER Use the graph to

Average Rainfall in Erie, PA Rainfall (in.)

classify the statement as true or false. 3. Rainfall increases each month

over the previous month. 4. The amount of rainfall is the

same in May and July.

4.5 4.0 3.5 3.0 2.5 2.0 0

5. The greatest amount of rainfall

occurs in August.

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

䊳 Source: National Oceanic and Atmospheric Administration

Practice and Applications GOLF In Exercises 6 and 7, use the table showing scores for two rounds

of golf. Player 1

Player 2

Player 3

Player 4

Round 1

90

88

79

78

Round 2

94

84

83

80

6. Make a table showing the average score of each player. HINT: Find each

average by adding the two scores and dividing by the number of rounds. 7. Which player has the lowest average? Which one has the highest average? 8. SCHOOL ENROLLMENT The table shows the number of students (in

millions) enrolled in school in the United States by age. Make a table showing the total number of students enrolled for each given year.


Example 1: Exs. 6–10 Example 2: Exs. 11–14 Example 3: Exs. 15–18

Age

1980

1985

1990

1995

2000

14–15 years old

7282

7362

6555

7651

8100

16–17 years old

7129

6654

6098

6997

7600

18–19 years old

3788

3716

4044

4274

4800

䊳 Source: U.S. Bureau of the Census; 2000 data are estimated by authors.

9. Which year had the least number of students enrolled? Which had the

greatest number of students enrolled? 10. Did the total enrollment increase for each 5 year period? Explain. 1.7

Tables and Graphs

45

BRAKING DISTANCE In Exercises 11–13, use the bar graph showing

Driving

average braking distances for medium sized cars. 11. Estimate the braking distance for

Average Braking Distance

a car traveling 50 miles per hour.

Speed (mi/h)

70

12. Does it take twice as far to stop

a car that is going twice as fast? Explain. 13. Explain why it would be dangerous

60 50 40 30 20

to follow another car too closely when driving at 70 miles per hour. 14. BRAKING DISTANCE is the distance it takes for a vehicle to come to a complete stop after the brakes have been activated. The length of a skid mark indicates the speed at which a vehicle was traveling.

0

The table shows the number of gallons of water needed to produce one pound of some foods. Make a bar graph of the data. Food (1 lb)

Lettuce

Tomatoes

Melons

Broccoli

Corn

21

29

40

42

119

Water (gallons)

䊳 Source: Water Education Foundation

MINIMUM WAGE

Federal Minimum Wage Wage (dollars per hour)

In Exercises 15–17, use the line graph showing the minimum wage for 1991–1999. 15. For how many years did the

minimum wage remain the same as it was in 1991?

6 5 4 3 2 1 0 19 91 19 92 19 93 19 94 19 95 19 96 19 97 19 98 19 99

16. Estimate the minimum wage

during 1992.

In June of year

17. In which year did the minimum

wage first increase to over $5? 18.


䊳 Source: U.S. Bureau of Labor Statistics

The table shows the population (in thousands) of California following the Gold Rush of 1849. Make a line graph of the data. Year

NE ER T

Extra help with problem solving in Ex. 18 is available at www.mcdougallittell.com

INT

100 200 300 400 Braking distance (ft)

Population

1850

1860

1870

1880

1890

93

380

560

865

1213

19. CRITICAL THINKING The table shows the average fuel efficiency for

INT

passenger cars for different years. Organize the data into a graph. Explain why you chose the type of graph you used.

46

Chapter 1

Year

1980

1985

1990

1995

2000

Fuel efficiency (miles per gallon)

24.3

27.6

28.0

28.6

29.2

NE ER T

DATA UPDATE of National Highway Traffic Safety Administration at www.mcdougallittell.com; 2000 data are estimated by authors.



20. MULTIPLE CHOICE Which way of organizing data is useful for showing

changes in data over time? A 

B 

Table

Line graph

C 

Circle graph

D 

None of these

MULTIPLE CHOICE In Exercises 21 and 22, use the bar graph showing

one household’s monthly electricity usage in kilowatt-hours (kWh). Monthly Electricity Use

kWh used

140 120 100 80 60 0 J

F

M

A

M

J J Month

A

S

O

N

D

21. Which month shows the greatest decrease in use from the prior month? F  H 

G  J 

May June

October November

22. About how many total kilowatt-hours were used for the months of January

through April? A 

B 

480

Mixed Review

C 

400

D 

550

600

Find the perimeter and area of the geometric figure.

(Lesson 1.1) 23.

24.

7 in.

25. 10 m

14 in.

14 in. 7 in.

8m

6m

10 m

24 ft

6m

CHECKING SOLUTIONS Check to see if x

25 ft

7 ft

5 is or is not a solution of

the equation or the inequality. (Lesson 1.4)

Maintaining Skills

26. 17 x < 12

27. x 3x ≥ 18

28. 5x 2 12.5

29. 2.5 > 1.2x 3

30. x2 25

31. (3x)2 ≤ 255

32. 3x 2x 25

33. 19 2x > 10

34. 16 ≤ 3x 1

COMPARING DECIMALS Compare using or .

(Skills Review p. 770)

䢇 ? 1.67 38. 1.666 䢇 ? 0.479 41. 0.48 䢇

35. 71.717 ? 77.117

䢇 ? 15.700 39. 15.7 䢇 ? 3.09 42. 3.11 䢇

䢇 ? 0.434 40. 0.4321 䢇 ? 9.540 43. 9.54 䢇

36. 2.6 ? 2.65

37. 0.01 ? 0.0001

1.7

Tables and Graphs

47

1.8

An Introduction to Functions What is the altitude of the balloon?

Goal Use four different ways to represent functions.

You are in a hot-air balloon. You rise at a steady rate of 20 feet per minute. In Example 2 you will use the relationship between time and height to find the altitude of the balloon after a given number of minutes.

Key Words • • • • • •

function input output input-output table domain range

A function is a rule that establishes a relationship between two quantities, called the input and the output. For each input, there is exactly one output—even though two different inputs may give the same output. One way to describe a function is to make an input-output table.

EXAMPLE

1

GEOMETRY LINK

Make an Input-Output Table The diagram shows the first six triangular numbers.

a. Using the first six figures, make an input-output table in which the input is

the figure number n and the output is the triangular number T. b. Does the table represent a function? Justify your answer.

Solution a. Use the diagram to make an input-output table, as shown below. Input n

1

2

3

4

5

6

Output T

1

3

6

10

15

21

b. This is a function, because for each input there is exactly one output.

48

Chapter 1


The collection of all input values is the domain of the function and the collection of all output values is the range of the function. The domain of the function in Example 1 is 1, 2, 3, 4, 5, 6; the range of the function is 1, 3, 6, 10, 15, 21.

DOMAIN AND RANGE

When you are given the rule for a function, you can prepare to graph the function by making a table showing numbers in the domain and their corresponding output values.

EXAMPLE

2

Use a Table to Graph a Function

BALLOONING You are at an altitude of 250 feet in a hot-air balloon. You turn on the burner and rise at a rate of 20 feet per minute for 5 minutes. Your altitude h in feet after you have risen for t minutes is given by

h ⫽ 250 ⫹ 20t, where t ≥ 0 and t ≤ 5. a. Use the function to find the output h in feet for several inputs. Then

organize the data into an input-output table. b. Use the data in the table to draw a graph that represents the function.

Solution a. Find the outputs for t ⫽ 0, 1, 2, 3, 4, and 5. Then make a table. INPUT (MINUTES)

FUNCTION

OUTPUT (FEET)

t⫽0

h ⫽ 250 ⫹ 20(0)

h ⫽ 250

t⫽1

h ⫽ 250 ⫹ 20(1)

h ⫽ 270

t⫽2

h ⫽ 250 ⫹ 20(2)

h ⫽ 290

t⫽3

h ⫽ 250 ⫹ 20(3)

h ⫽ 310

t⫽4

h ⫽ 250 ⫹ 20(4)

h ⫽ 330

t⫽5

h ⫽ 250 ⫹ 20(5)

h ⫽ 350

Input t Output h b.


To plot the first point (t, h) find t ⫽ 0 on the horizontal axis. Then find h ⫽ 250 on the vertical axis. Mark the point (0, 250).

0

1

2

3

4

5

250

270

290

310

330

350

Let the horizontal axis represent the input t (in minutes). Label the axis from 0 to 5. Let the vertical axis represent the output h (in feet). Label the axis from 0 to 400. Plot the data points given in the table. Finally, connect the points. The graph shows that as the time increases, the height of the balloon increases. The graph represents the function h ⫽ 250 ⫹ 20t, where t ≥ 0 and t ≤ 5.

1.8

An Introduction to Functions

49

MORE EXAMPLES

Write an Equation to Represent a Function

As you dive deeper and deeper into the ocean, the pressure of the water on your body steadily increases. The pressure at the surface of the water is 14.7 pounds per square inch (psi). The pressure increases at a rate of 0.445 psi for each foot you descend. Write an equation to represent the pressure P as a function of the depth d for every 20 feet you descend until you reach a depth of 60 feet.

SCUBA DIVING

NE ER T


INT

3

EXAMPLE

Student Help

Solution Pressure at Diving Pressure Rate of change p given depth depth in pressure at surface

VERBAL MODEL

LABELS

(psi)

Pressure at surface 14.7

(psi)

Rate of change in pressure 0.445

(psi per foot of depth)

Diving depth d

(feet)

P 14.7 0.445 d where d ≥ 0 and d ≤ 60

ALGEBRAIC MODEL

ANSWER 䊳

Pressure at given depth P

The function can be represented by the equation P 14.7 0.445d, where d ≥ 0 and d ≤ 60.

Represent a Function Use the algebraic model from Example 3. 1. Make an input-output table for the function. Use d 0, 20, 40, and 60. 2. Draw a graph that represents the function.

SUMMARY Four Ways to Represent Functions INPUT-OUTPUT TABLE

50

Input n

Output P

1

1

2

3

3

6

4

10

5

15

6

21

Chapter 1

WORDS

You are in a hot-air balloon at a height of 250 feet. You begin to rise higher at a rate of 20 feet per minute for a period of 5 minutes.


EQUATION

h 250 20t, where t ≥ 0 and t≤5

GRAPH


Complete the sentence. 1. A function is a relationship between two quantities, called the ____ ? and

the ____ ? . 2. The collection of all input values is the ____ ? of the function. 3. The collection of all output values is the ____ ? of the function.

Skill Check

CAMPING In Exercises 4–6, use the following information.

You are going camping. The cost for renting a cabin at Shady Knoll Campground is $65.00 plus $12.00 per person. The cost in dollars is C 65 12n, where n is the number of people. 4. Copy and complete the input-output table. Input n

1

2

3

4

5

6

Output C

?

?

?

?

?

?

5. Draw a graph that is made up of isolated points representing the cost of

renting a cabin. 6. Determine the range of the function from the given input values in the

input-output table.

Practice and Applications INPUT-OUTPUT TABLES Make an input-output table for the function. Use

0, 1, 2, 3, 4, and 5 as values for x. 7. y 6x 5

8. y 26 2x

9. y (x 3) p 7

10. y 85 15x

11. y 5(15 x)

12. y 2(6x 10)

LINE GRAPHS Draw a line graph to represent the function given by the

input-output table.

Student Help

13.

HOMEWORK HELP

Example 1: Exs. 7–12, 15–20, 23, 24 Example 2: Exs. 13–15, 21, 23–25 Example 3: Exs. 23, 24

14.

Input x

1

2

3

4

5

6

Output y

14

12

10

8

6

4

Input x

1

2

3

4

5

6

Output y

8

11

14

17

20

23

1.8

An Introduction to Functions

51

15.

Student Help

CRITICAL THINKING Determine whether the table represents a function.

HOMEWORK HELP

16.

NE ER T


INT

The distance d (in miles) that sound travels in air in time t (in seconds) is represented by the function d 0.2t. Make a table of the input t and the output d. Use t values of 0, 5, 10, 15, 20, 25, and 30. Use your table to help you draw the graph of the function.

18.

Input

Output

1

17.

Input

Output

3

1

2

2

4

3

3

3

5

3

4

Input

Output

Input

Output

2

2

1

3

3

4

1

4

4

6

2

5

19.

CAR RACING In Exercises 20–22, use the following information.

The fastest winning speed in the Daytona 500 is about 178 miles per hour. In the table below, calculate the distance traveled d (in miles) after time t (in hours) using the equation d 178t. 20. Copy and complete the input-output table. Time (hours)

0.25

0.50

0.75

1.00

1.25

1.50

?

?

?

?

?

?

Distance traveled (miles) 21. Use the data to draw a graph.

History

22. For what values of t does the formula d 178t correspond to the situation

being modeled? 23.

In 1866 Texas cowhands used the Chisholm Trail to drive cattle north to the railroads in Kansas. The average rate r that the cattle could be moved along the trail was 11 miles per day. a. Write an equation, where d is

distance and t is time in days. b. Make a table of input t and

output d for t 7, 14, and 28. Then graph the data. JESSE CHISHOLM , the person for whom the Chisholm Trail is named, was a trader who was part Cherokee. 52

Chapter 1

c. The distance d from San Antonio

to Abilene was about 1100 miles. How long did it take to drive cattle the entire length of the trail? Connections to Algebra

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