0321673484 Beg Algebra b

11 EDITIONTHBEGINNINGALGEBRA

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Margaret L. LialAmerican River College

John HornsbyUniversity of New Orleans

Terry McGinnis

Addison-WesleyBoston Columbus Indianapolis New York San Francisco Upper Saddle River

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Library of Congress Cataloging-in-Publication DataLial, Margaret L.

Beginning algebra/Margaret L. Lial, John Hornsby, Terry McGinnis.11th ed.p. cm.

Includes bibliographical references and index.ISBN-13: 978-0-321-67348-0(student edition)ISBN-10: 0-321-67348-4(student edition)

1. Algebra. I. Hornsby, E. John. II. McGinnis, Terry. III. Title.QA152.3.L5 2012512.9dc22

2010002280

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

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protected by U.S.copyright laws and is provided solely forthe use of college in-structors in reviewingcourse materials forclassroom use. Dis-semination or sale ofthis work, or any part(including on theWorld Wide Web),will destroy the in-tegrity of the workand is not permitted.The work and materi-als from it shouldnever be made avail-able to students ex-cept by instructorsusing the accompany-ing text in theirclasses. All recipi-ents of this work areexpected to abide bythese restrictionsand to honor the in-tended pedagogicalpurposes and theneeds of other in-structors who rely onthese materials.

To Margaret, Cody, and FinleyP.P.J.

To PapaT.

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Contents

The Real Number System 11

Preface xi

STUDY SKILLS Using Your Math Textbook xx

1.1 Fractions 2STUDY SKILLS Reading Your Math Textbook 14

1.2 Exponents, Order of Operations, and Inequality 15STUDY SKILLS Taking Lecture Notes 22

1.3 Variables, Expressions, and Equations 221.4 Real Numbers and the Number Line 28STUDY SKILLS Tackling Your Homework 36

1.5 Adding and Subtracting Real Numbers 37STUDY SKILLS Using Study Cards 48

1.6 Multiplying and Dividing Real Numbers 49SUMMARY EXERCISES on Operations with Real Numbers 59

1.7 Properties of Real Numbers 601.8 Simplifying Expressions 69STUDY SKILLS Reviewing a Chapter 75

Chapter 1 Summary 76

Chapter 1 Review Exercises 79

Chapter 1 Test 83

Linear Equations and Inequalities in One Variable 8522.1 The Addition Property of Equality 862.2 The Multiplication Property of Equality 922.3 More on Solving Linear Equations 97SUMMARY EXERCISES on Solving Linear Equations 106

STUDY SKILLS Using Study Cards Revisited 107

2.4 An Introduction to Applications of Linear Equations 1082.5 Formulas and Additional Applications from Geometry 1202.6 Ratio, Proportion, and Percent 1302.7 Further Applications of Linear Equations 1392.8 Solving Linear Inequalities 151STUDY SKILLS Taking Math Tests 163



Chapter 2 Test 171

Chapters 12 Cumulative Review Exercises 172

vii

viii Contents

Linear Equations and Inequalities in Two Variables;Functions 1753

3.1 Linear Equations in Two Variables; The Rectangular Coordinate System 176STUDY SKILLS Managing Your Time 187

3.2 Graphing Linear Equations in Two Variables 1883.3 The Slope of a Line 1993.4 Writing and Graphing Equations of Lines 211SUMMARY EXERCISES on Linear Equations and Graphs 222

3.5 Graphing Linear Inequalities in Two Variables 2233.6 Introduction to Functions 229STUDY SKILLS Analyzing Your Test Results 237



Chapter 3 Test 244


Systems of Linear Equations and Inequalities 24744.1 Solving Systems of Linear Equations by Graphing 2484.2 Solving Systems of Linear Equations by Substitution 2574.3 Solving Systems of Linear Equations by Elimination 264SUMMARY EXERCISES on Solving Systems of Linear Equations 270

4.4 Applications of Linear Systems 2724.5 Solving Systems of Linear Inequalities 281



Chapter 4 Test 291


Exponents and Polynomials 29555.1 The Product Rule and Power Rules for Exponents 2965.2 Integer Exponents and the Quotient Rule 303SUMMARY EXERCISES on the Rules for Exponents 311

5.3 An Application of Exponents: Scientific Notation 3125.4 Adding and Subtracting Polynomials; Graphing Simple Polynomials 3205.5 Multiplying Polynomials 3295.6 Special Products 3355.7 Dividing Polynomials 340



Chapter 5 Test 355


Contents ix

Factoring and Applications 35966.1 The Greatest Common Factor; Factoring by Grouping 3606.2 Factoring Trinomials 3686.3 More on Factoring Trinomials 3736.4 Special Factoring Techniques 381SUMMARY EXERCISES on Factoring 389

6.5 Solving Quadratic Equations by Factoring 3926.6 Applications of Quadratic Equations 400



Chapter 6 Test 416


Rational Expressions and Applications 41977.1 The Fundamental Property of Rational Expressions 4207.2 Multiplying and Dividing Rational Expressions 4297.3 Least Common Denominators 4357.4 Adding and Subtracting Rational Expressions 4407.5 Complex Fractions 4487.6 Solving Equations with Rational Expressions 456SUMMARY EXERCISES on Rational Expressions and Equations 465

7.7 Applications of Rational Expressions 4677.8 Variation 475STUDY SKILLS Preparing for Your Math Final Exam 482



Chapter 7 Test 490


Roots and Radicals 49388.1 Evaluating Roots 4948.2 Multiplying, Dividing, and Simplifying Radicals 5048.3 Adding and Subtracting Radicals 5138.4 Rationalizing the Denominator 5178.5 More Simplifying and Operations with Radicals 523SUMMARY EXERCISES on Operations with Radicals 530

8.6 Solving Equations with Radicals 5318.7 Using Rational Numbers as Exponents 540



Chapter 8 Test 549


x Contents

Quadratic Equations 55399.1 Solving Quadratic Equations by the Square Root Property 5549.2 Solving Quadratic Equations by Completing the Square 5609.3 Solving Quadratic Equations by the Quadratic Formula 567SUMMARY EXERCISES on Quadratic Equations 573

9.4 Complex Numbers 5749.5 More on Graphing Quadratic Equations; Quadratic Functions 580



Chapter 9 Test 593


Appendix A Sets 597Appendix B An Introduction to Calculators 603

Answers to Selected Exercises A-1

Glossary G-1

Credits C-1

Index I-1

xi

PrefaceIt is with pleasure that we offer the eleventh edition of Beginning Algebra. With eachnew edition, the text has been shaped and adapted to meet the changing needs of bothstudents and educators, and this edition faithfully continues that process. As always,we have taken special care to respond to the specific suggestions of users and re-viewers through enhanced discussions, new and updated examples and exercises,helpful features, updated figures and graphs, and an extensive package of supple-ments and study aids. We believe the result is an easy-to-use, comprehensive text thatis the best edition yet.

Students who have never studied algebraas well as those who require further reviewof basic algebraic concepts before taking additional courses in mathematics, busi-ness, science, nursing, or other fieldswill benefit from the texts student-orientedapproach. Of particular interest to students and instructors will be the StudySkills activities and Now Try Exercises.

This text is part of a series that also includes the following books:

N Intermediate Algebra, Eleventh Edition, by Lial, Hornsby, and McGinnis

N Beginning and Intermediate Algebra, Fifth Edition, by Lial, Hornsby, andMcGinnis

N Algebra for College Students, Seventh Edition, by Lial, Hornsby, and McGinnis

In this edition of the text, we are pleased to offer the following new student-orientedfeatures:

Study Skills Poor study skills are a major reason why students do not succeed inmathematics. In these short activities, we provide helpful information, tips, andstrategies on a variety of essential study skills, including Reading Your Math Text-book, Tackling Your Homework, Taking Math Tests, and Managing Your Time. Whilemost of the activities are concentrated in the early chapters of the text, each has beendesigned independently to allow flexible use with individuals or small groups of stu-dents, or as a source of material for in-class discussions. (See pages 48 and 163.)

Now Try Exercises To actively engage students in the learning process, we now include a parallel margin exercise juxtaposed with each numbered example. These all-new exercises enable students to immediately apply and reinforce the concepts andskills presented in the corresponding examples. Answers are conveniently located onthe same page so students can quickly check their results. (See pages 3 and 87.)

Revised Exposition As each section of the text was being revised, we paid specialattention to the exposition, which has been tightened and polished. (See Section 1.4Real Numbers and the Number Line, for example.) We believe this has improved dis-cussions and presentations of topics.

NEW IN THIS EDITION

NEW

Specific Content Changes These include the following:

N We gave the exercise sets special attention. There are approximately 1,100 newand updated exercises, including problems that check conceptual understanding,focus on skill development, and provide review. We also worked to improve theeven-odd pairing of exercises.

N Real-world data in over 185 applications in the examples and exercises has beenupdated.

N There is an increased emphasis on the difference between expressions and equa-tions, including a new Caution at the beginning of Section 2.1. Throughout thetext, we have reformatted many example solutions to use a drop down layout inorder to further emphasize for students the difference between simplifying expressions and solving equations.

N We increased the emphasis on checking solutions and answers, as indicated bythe new CHECK tag and in the exposition and examples.

N The presentation on solving linear equations in Sections 2.12.3 now includesfive new examples and corresponding exercises.

N Section 2.6 includes entirely new discussion and examples on percent, percentequations, and percent applications, plus corresponding exercises.

N Section 3.4 on writing and graphing equations of lines provides increased devel-opment and coverage of the slope-intercept form, including two new examples.

N Presentations of the following topics have also been enhanced and expanded:

Dividing real numbers involving zero (Section 1.6)Solving applications involving consecutive integers and finding angle measures

(Section 2.4)Solving formulas for specified variables (Sections 2.5 and 7.7)Using interval notation (Section 2.8)Graphing linear equations in two variables (Section 3.2)Solving systems of equations with decimal coefficients (Section 4.2)Dividing polynomials (Section 5.7)Factoring trinomials (Section 6.2)Solving quadratic equations by factoring (Section 6.6)

We have included the following helpful features, each of which is designed to in-crease ease-of-use by students and/or instructors.

Annotated Instructors Edition For convenient reference, we include answersto the exercises on page in the Annotated Instructors Edition, using an enhanced,easy-to-read format. In addition, we have added approximately 35 new Teaching Tipsand over 50 new and updated Classroom Examples.

Relevant Chapter Openers In the new and updated chapter openers, we featurereal-world applications of mathematics that are relevant to students and tied to spe-cific material within the chapters. Examples of topics include the Olympics, studentcredit card debt, and popular movies. Each opener also includes a section outline.(See pages 85, 175, and 247.)

HALLMARK FEATURES

xii Preface

Helpful Learning Objectives We begin each section with clearly stated, numberedobjectives, and the included material is directly keyed to these objectives so that studentsand instructors know exactly what is covered in each section. (See pages 2 and 130.)

Popular Cautions and Notes One of the most popular features of previous editions, we include information marked and NOTE to warn studentsabout common errors and emphasize important ideas throughout the exposition. Theupdated text design makes them easy to spot. (See pages 2 and 56.)

Comprehensive Examples The new edition of this text features a multitude ofstep-by-step, worked-out examples that include pedagogical color, helpful side com-ments, and special pointers. We give increased attention to checking example solutionsmore checks, designated using a special CHECK tag, are included than inpast editions. (See pages 87 and 396.)

More Pointers Well received by both students and instructors in the previous edi-tion, we incorporate more pointers in examples and discussions throughout thisedition of the text. They provide students with important on-the-spot reminders andwarnings about common pitfalls. (See pages 204 and 345.)

Updated Figures, Photos, and Hand-Drawn Graphs Todays students aremore visually oriented than ever. As a result, we have made a concerted effort to in-clude appealing mathematical figures, diagrams, tables, and graphs, including ahand-drawn style of graphs, whenever possible. (See pages 182 and 188.) Many ofthe graphs also use a style similar to that seen by students in todays print and elec-tronic media. We have incorporated new photos to accompany applications in exam-ples and exercises. (See pages 109 and 593.)

Relevant Real-Life Applications We include many new or updated applicationsfrom fields such as business, pop culture, sports, technology, and the life sciencesthat show the relevance of algebra to daily life. (See pages 277 and 409.)

Emphasis on Problem-Solving We introduce our six-step problem-solvingmethod in Chapter 2 and integrate it throughout the text. The six steps, Read, Assigna Variable, Write an Equation, Solve, State the Answer, and Check, are emphasized in boldface type and repeated in examples and exercises to reinforce the problem-solving process for students. (See pages 108 and 272.) We also provide students with

boxes that feature helpful problem-solving tips andstrategies. (See pages 139 and 401.)

Connections We include these to give students another avenue for making connec-tions to the real world, graphing technology, or other mathematical concepts, as wellas to provide historical background and thought-provoking questions for writing,class discussion, or group work. (See pages 195 and 315.)

Ample and Varied Exercise Sets One of the most commonly mentioned strengthsof this text is its exercise sets. We include a wealth of exercises to provide studentswith opportunities to practice, apply, connect, review, and extend the algebraic con-cepts and skills they are learning. We also incorporate numerous illustrations, tables,graphs, and photos to help students visualize the problems they are solving. Problemtypes include writing , graphing calculator , multiple-choice, true/false, matching,and fill-in-the-blank problems, as well as the following:

N Concept Check exercises facilitate students mathematical thinking and concep-tual understanding. (See pages 96 and 196.)

Preface xiii

CAUTION

PROBLEM-SOLVING HINT

N WHAT WENT WRONG? exercises ask students to identify typical errors in solu-tions and work the problems correctly. (See pages 208 and 398.)

N Brain Busters exercises challenge students to go beyond the section examples.(See pages 119 and 455.)

N exercises help students tie together topics and developproblem-solving skills as they compare and contrast ideas, identify and describe patterns, and extend concepts to new situations. These exercises make great collabo-rative activities for pairs or small groups of students. (See pages 209 and 539.)

N exercises provide an opportunity for students tointerpret typical results seen on graphing calculator screens. Actual screens fromthe TI-83/84 Plus graphing calculator are featured. (See pages 256 and 263.)

N allow students to review previously-studied conceptsand preview skills needed for the upcoming section. These make good oral warm-up exercises to open class discussions. (See pages 257 and 367.)

Special Summary Exercises We include a set of these popular in-chapter exer-cises in every chapter. They provide students with the all-important mixed reviewproblems they need to master topics and often include summaries of solution meth-ods and/or additional examples. (See pages 311 and 465.)

Extensive Review Opportunities We conclude each chapter with the followingreview components:

N A Chapter Summary that features a helpful list of Key Terms, organized bysection, New Symbols, Test Your Word Power vocabulary quiz (with answersimmediately following), and a Quick Review of each sections contents, com-plete with additional examples (See pages 238241.)

N A comprehensive set of Chapter Review Exercises, keyed to individual sectionsfor easy student reference, as well as a set of Mixed Review Exercises that helpsstudents further synthesize concepts (See pages 241244.)

N A Chapter Test that students can take under test conditions to see how well theyhave mastered the chapter material (See pages 244245.)

N A set of Cumulative Review Exercises (beginning in Chapter 2) that covers ma-terial going back to Chapter 1 (See page 246.)

Glossary For easy reference at the back of the book, we include a comprehensive glos-sary featuring key terms and definitions from throughout the text. (See pages G-1 to G-5.)

For a comprehensive list of the supplements and study aids that accompanyBeginning Algebra, Eleventh Edition, see pages xvixviii.

The comments, criticisms, and suggestions of users, nonusers, instructors, and stu-dents have positively shaped this textbook over the years, and we are most grateful forthe many responses we have received. Thanks to the following people for their reviewwork, feedback, assistance at various meetings, and additional media contributions:

ACKNOWLEDGMENTS

SUPPLEMENTS

PREVIEW EXERCISES

TECHNOLOGY INSIGHTS

RELATING CONCEPTS

xiv Preface

Barbara Aaker, Community College of DenverKim Bennekin, Georgia Perimeter CollegeDixie Blackinton, Weber State UniversityCallie Daniels, St. Charles Community CollegeCheryl Davids, Central Carolina Technical CollegeRobert Diaz, Fullerton CollegeChris Diorietes, Fayetteville Technical Community CollegeSylvia Dreyfus, Meridian Community CollegeSabine Eggleston, Edison State CollegeLaTonya Ellis, Bishop State Community CollegeBeverly Hall, Fayetteville Technical Community CollegeSandee House, Georgia Perimeter CollegeJoe Howe, St. Charles Community CollegeLynette King, Gadsden State Community CollegeLinda Kodama, Windward Community CollegeCarlea McAvoy, South Puget Sound Community CollegeJames Metz, Kapiolani Community CollegeJean Millen, Georgia Perimeter CollegeMolly Misko, Gadsden State Community CollegeJane Roads, Moberly Area Community CollegeMelanie Smith, Bishop State Community CollegeErik Stubsten, Chattanooga State Technical Community CollegeTong Wagner, Greenville Technical CollegeSessia Wyche, University of Texas at Brownsville

Special thanks are due all those instructors at Broward Community College for theirinsightful comments.

Over the years, we have come to rely on an extensive team of experienced professionals.Our sincere thanks go to these dedicated individuals at Addison-Wesley, who workedlong and hard to make this revision a success: Chris Hoag, Maureen OConnor, MichelleRenda, Adam Goldstein, Kari Heen, Courtney Slade, Kathy Manley, Lin Mahoney, andMary St. Thomas.

We are especially grateful to Callie Daniels for her excellent work on the new NowTry Exercises. Abby Tanenbaum did a terrific job helping us revise real-data applica-tions. Kathy Diamond provided expert guidance through all phases of production andrescued us from one snafu or another on multiple occasions. Marilyn Dwyer and Nesbitt Graphics, Inc. provided some of the highest quality production work we haveexperienced on the challenging format of these books.

Special thanks are due Jeff Cole, who continues to supply accurate, helpful solutionsmanuals; David Atwood, who wrote the comprehensive Instructors Resource Manualwith Tests; Beverly Fusfield, who provided the new MyWorkBook; Beth Anderson,who provided wonderful photo research; and Lucie Haskins, for yet another accurate,useful index. De Cook, Shannon dHemecourt, Paul Lorczak, and Sarah Sponholz dida thorough, timely job accuracy checking manuscript and page proofs. It has indeedbeen a pleasure to work with such an outstanding group of professionals.

As an author team, we are committed to providing the best possible text and supple-ments package to help instructors teach and students succeed. As we continue towork toward this goal, we would welcome any comments or suggestions you mighthave via e-mail to [email protected].

Margaret L. Lial

John Hornsby

Terry McGinnis

Preface xv

xvi Preface

STUDENT SUPPLEMENTS INSTRUCTOR SUPPLEMENTS

Annotated Instructors EditionN Provides on-page answers to all text exercises in

an easy-to-read margin format, along with TeachingTips and extensive Classroom Examples

N Includes icons to identify writing and calculator exercises. These are in Student Edition also.

ISBNs: 0-321-67585-1, 978-0-321-67585-9

Instructors Solutions ManualN By Jeffery A. Cole, Anoka-Ramsey Community CollegeN Provides complete answers to all text exercises,

including all Classroom Examples and Now TryExercises

ISBNs: 0-321-70243-3, 978-0-321-70243-2

Instructors Resource Manual with TestsN By David Atwood, Rochester Community and Techni-

cal College

N Contains two diagnostic pretests, four free-responseand two multiple-choice test forms per chapter, andtwo final exams

N Includes a mini-lecture for each section of the textwith objectives, key examples, and teaching tips

N Provides a correlation guide from the tenth to theeleventh edition

ISBNs: 0-321-69116-4, 978-0-321-69116-3

PowerPoint Lecture SlidesN Present key concepts and definitions from the textN Available for download at www.pearsonhighered.comISBNs: 0-321-70248-4, 978-0-321-70248-7

TestGen (www.pearsonhighered.com/testgen)N Enables instructors to build, edit, print, and adminis-

ter tests using a computerized bank of questions developed to cover all text objectives

N Allows instructors to create multiple but equivalentversions of the same question or test with the clickof a button

N Allows instructors to modify test bank questions oradd new questions

N Available for download from Pearson Educationsonline catalog

ISBNs: 0-321-70244-1, 978-0-321-70244-9

Students Solutions ManualN By Jeffery A. Cole, Anoka-Ramsey Community CollegeN Provides detailed solutions to the odd-numbered,

section-level exercises and to all Now Try Exercises,Relating Concepts, Summary, Chapter Review,Chapter Test, and Cumulative Review Exercises

ISBNs: 0-321-70245-X, 978-0-321-70245-6

Lial Video LibraryThe Lial Video Library, available in MyMathLab and onthe Video Resources on DVD, provides students with awealth of video resources to help them navigate theroad to success! All video resources in the library in-clude optional subtitles in English. The Lial Video Library includes the following resources:

N Section Lecture Videos offer a new navigation menuthat allows students to easily focus on the key exam-ples and exercises that they need to review in eachsection. Optional Spanish subtitles are available.

N Solutions Clips show an instructor working throughthe complete solutions to selected exercises fromthe text. Exercises with a solution clip are marked inthe text and e-book with a DVD icon .

N Quick Review Lectures provide a short summary lec-ture of each key concept from the Quick Reviews atthe end of every chapter in the text.

N The Chapter Test Prep Videos provide step-by-stepsolutions to all exercises from the Chapter Tests.These videos provide guidance and support whenstudents need it the most: the night before anexam. The Chapter Test Prep Videos are also avail-able on YouTube (searchable using author name and book title).

MyWorkBookN Provides Guided Examples and corresponding Now

Try Exercises for each text objective

N Refers students to correlated Examples, LectureVideos, and Exercise Solution Clips

N Includes extra practice exercises for every section ofthe text with ample space for students to show theirwork

N Lists the learning objectives and key vocabularyterms for every text section, along with vocabularypractice problems

ISBNs: 0-321-70251-4, 978-0-321-70251-7

NEW

NEW

STUDENT SUPPLEMENTS

N Includes an interactive guided solution for each ex-ercise that gives helpful feedback when an incorrectanswer is entered

N Enables students to view the steps of a worked-outsample problem similar to the one being worked on

INSTRUCTOR SUPPLEMENTS

N Staffed by qualified instructors with more than 50 years of combined experience at both thecommunity college and university levels

Assistance is provided for faculty in the followingareas:

N Suggested syllabus consultationN Tips on using materials packed with your bookN Book-specific content assistanceN Teaching suggestions, including advice on classroom

strategies

Available for Students and Instructors

MyMathLab Online Course (Access code required.)

MyMathLab is a text-specific, easily customizable online course that integrates interactive multimedia instruction with textbook content. MyMathLab gives instruc-tors the tools they need to deliver all or a portion of their course online, whether theirstudents are in a lab setting or working from home.

N Interactive homework exercises, correlated to the textbook at the objectivelevel, are algorithmically generated for unlimited practice and mastery. Most exercises are free-response and provide guided solutions, sample problems, andtutorial learning aids for extra help.

N Personalized homework assignments can be designed to meet the needs ofthe class. MyMathLab tailors the assignment for each student based on their testor quiz scores so that each students homework assignment contains only theproblems they still need to master.

N Personalized Study Plan, generated when students complete a test or quiz orhomework, indicates which topics have been mastered and links to tutorial exer-cises for topics students have not mastered. Instructors can customize the StudyPlan so that the topics available match their course content.

N Multimedia learning aids, such as video lectures and podcasts, animations,and a complete multimedia textbook, help students independently improve theirunderstanding and performance. Instructors can assign these multimedia learn-ing aids as homework to help their students grasp the concepts.

N Homework and Test Manager lets instructors assign homework, quizzes,and tests that are automatically graded. They can select just the right mix of ques-tions from the MyMathLab exercise bank, instructor-created custom exercises,and/or TestGen test items.

N Gradebook, designed specifically for mathematics and statistics, automati-cally tracks students results, lets instructors stay on top of student performance,and gives them control over how to calculate final grades. They can also addoffline (paper-and-pencil) grades to the gradebook.

Preface xvii

InterAct Math Tutorial Website www.interactmath.comN Provides practice and tutorial help onlineN Provides algorithmically generated practice exercises

that correlate directly to the exercises in the textbook

N Allows students to retry an exercise with new valueseach time for unlimited practice and mastery

Pearson Math Adjunct Support Center(http://www.pearsontutorservices.com/math-adjunct.html)

N MathXL Exercise Builder allows instructors to create static and algorithmicexercises for their online assignments. They can use the library of sample exer-cises as an easy starting point, or they can edit any course-related exercise.

N Pearson Tutor Center (www.pearsontutorservices.com) access is automati-cally included with MyMathLab. The Tutor Center is staffed by qualified mathinstructors who provide textbook-specific tutoring for students via toll-freephone, fax, email, and interactive Web sessions.

Students do their assignments in the Flash-based MathXL Player, which is compat-ible with almost any browser (Firefox, SafariTM, or Internet Explorer) on almost anyplatform (Macintosh or Windows). MyMathLab is powered by CourseCompassTM,Pearson Educations online teaching and learning environment, and by MathXL, ouronline homework, tutorial, and assessment system. MyMathLab is available to quali-fied adopters. For more information, visit our website at www.mymathlab.com orcontact your Pearson representative.

MathXL Online Course (access code required)MathXL is an online homework, tutorial, and assessment system that accompaniesPearsons textbooks in mathematics or statistics.

N Interactive homework exercises, correlated to your textbook at the objec-tive level, are algorithmically generated for unlimited practice and mastery. Mostexercises are free-response and provide guided solutions, sample problems, andlearning aids for extra help.

N Personalized homework assignments are designed by the instructor to meetthe needs of the class, and then personalized for each student based on their testor quiz results. As a result, each student receives a homework assignment thatcontains only the problems they still need to master.

N Personalized Study Plan, generated when students complete a test or quiz orhomework, indicates which topics have been mastered and links to tutorial exer-cises for topics students have not mastered. Instructors can customize the avail-able topics in the study plan to match their course concepts.

N Multimedia learning aids, such as video lectures and animations, help stu-dents independently improve their understanding and performance. These areassignable as homework, to further encourage their use.

N Gradebook, designed specifically for mathematics and statistics, automati-cally tracks students results, lets instructors stay on top of student performance,and gives them control over how to calculate final grades.

N MathXL Exercise Builder allows instructors to create static and algorithmicexercises for their online assignments. They can use the library of sample exer-cises as an easy starting point or the Exercise Builder to edit any of the course-related exercises.

N Homework and Test Manager lets instructors create online homework,quizzes, and tests that are automatically graded. They can select just the right mixof questions from the MathXL exercise bank, instructor-created custom exer-cises, and/or TestGen test items.

The new, Flash-based MathXL Player is compatible with almost any browser (Firefox, SafariTM, or Internet Explorer) on almost any platform (Macintosh orWindows). MathXL is available to qualified adopters. For more information, visitour website at www.mathxl.com, or contact your Pearson representative.

xviii Preface

STUDY SKILLS

Your textbook is a valuable resource. You will learn more if you fully make use of thefeatures it offers.

General Features

N Table of Contents Find this at the front of the text. Markthe chapters and sections you will cover, as noted on yourcourse syllabus.

N Answer Section Tab this section at the back of the bookso you can refer to it frequently when doing homework.Answers to odd-numbered section exercises are provided.Answers to ALL summary, chapter review, test, and cumu-lative review exercises are given.

N Glossary Find this feature after the answer section at theback of the text. It provides an alphabetical list of thekey terms found in the text, with definitions and sectionreferences.

N List of Formulas Inside the back cover of the text is ahelpful list of geometric formulas, along with reviewinformation on triangles and angles. Use these for refer-ence throughout the course.

Specific Features

N Objectives The objectives are listed at the beginning ofeach section and again within the section as the corre-sponding material is presented. Once you finish a sec-tion, ask yourself if you have accomplished them.

N Now Try Exercises These margin exercises allow you to immediately practice thematerial covered in the examples and prepare you for the exercises. Check yourresults using the answers at the bottom of the page.

N Pointers These small shaded balloons provide on-the-spot warnings and reminders,point out key steps, and give other helpful tips.

N Cautions These provide warnings about common errors that students often makeor trouble spots to avoid.

N Notes These provide additional explanations or emphasize important ideas.

N Problem-Solving Hints These green boxes give helpful tips or strategies to usewhen you work applications.

Find an example of each of these features in your textbook.

Using Your Math Textbook

OBJECTIVES

296 CHAPTER 5 Exponents and Polynomials

OBJECTIVE 1 Use exponents. Recall from Section 1.2 that in the expressionthe number 5 is the base and 2 is the exponent, or power. The expression iscalled an exponential expression. Although we do not usually write the exponentwhen it is 1, in general, for any quantity a,

Using ExponentsWrite in exponential form and evaluate.Since 3 occurs as a factor four times, the base is 3 and the exponent is 4. The exponential expression is , read 3 to the fourth power or simply 3 to the fourth.

3 # 3 # 3 # 3 = 34 = 8134

3 # 3 # 3 # 3EXAMPLE 1

a1 a.

5252,

The Product Rule and Power Rules for Exponents

5.1

1 Use exponents.2 Use the product

rule for exponents.3 Use the rule

4 Use the rule

5 Use the rule

6 Use combinations of rules.

7 Use the rules forexponents in ageometryapplication.

A ab Bm = ambm.

1ab2m = ambm.

1am2n = amn.

NOW TRY EXERCISE 1

Write in exponentialform and evaluate.

4 # 4 # 4

NOW TRY ANSWERS1.2. (a) 81; ; 4 (b) ; 3; 4-81-3

43 = 64

NOW TRY EXERCISE 2

Evaluate. Name the base andthe exponent.(a) (b) -341-324

CAUTION Note the differences between Example 2(b) and 2(c). In , theabsence of parentheses shows that the exponent 4 applies only to the base 5, not In , the parentheses show that the exponent 4 applies to the base In sum-mary, and are not necessarily the same.

1-a2n-an -5.1-524 -5.

-54

Expression Base Exponent

5 45 4

4-51-524

-5454

The base is 5.

(b)

(c)

NOW TRY

1-524 = 1-521-521-521-52 = 625

-54 = -1 # 54 = -1 # 15 # 5 # 5 # 52 = -625

OBJECTIVE 2 Use the product rule for exponents. To develop the productrule, we use the definition of exponents.

4 factors 3 factors

factors= 27

4 + 3 = 7

= 2 # 2 # 2 # 2 # 2 # 2 # 2

24 # 23 = 12 # 2 # 2 # 2212 # 2 # 22

4 factors of 3NOW TRY

Evaluating Exponential ExpressionsEvaluate. Name the base and the exponent.(a) 54 = 5 # 5 # 5 # 5 = 625

EXAMPLE 2

Expression Base Exponent Examplea n

n 1-322 = 1-321-32 = 9-a1-a2n

-32 = -13 # 32 = -9-an

xx Preface

The personal savings rate of Americans has fluctuated over time. It stood at a hefty

10.8% of after-tax income in 1984, but dropped to by 2005 when Americans

actually spent more than they earned. This was the first negative savings rate since

the Great Depression of the 1930s. In recent years, Americans have spent less and

saved more, and personal savings rates have returned to positive territory, reaching

6.9% in May 2009. (Source: U.S. Bureau of Economic Analysis.)

In this chapter, we examine signed numbers and apply them to situations such

as the personal savings rate of Americans in Exercise 115 of Section 1.5.

-0.5%

The Real Number System

1.1 Fractions

1.2 Exponents, Order ofOperations, andInequality

1.3 Variables,Expressions, and Equations

1.4 Real Numbers andthe Number Line

1.5 Adding andSubtracting RealNumbers

1.6 Multiplying andDividing RealNumbers

Summary Exercises onOperations with RealNumbers

1.7 Properties of RealNumbers

1.8 SimplifyingExpressions

1

CHAPTER 1

The fraction bar represents divisionA fraction is classified as being either a proper fraction or an improper fraction.

Proper fractions

Improper fractions32

, 55

, 117

, 284

15

, 27

, 910

, 2325

Aab a b B .

2 CHAPTER 1 The Real Number System

OBJECTIVES In everyday life, the numbers seen most often are the natural numbers,

the whole numbers,

and fractions, such as

and

The parts of a fraction are named as shown.

Fraction bar47

157

.23

,12

,

0, 1, 2, 3, 4, ,

1, 2, 3, 4, ,

Fractions1.1

1 Learn the definitionof factor.

2 Write fractions inlowest terms.

3 Multiply and dividefractions.

4 Add and subtractfractions.

5 Solve appliedproblems thatinvolve fractions.

6 Interpret data in acircle graph.

Numerator

Denominator

Numerator is less than denominator.Value is less than 1.

A mixed number is a single number that represents the sum of a natural number anda proper fraction.

Mixed number

OBJECTIVE 1 Learn the definition of factor. In the statement the numbers 3 and 6 are called factors of 18. Other factors of 18 include 1, 2, 9, and 18.The result of the multiplication, 18, is called the product. We can represent the prod-uct of two numbers, such as 3 and 6, in several ways.

Products

We factor a number by writing it as the product of two or more numbers. Fac-toring is the reverse of multiplying two numbers to get the product.

Multiplication Factoring

Factors Product Product Factors

NOTE In algebra, a raised dot is often used instead of the symbol to indicatemultiplication because may be confused with the letter x.

A natural number greater than 1 is prime if it has only itself and 1 as factors.Factors are understood here to mean natural number factors.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 First dozen prime numbers

**#

18 = 3 # 63 # 6 = 18

31621326,132162,3 # 6,3 * 6,

3 * 6 = 18,

5 34

= 5 +34

Numerator is greater than or equalto denominator. Value is greaterthan or equal to 1.

A natural number greater than 1 that is not prime is called a composite number.

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 First dozen composite numbers

By agreement, the number 1 is neither prime nor composite.Sometimes we must find all prime factors of a numberthose factors which are

prime numbers.

Factoring Numbers

Write each number as the product of prime factors.

(a) 35Write 35 as the product of the prime factors 5 and 7, or as

(b) 24We show a factor tree on the right. The prime factors are circled.

24Divide by the least prime factor of 24, which is 2.

Divide 12 by 2 to find twofactors of 12.

Now factor 6 as

All factors are prime. NOW TRY

2 # 324 = 2 # 2 # 2 # 32 # 3.

2 # 624 = 2 # 2 # 6

2 # 1224 = 2 # 12

35 = 5 # 7.

EXAMPLE 1

SECTION 1.1 Fractions 3

NOW TRYEXERCISE 1

Write 60 as the product ofprime factors.

Basic Principle of Fractions

If the numerator and denominator of a fraction are multiplied or divided by thesame nonzero number, the value of the fraction is not changed.

NOTE When factoring, we need not start with the least prime factor. No matterwhich prime factor we start with, we will always obtain the same prime factorization.Verify this in Example 1(b) by starting with 3 instead of 2.

OBJECTIVE 2 Write fractions in lowest terms. Recall the following basicprinciple of fractions, which is used to write a fraction in lowest terms.

A fraction is in lowest terms when the numerator and denominator have no fac-tors in common (other than 1).

NOW TRY ANSWER1. 2 # 2 # 3 # 5

Writing a Fraction in Lowest Terms

Step 1 Write the numerator and the denominator as the product of primefactors.

Step 2 Divide the numerator and the denominator by the greatest commonfactor, the product of all factors common to both.

Writing Fractions in Lowest Terms

Write each fraction in lowest terms.

(a)

The factored form shows that 5 is the greatest common factor of 10 and 15.Dividing both numerator and denominator by 5 gives in lowest terms as

(b)

By inspection, the greatest common factor of 15 and 45 is 15.

1545

=15

3 # 15 =1

3 # 1 =13

1545

23 .

1015

1015

=2 # 53 # 5 =

2 # 13 # 1 =

23

EXAMPLE 2


NOW TRY EXERCISE 2

Write in lowest terms.3042

Remember to write 1in the numerator.

If the greatest common factor is not obvious, factor the numerator and denominatorinto prime factors.

The same answer results.

NOW TRY

1545

=3 # 5

3 # 3 # 5 =1 # 1

3 # 1 # 1 =13

CAUTION When writing fractions like from Example 2(b) in lowest terms,be sure to include the factor 1 in the numerator.

1545

OBJECTIVE 3 Multiply and divide fractions.

Multiplying Fractions

If and are fractions, then

That is, to multiply two fractions, multiply their numerators and then multiplytheir denominators.

a

b# c

d

a # cb # d .

c

d

a

b

Multiplying Fractions

Find each product, and write it in lowest terms.

(a)Multiply numerators.Multiply denominators.

Factor the denominator.

Divide numerator and denominator by 3 4, or 12.

Lowest terms =16

=1

2 # 3

=3 # 4

2 # 4 # 3 # 3

38# 4

9=

3 # 48 # 9

EXAMPLE 3

Remember towrite 1 in thenumerator.

NOW TRY ANSWER2. 57


NOW TRYEXERCISE 3

Find each product, and writeit in lowest terms.

(a) (b) 3 25# 6 2

347# 5

8(b)

Write each mixed number as an improper fraction.

Multiply numerators.Multiply denominators.

Factor the numerator.

or Write in lowest termsand as a mixed number.12 14

=494

,

=7 # 3 # 7

3 # 4

=7 # 213 # 4

2 13# 5 1

4=

73# 21

4

Think: , and, so 5 14 =

214 .20 + 1 = 21

4 # 5 = 20

Think: means

124

gives9

18

12 14.449

49 , 4.494

NOW TRY

NOTE Some students prefer to factor and divide out any common factors beforemultiplying.

Example 3(a)

Divide out common factors. Multiply.

The same answer results. =16

=1

2 # 3

38# 4

9=

32 # 4

# 43 # 3

Two fractions are reciprocals of each other if their product is 1. See the table inthe margin. Because division is the opposite (or inverse) of multiplication, we use re-ciprocals to divide fractions.

A number and itsreciprocal have a productof 1. For example,

34# 4

3 =1212 = 1.

Number Reciprocal

5, or

9, or 1991

51

15

711

117

43

34

Dividing Fractions


That is, to divide by a fraction, multiply by its reciprocal.

a

b

c

d

a

b# d

c.

c

d

a

b

As an example of why this method works, we know that and also thatThe answer to a division problem is called a quotient. For example, the

quotient of 20 and 10 is 2.20 # 110 = 2.

20 , 10 = 2

Dividing Fractions

Find each quotient, and write it in lowest terms.

(a)

Multiply by the reciprocal of the second fraction.

34

,85

=34# 5

8=

3 # 54 # 8 =

1532

EXAMPLE 4

Make sure the answeris in lowest terms.

(b) or 1 15

34

,58

=34# 8

5=

3 # 84 # 5 =

3 # 4 # 24 # 5 =

65

,

Think: and

so 2 13 =73 .

6 + 1 = 7,3 # 2 = 6,

NOW TRY ANSWERS3. (a) (b) or 22 23

683 ,

514

(c)

Write 10 as 101 .

58

, 10 =58

,101

=58# 1

10=

5 # 18 # 10 =

5 # 18 # 5 # 2 =

116


NOW TRY ANSWERS4. (a) (b)

5. 12

78

928

Rememberto write1 in the

numerator.

(d)Write each mixed number as an improper fraction.

Multiply numerators.Multiply denominators. NOW TRY

=1027

=53# 2

9

1 23

, 4 12

=53

,92

OBJECTIVE 4 Add and subtract fractions. The result of adding two numbers iscalled the sum of the numbers. For example, so 5 is the sum of 2 and 3.2 + 3 = 5,

Adding Fractions


That is, to find the sum of two fractions having the same denominator, add thenumerators and keep the same denominator.

a

b

c

b

a c

b.

c

b

a

b

Adding Fractions with the Same Denominator

Find each sum, and write it in lowest terms.

(a)Add numerators.Keep the same denominator.

(b) Write in lowest terms. NOW TRY

If the fractions to be added do not have the same denominators, we must firstrewrite them with a common denominator. For example, to rewrite as an equivalentfraction with denominator 32, think,

We must find the number that can be multiplied by 4 to give 32. Since we multiply numerator and denominator by 8.

34

=3 # 84 # 8 =

2432

4 # 8 = 32,

34

=?

32.

34

210

+310

=2 + 3

10=

510

=12

37

+27

=3 + 2

7=

57

EXAMPLE 5NOW TRY EXERCISE 5

Find the sum, and write it inlowest terms.

18

+38

and are equivalent fractions.243234

Finding the Least Common Denominator

To add or subtract fractions with different denominators, find the least commondenominator (LCD) as follows.

Step 1 Factor each denominator.

Step 2 For the LCD, use every factor that appears in any factored form. If afactor is repeated, use the largest number of repeats in the LCD.

NOW TRYEXERCISE 4

Find each quotient, and writeit in lowest terms.

(a) (b) 3 34

, 4 27

27

,89

Multiply by the reciprocal of thesecond fraction.

Adding Fractions with Different Denominators

Find each sum, and write it in lowest terms.

(a)

To find the least common denominator, first factor both denominators.

and

Since 5 and 3 appear as factors, and 3 is a factor of 9 twice, the LCD is

15 9

, or 45.

Write each fraction with 45 as denominator.

and

Add the two equivalent fractions.415

+59

=1245

+2545

=3745

59

=5 # 59 # 5 =

2545

415

=4 # 315 # 3 =

1245

5 # 3 # 3

9 = 3 # 315 = 5 # 3

415

+59

EXAMPLE 6


NOW TRYEXERCISE 6

Find each sum, and write it inlowest terms.

(a) (b) 3 14

+ 5 58

512

+38

NOW TRY ANSWERS6. (a) (b) 718 , or 8

78

1924

At this stage, thefractions are notin lowest terms.

(b)

Method 1 Write each mixed numberas an improper fraction.

Add. Write as a mixed number. =254

, or 6 14

=144

+114

3 12

+ 2 34

=72

+114

3 12

+ 2 34

Think: 7 # 22 # 2 =

144

Method 2

or NOW TRY254

5 54

= 5 + 1 14

= 6 14

,

+ 2 34

= 2 34

3 12

= 3 24 Write as Then add vertically.

Add the whole numbers and the fractions separately.

3 24 .3 12

The difference between two numbers is found by subtracting the numbers. Forexample, so the difference between 9 and 5 is 4.9 - 5 = 4,

Subtracting Fractions


That is, to find the difference between two fractions having the same denomina-tor, subtract the numerators and keep the same denominator.

a

b

c

b

a c

b.

c

b

a

b

Find a common denominator.The LCD is 4.

Subtracting Fractions

Find each difference, and write it in lowest terms.

(a)Subtract numerators.Keep the same denominator.

Write in lowest termsand as a mixed number.

(b) and sothe LCD is

Write the equivalent fractions.

Subtract. The answer is inlowest terms.

(c)

Since and there are no common factors.The LCD is

Find a common denominator.

Write the equivalent fractions.

Subtract numerators.Keep the common denominator.

(d)

Method 1Write each mixed number as animproper fraction.

Subtract. Write as a mixed number. =114

, or 2 34

=184

-74

4 12

- 1 34

=92

-74

4 12

- 1 34

=3231440

=6751440

-3521440

1532

-1145

=15 # 4532 # 45 -

11 # 3245 # 32

32 # 45 = 1440.45 = 3 # 3 # 5,32 = 2 # 2 # 2 # 2 # 2

1532

-1145

=1190

=3590

-2490

2 # 3 # 3 # 5 = 90.15 = 3 # 5,18 = 2 # 3 # 3 7

18-

415

=7 # 5

2 # 3 # 3 # 5 -4 # 2 # 3

2 # 3 # 3 # 5

=32

, or 1 12

=128

158

-38

=15 - 3

8

EXAMPLE 7


Think: 9# 2

2 # 2 =184

NOW TRYEXERCISE 7

Find each difference, andwrite it in lowest terms.

(a) (b) 4 13

- 2 56

511

-29

NOW TRY ANSWERS7. (a) (b) 32 , or 1

12

2399

Method 2

or NOW TRY114

2 34

,

-1 34

= 1 34

= 1 34

4 24 = 3 + 1 +24 = 3 +

44 +

24 = 3

64 4

12

= 4 24

= 3 64

Find a common denominator.The LCD is 4.

Adding Fractions to Solve an Applied Problem

The diagram in FIGURE 1 appears in the book Woodworkers 39 Sure-Fire Projects.Find the height of the bookcase/desk to the top of the writing surface.

EXAMPLE 8


NOW TRYEXERCISE 8

A board is ft long. If itmust be divided into fourpieces of equal length forshelves, how long must eachpiece be?

10 12

NOW TRY ANSWER8. 2 58 ft

4

9

Cut 3 leg sectionsfrom ready-madeturned leg.

"

"

9 "

4 "

"3412

12

12

12

"34

"34

WritingSurface

FIGURE 1

26 174

4 24

=+ 4 12

34

34

9 24

=9 12

34

34

9 24

=9 12

4 24

=4 12

34

34

We must addthese measures.

( means inches.)

Since The height is in. NOW TRY

OBJECTIVE 6 Interpret data in a circle graph. In a circle graph, or pie chart,a circle is used to indicate the total of all the data categories represented. The circleis divided into sectors, or wedges, whose sizes show the relative magnitudes of thecategories. The sum of all the fractional parts must be 1 (for 1 whole circle).

Using a Circle Graph to Interpret Information

Recently there were about 970 million Internet users worldwide. The circle graph inFIGURE 2 shows the fractions of these users living in various regions of the world.

EXAMPLE 9

30 1426 174 = 26 + 4

14 = 30

14 .

174 = 4

14 ,

Because is animproper fraction,this is not the final

answer.

174

Think: means 17 , 4.174

NorthAmerica

231007

20

Europe

Asia

310

Other325

Source: www.internetworldstats.com

Worldwide Internet UsersBy Region

FIGURE 2

Use Method 2 fromExample 6(b).The commondenominator is 4.

OBJECTIVE 5 Solve applied problems that involve fractions.

(a) Which region had the largest share of Internet users? What was that share?The sector for Asia is the largest, so Asia had the largest share of Internet users,

(b) Estimate the number of Internet users in North America.A share of can be rounded to or and the total number of Internet users,

970 million, can be rounded to 1000 million (1 billion). We multiply by 1000. Thenumber of Internet users in North America would be about

(c) How many actual Internet users were there in North America?We multiply the actual fraction from the graph for North America, by the

number of users, 970 million.

23100

19702 = 23100

# 9701

=22,310

100= 223

110

23100 ,

14

110002 = 250 million.

14

14 ,

25100 ,

23100

720 .


This is reasonable,given our estimate

in part (b).

Thus, or 223,100,000 since million

people in North America used the Internet. NOW TRY

100,0002,= 110 # 1,000,000 =1101223 110 million,

Complete solution availableon the Video Resources on DVD

1.1 EXERCISES

Concept Check Decide whether each statement is true or false. If it is false, say why.

1. In the fraction 5 is the numerator and8 is the denominator.

2. The mixed number equivalent of is6 15 .

315

58 ,

5. The fraction is in lowest terms. 6. The reciprocal of is 31 .62

1339

3. The fraction is proper.77 4. The number 1 is prime.

7. The product of 10 and 2 is 12. 8. The difference between 10 and 2 is 5.

Identify each number as prime, composite, or neither. If the number is composite, write it asthe product of prime factors. See Example 1.

9. 19 10. 31 11. 30 12. 50

13. 64 14. 81 15. 1 16. 0

17. 57 18. 51 19. 79 20. 83 21. 124

22. 138 23. 500 24. 700 25. 3458 26. 1025

Write each fraction in lowest terms. See Example 2.

27. 28. 29. 30. 31.

32. 33. 34. 35. 36.

37. Concept Check Which choice shows the correct way to write in lowest terms?

A. B.

C. D.1624

=14 + 221 + 3

=23

1624

=8 # 28 # 3 =

23

1624

=4 # 44 # 6 =

46

1624

=8 + 88 + 16

=816

=12

1624

13277

144120

1664

1890

55200

64100

1620

1518

412

816

NOW TRY ANSWERS9. (a) other

(b) 333 million

(c) million, or339,500,000339 12

A 720 is about 13 . B

NOW TRYEXERCISE 9

Refer to the circle graph inFIGURE 2 on the preceding page.

(a) Which region had the leastnumber of Internet users?

(b) Estimate the number ofInternet users in Asia.

(c) How many actual Internetusers were there in Asia?

38. Concept Check Which fraction is not equal to

A. B. C. D.

Find each product or quotient, and write it in lowest terms. See Examples 3 and 4.

39. 40. 41. 42.

43. 44. 45. 46.

47. 48. 49. 50.

51. 52. 53. 54.

55. 56. 57. 58.

59. 60. 61. 62.

63. 64. 65. 66.

67. Concept Check For the fractions and which one of the following can serve as acommon denominator?

A. B. C. D.

68. Concept Check Write a fraction with denominator 24 that is equivalent to

Find each sum or difference, and write it in lowest terms. See Examples 57.

69. 70. 71. 72.

73. 74. 75. 76.

77. 78. 79. 80.

81. 82. 83. 84.

85. 86. 87. 88.

89. 90. 91. 92.

Use the table to answer Exercises 93 and 94.

5 13

- 4 12

6 14

- 5 13

3 45

- 1 49

4 34

- 1 25

1116

-1

12712

-19

56

-12

712

-13

1112

-3

121315

-315

811

-311

79

-29

5 34

+ 1 13

3 14

+ 1 45

4 23

+ 2 16

3 18

+ 2 14

56

+29

38

+56

415

+15

59

+13

316

+5

16712

+112

29

+59

715

+415

58 .

p + rp # rq + sq # s

rs ,

pq

2 310

, 1 45

2 58

, 1 1532

2 29

, 1 25

2 12

, 1 57

5 35

,710

6 34

,38

8 ,49

6 ,35

25

, 3034

, 12247

,621

325

,815

75

,310

54

,38

3 35# 7 1

62

38# 3 1

5

2 23# 1 3

53

14# 1 2

336 # 4

921 # 3

7

218

# 47

154

# 825

18# 10

7110

# 125

35# 20

2123# 15

1659# 2

745# 6

7

5599

4074

3054

1527

59?


Microwave Stove Top

Servings 1 1 4 6

Water cup 1 cup 3 cups 4 cups

Grits 3 Tbsp 3 Tbsp cup 1 cup

Salt (optional) Dash Dash tsp tsp1214

34

34

Source: Package of Quaker Quick Grits.

93. How many cups of water wouldbe needed for eight microwaveservings?

94. How many teaspoons of saltwould be needed for five stove-topservings? (Hint: 5 is halfway between 4 and 6.)

The Pride Golf Tee Company, the only U.S.manufacturer of wooden golf tees, has createdthe Professional Tee System, shown in thefigure. Use the information given to workExercises 95 and 96. (Source: The Gazette.)

95. Find the difference in length between theProLength Plus and the once-standardShortee.

96. The ProLength Max tee is the longesttee allowed by the U.S. Golf Associa-tions Rules of Golf. How much longeris the ProLength Max than the Shortee?

Solve each problem. See Example 8.

97. A hardware store sells a 40-piece socket wrench set. The measure of the largest socket isin. The measure of the smallest is in. What is the difference between these measures?

98. Two sockets in a socket wrench set have measures of in. and in. What is the differ-ence between these two measures?

99. A piece of property has an irregular shape, with fivesides, as shown in the figure. Find the total distancearound the piece of property. (This distance is called theperimeter of the figure.)

38

916

316

34


ProLength Max

ProLength Plus

ProLength

Shortee

4 in.

3 in.1 4

2 in.3 4

2 in.1 8

100

196 76

98

146

Measurements in feet

43

21

87

85

5 7

10

ft ft

ft81

41

21

100. Find the perimeter of the triangle in thefigure.

101. A board is in. long. If it must bedivided into three pieces of equallength, how long must each piece be?

in.15 85

15 58

102. Paul Beaulieus favorite recipe for barbecue sauce calls for cups of tomato sauce. Therecipe makes enough barbecue sauce to serve seven people. How much tomato sauce isneeded for one serving?

103. A cake recipe calls for of sugar. A caterer has cups of sugar on hand. Howmany cakes can he make?

104. Kyla Williams needs of fabric to cover a chair. How many chairs can she coverwith of fabric?

105. It takes of fabric to make a costume for a school play. How much fabric would beneeded for seven costumes?

106. A cookie recipe calls for cups of sugar. How much sugar would be needed to makefour batches of cookies?

107. First published in 1953, the digest-sized TV Guide has changed to a full-sized magazine. The full-sizedmagazine is 3 in. wider than the oldguide. What is the difference in theirheights? (Source: TV Guide.)

2 23

2 38 yd

23 23 yd2 14 yd

15 121 34 cups

2 13

8 in.New

5 in.Old

7 in.1 8

10 in.1 2

631 26560861 4

631 26560861 4

108. Under existing standards, most of the holes in Swisscheese must have diameters between and in. To ac-commodate new high-speed slicing machines, the U.S.Department of Agriculture wants to reduce the minimumsize to in. How much smaller is in. than in.? (Source:U.S. Department of Agriculture.)

Approximately 38 million people living in theUnited States in 2006 were born in othercountries. The circle graph gives the fractionalnumber from each region of birth for thesepeople. Use the graph to answer each question.See Example 9.

109. What fractional part of the foreign-born population was from other regions?

110. What fractional part of the foreign-born population was from Latin America or Asia?

111. How many people (in millions) were born inEurope?

112. At the conclusion of the Pearson Education softball league season, batting statistics forfive players were as follows:

1116

38

38

1316

1116


113. For each description, write a fraction in lowest terms that represents the region described.

(a) The dots in the rectangle as a part of the dots in theentire figure

(b) The dots in the triangle as a part of the dots in the entirefigure

(c) The dots in the overlapping region of the triangle and the rectangle as a part of thedots in the triangle alone

(d) The dots in the overlapping region of the triangle and the rectangle as a part of thedots in the rectangle alone

114. Concept Check Estimate the best approximation for the sum.

A. 6 B. 7 C. 5 D. 8

1426

+9899

+10051

+9031

+1327

Source: U.S. Census Bureau.

Other

Asia

Europe

LatinAmerica

27100

750

2750

U.S. Foreign-Born PopulationBy Region of Birth

Player At-Bats Hits Home Runs

Courtney Slade 36 12 3

Kari Heen 40 9 2

Adam Goldstein 11 5 1

Nathaniel Koven 16 8 0

Jonathan Wooding 20 10 2

Use the table to answer each question. Estimate as necessary.

(a) Which player got a hit in exactly of his or her at-bats?

(b) Which player got a hit in just less than of his or her at-bats?

(c) Which player got a home run in just less than of his or her at-bats?

(d) Which player got a hit in just less than of his or her at-bats?

(e) Which two players got hits in exactly the same fractional parts of their at-bats? Whatwas the fractional part, expressed in lowest terms?

14

110

12

13


STUDY SKILLS

Take time to read each section and its examples before doing your homework. Youwill learn more and be better prepared to work the exercises your instructor assigns.

Approaches to Reading Your Math Textbook

Student A learns best by listening to her teacher explainthings. She gets it when she sees the instructor work prob-lems. She previews the section before the lecture, so she knowsgenerally what to expect. Student A carefully reads the sec-tion in her text AFTER she hears the classroom lecture on thetopic.

Student B learns best by reading on his own. He reads the sec-tion and works through the examples before coming to class.That way, he knows what the teacher is going to talk aboutand what questions he wants to ask. Student B carefullyreads the section in his text BEFORE he hears the classroomlecture on the topic.

Which reading approach works best for youthat of Student A or Student B?

Tips for Reading Your Math Textbook

N Turn off your cell phone. You will be able to concen-trate more fully on what you are reading.

N Read slowly. Read only one sectionor even part of a sectionat a sitting, withpaper and pencil in hand.

N Pay special attention to important information given in colored boxes or set inboldface type.

N Study the examples carefully. Pay particular attention to the blue side commentsand pointers.

N Do the Now Try exercises in the margin on separate paper as you go. These mir-ror the examples and prepare you for the exercise set. The answers are given atthe bottom of the page.

N Make study cards as you read. (See page 48.) Make cards for new vocabulary,rules, procedures, formulas, and sample problems.

N Mark anything you dont understand. ASK QUESTIONS in classeveryone willbenefit. Follow up with your instructor, as needed.

Select several reading tips to try this week.

Reading Your Math Textbook

SECTION 1.2 Exponents, Order of Operations, and Inequality 15

OBJECTIVES

Exponents, Order of Operations, and Inequality1.2

1 Use exponents.

2 Use the rules fororder of operations.

3 Use more than onegrouping symbol.

4 Know the meaningsof , , , , and .

5 Translate wordstatements tosymbols.

6 Write statementsthat change thedirection ofinequality symbols.

76Z

OBJECTIVE 1 Use exponents. Consider the prime factored form of 81.

The factor 3 appears four times.

In algebra, repeated factors are written with an exponent, so the product is written as and read as 3 to the fourth power.

Exponent

4 factors of 3 Base

The number 4 is the exponent, or power, and 3 is the base in the exponential expres-sion A natural number exponent, then, tells how many times the base is used as afactor. A number raised to the first power is simply that number. For example,

and

Evaluating Exponential Expressions

Find the value of each exponential expression.

(a)

5 is used as a factor 2 times.

Read as 5 to the second power or, more commonly, 5 squared.

(b)

6 is used as a factor 3 times.

Read as 6 to the third power or, more commonly, 6 cubed.

(c) 2 is used as a factor 5 times.Read as 2 to the fifth power.

(d) is used as a factor 3 times.

(e) 0.3 is used as a factor 2 times. NOW TRY10.322 = 0.310.32 = 0.0923a23 b

3

=23# 2

3# 2

3=

827

2525 = 2 # 2 # 2 # 2 # 2 = 32

63

63 = 6 # 6 # 6 = 21652

52 = 5 # 5 = 25

EXAMPLE 1

a 12b1 = 1

2 .51 = 5

34.

3 # 3 # 3 # 3 = 34

343 # 3 # 3 # 3

81 = 3 # 3 # 3 # 3

NOW TRYEXERCISE 1Find the value of each exponential expression.

(a) (b) a45b362

NOW TRY ANSWERS1. (a) 36 (b) 64125

CAUTION Squaring, or raising a number to the second power, is NOT thesame as doubling the number. For example,

means not

Thus not 6. Similarly, cubing, or raising a number to the third power, doesnot mean tripling the number.

OBJECTIVE 2 Use the rules for order of operations. When a problem in-volves more than one operation, we often use grouping symbols, such as parentheses

, to indicate the order in which the operations should be performed.Consider the expression To show that the multiplication should be

performed before the addition, we use parentheses to group

equals or 11.5 + 6,5 + 12 # 322 # 3.

5 + 2 # 3.1 2

32 = 9,2 # 3.3 # 3,32

If addition is to be performed first, the parentheses should group

equals or 21.

Other grouping symbols are brackets braces and fraction bars. (Forexample, in the expression is grouped in the numerator.)

To work problems with more than one operation, we use the following order ofoperations. This order is used by most calculators and computers.

8 - 28 - 23 ,5 6,3 4,

7 # 3,15 + 22 # 35 + 2.


Order of Operations

If grouping symbols are present, simplify within them, innermost first (andabove and below fraction bars separately), in the following order.

Step 1 Apply all exponents.

Step 2 Do any multiplications or divisions in the order in which they occur,working from left to right.

Step 3 Do any additions or subtractions in the order in which they occur,working from left to right.

If no grouping symbols are present, start with Step 1.

NOTE In expressions such as or multiplication is understood.

Using the Rules for Order of Operations

Find the value of each expression.

(a)

Multiply.

Add. = 34 = 4 + 30

4 + 5 # 6

EXAMPLE 2

1-521-42,3172

Be careful!Multiply first.

(b)Work inside parentheses.

Multiply.

(c)Multiply, working from left to right.

Add. = 58 = 48 + 10

6 # 8 + 5 # 2 = 153 = 91172

916 + 112

(d)

Work inside parentheses.

Multiply.

Add. = 43 = 22 + 21 = 21112 + 7 # 3

215 + 62 + 7 # 3

(e)

Apply the exponent.

Multiply.

Subtract.

Add. = 6 = 1 + 5 = 9 - 8 + 5 = 9 - 2 # 2 # 2 + 5

9 - 23 + 5not 2 # 3.23 = 2 # 2 # 2,

(f)

Apply the exponents.

Divide.

Multiply.

Add.

Subtract. = 113 = 140 - 27 = 108 + 32 - 27

= 36 # 3 + 4 # 8 - 27 = 72 , 2 # 3 + 4 # 8 - 27

72 , 2 # 3 + 4 # 23 - 33

SECTION 1.2 Exponents, Order of Operations, and Inequality 17

NOW TRYEXERCISE 2

Find the value of eachexpression.

(a)

(b)

(c) 8 # 10 , 4 - 23 + 3 # 42612 + 42 - 7 # 515 - 2 # 6

Think: 33 = 3 # 3 # 3

Multiplications and divisions are done from left to right as they appear. Thenadditions and subtractions are done from left to right as they appear. NOW TRY

OBJECTIVE 3 Use more than one grouping symbol. In an expression such aswe often use brackets, in place of one pair of parentheses.

Using Brackets and Fraction Bars as Grouping Symbols

Simplify each expression.

(a)

Add inside parentheses.

Multiply inside brackets.

Add inside brackets.

Multiply. = 82 = 23414 = 238 + 334 = 238 + 311124

238 + 316 + 524

EXAMPLE 3

3 4,218 + 316 + 522,NOW TRYEXERCISE 3

Simplify each expression.

(a)

(b)9114 - 42 - 2

4 + 3 # 6

73132 - 12 + 44

NOW TRY ANSWERS2. (a) 3 (b) 1 (c) 603. (a) 84 (b) 4

(b) Simplify the numerator and denominator separately.

Work inside parentheses.

Multiply.

or 7 Add and subtract. Then divide. NOW TRY =355

,

=32 + 36 - 1

=4182 + 32132 - 1

415 + 32 + 32132 - 1

NOTE The expression in Example 3(b) can be written as the quotient

which shows that the fraction bar groups the numerator and denominator separately.

3415 + 32 + 34 , 32132 - 14,415 + 32 + 3

2132 - 1

OBJECTIVE 4 Know the meanings of , , , and . So far, we haveused the equality symbol The symbols and are used to express aninequality, a statement that two expressions may not be equal. The equality symbolwith a slash through it, means is not equal to.

7 is not equal to 8.

If two numbers are not equal, then one of the numbers must be less than the other.The symbol represents is less than.

7 is less than 8.7 6 86

7 Z 8Z ,

,7 ,6 ,Z ,= .

The symbol means is greater than.

8 is greater than 2.

To keep the meanings of the symbols and clear, remember that the symbolalways points to the lesser number.

Lesser number

Lesser number

The symbol means is less than or equal to.

5 is less than or equal to 9.

If either the part or the part is true, then the inequality is true. The state-ment is true, since is true.

The symbol means is greater than or equal to.

9 is greater than or equal to 5.

Using Inequality Symbols

Determine whether each statement is true or false.

(a) This statement is false because

(b) The statement is true, since .

(c) The statement is true, since

(d) Both and are false, so is false.

(e) Since this statement is true.

(f) Since this statement is false.

(g)

Get a common denominator.

Both statements and are false. Therefore, is false.

NOW TRY

OBJECTIVE 5 Translate word statements to symbols.

Translating from Words to Symbols

Write each word statement in symbols.

EXAMPLE 5

615

23

615 =

1015

615 7

1015

615

1015

615

23

9 = 9,9 6 912 = 12,12 12

25 3025 = 3025 7 3025 3015 6 40.15 20 # 215 20 # 28 6 195 + 3 6 195 + 3 6 19

6 = 5 + 1.6 Z 5 + 1

EXAMPLE 4

9 5

5 6 95 96 6 10

C,Ax By CAx By

OBJECTIVE 1 Graph linear inequalities in two variables. Consider thegraph in FIGURE 35. The graph of the line divides the points in the rectan-gular coordinate system into three sets:

1. Those points that lie on the line itself and satisfy the equation likeand

2. Those that lie in the region above the line and satisfy the inequality like and

3. Those that lie in the region below the line and satisfy the inequality like and , .

The graph of the line is called the boundary line for the inequalities

and

Graphs of linear inequalities in two variables are regions in the real number planethat may or may not include boundary lines.

Graphing a Linear Inequality

Graph The inequality means that

We begin by graphing the equation a line with intercepts andas shown in FIGURE 36. This boundary line divides the plane into two regions,

one of which satisfies the inequality. A test point gives a quick way to find the cor-rect region. We choose any point not on the boundary line and substitute it into thegiven inequality to see whether the resulting statement is true or false. The point

is a convenient choice.

Original inequality

Let and

True

Since the last statement is true, we shade the region that includes the test pointSee FIGURE 36. The shaded region, along with the boundary line, is the

desired graph.10, 02.

0 60 + 0 ? 6

y = 0.x = 02102 + 3102 ? 62x + 3y 6

10, 02

13, 02, 10, 222x + 3y = 6,2x + 3y 6 6 or 2x + 3y = 6.

2x + 3y 62x + 3y 6.

EXAMPLE 1

x + y 6 5.x + y 7 5x + y = 5

-1241-310, 023 x + y 6 512, 42415, 323 x + y 7 515, 02410, 52, 12, 32, 3x y 5

x + y = 5

224 CHAPTER 3 Linear Equations and Inequalities in Two Variables; Functions

NOW TRYEXERCISE 1

Graph x + 3y 6.

x

y

x + y = 5 (2, 4)

(0, 0)(3, 1)

(5, 3)

x + y > 5

x + y < 5

FIGURE 35

Use asa test point.10, 02

(0, 2)

(3, 0)

yy

x

2x 3y 6

Boundaryline

Testpoint

2x 3y 6(0, 0)

FIGURE 36 NOW TRY

NOW TRY ANSWER1.

x

y

0x + 3y 6

2

6

NOTE Alternatively in Example 1, we can find the required region by solving thegiven inequality for y.

Inequality from Example 1

Subtract 2x.

Divide by 3.

Ordered pairs in which y is equal to are on the boundary line, so pairs in

which y is less than will be below that line. As we move down vertically, they-values decrease.) This gives the same region that we shaded in FIGURE 36. (Ordered

pairs in which y is greater than will be above the boundary line.)


Graph This inequality does not include the equals symbol. Therefore, the points on the

line do not belong to the graph. However, the line still serves as a bound-ary for two regions, one of which satisfies the inequality.

To graph the inequality, first graph the equation Use a dashed line toshow that the points on the line are not solutions of the inequality SeeFIGURE 37.

Now choose a test point to see which side of the line satisfies the inequality.

Original inequality

Let and

False 0 7 5y = 0.x = 0 0 - 0 7? 5

x - y 7 5

x - y 7 5.x - y = 5.

x - y = 5

x - y 7 5.

EXAMPLE 2

- 23 x + 2

- 23 x + 2- 23 x + 2

y - 23

x + 2

3y -2x + 6 2x + 3y 6

SECTION 3.5 Graphing Linear Inequalities in Two Variables 225

is aconvenienttest point.

10, 02

Since is false, the graph of the inequality is the region that does not containShade the other region, as shown in FIGURE 37, to obtain the required graph.10, 02.0 7 5

x

y

(0, 0)

(4, 3)

05

5

x y = 5

x y > 5

FIGURE 37

NOW TRYEXERCISE 2

Graph .2x - 4y 7 8

NOW TRY ANSWER2.

x

y

02

4

2x 4y > 8

To check that the correct region is shaded, we test a point in the shaded region.For example, use from the shaded region as follows.

CHECK Original inequality

Let and

TrueThis true statement verifies that the correct region was shaded in FIGURE 37.

NOW TRY

7 7 5y = -3.x = 4 4 - 1-32 7? 5

x - y 7 514, -32

Use parenthesesto avoid errors.

Graphing a Linear Inequality with a Vertical Boundary Line

Graph First, we graph a vertical line passing through the point We use a

dashed line (why?) and choose as a test point.

Original inequality

Let

True

Since is true, we shade the region containing as in FIGURE 38.

NOW TRY

OBJECTIVE 2 Graph an inequality with a boundary line through the origin. If the graph of an inequality has a boundary line that goes through the ori-gin, cannot be used as a test point.

Graphing a Linear Inequality with a Boundary Linethrough the Origin

Graph Graph using a solid line. Some ordered pairs that can be used to graph

this line are and Since is on the line it cannot beused as a test point. Instead, we choose a test point off the line, say

Original inequality

Let and

True 1 6y = 3.x = 1 1 ? 2132

x 2y

11, 32.x = 2y,10, 0214, 22.16, 32,10, 02,x = 2y,

x 2y.

EXAMPLE 4

10, 02

10, 02,0 6 3 0 6 3

x = 0. 0 6? 3 x 6 3

10, 02 13, 02.x = 3,x 6 3.

EXAMPLE 3


NOW TRYEXERCISE 3

Graph .x 7 2

x

y

03

x = 3x < 3

(0, 0)

FIGURE 38

NOW TRY ANSWER3.

x

y

0 2

x > 2

Be careful to drawa dashed line.


Step 1 Graph the boundary. Graph the line that is the boundary of the region. Use the methods of Section 3.2. Draw a solid line if the in-equality has or because of the equality portion of the symbol.Draw a dashed line if the inequality has or

Step 2 Shade the appropriate region. Use any point not on the line as atest point. Substitute for x and y in the inequality. If a true statementresults, shade the region containing the test point. If a false state-ment results, shade the other region.

7 .6

Since is true, shade the region containing the test point See FIGURE 39.11, 32.1 6SECTION 3.5 Graphing Linear Inequalities in Two Variables 227

x

y

(1, 3)

0

3

6

x = 2yx 2y

FIGURE 39

cannot be usedas a test point.

10, 02

NOW TRY

Graphing calculators have a feature that allows us to shade regions in the plane,so they can be used to graph a linear inequality in two variables. The calculator willnot draw the graph as a dashed line, so it is still necessary to understand what is andwhat is not included in the solution set.

To solve the inequalities in one variable, and weuse the graph of

in FIGURE 40. For we want the values of x such that sothat the line is above the x-axis. From FIGURE 40, we see that this is the case for

Thus, the solution set is Similarly, the solution set ofis because the line is below the x-axis when

For Discussion or Writing

Use a graphing calculator to solve the following inequalities in one variable fromSection 2.8.

1. (Example 4, page 156)

2. (Use the result from part (a).)

3. (Example 8(a), page 158)4 6 3x - 5 103x + 2 - 5 6 -x + 7 + 2x3x + 2 - 5 7 -x + 7 + 2x

x 7 2.12, q2,y = -2x + 4 6 0 1-q , 22.x 6 2.

y 7 0,y = -2x + 4 7 0,y = -2x + 4

-2x + 4 6 0,-2x + 4 7 0

CONNECTIONS

10

10

10 10

FIGURE 40

NOW TRY ANSWER4.

x

y

0y 2x

Complete solution availableon the Video Resources on DVD

3.5 EXERCISES

Concept Check The following statements each include one or more phrases that can besymbolized with one of the inequality symbols In Exercises 16, give the inequality symbol for the boldface italic words.

1. Since it was recognized in 1981, HIV/AIDS has killed more than 25 million people world-wide and infected more than 60 million, about two-thirds of whom live in Africa. (Source:The Presidents Emergency Plan for AIDS Relief, February, 2008.)

2. The average national automobile insurance premium of $1896 in 2007 was $20 less thanthe 2006 average premium. (Source: 2007 Mid-Year Auto Insurance Pricing Report.)

6 , , 7 , or .

NOW TRYEXERCISE 4

Graph .y -2x

3. As of December 2007, airline passengers were allowed one carry-on bag, with dimen-sions totaling at most 45 in. (Source: The Gazette.)

4. As of February 2008, all major airlines except US Airways award at least 500 frequentflier miles per flight. (Source: USA Today.)

5. By 1937, a population of as many as a million Attwaters prairie chickens had been cut toless than 9000. (Source: National Geographic, March 2002.)

6. [Easter Islands] nearly 1000 statues, some almost 30 feet tall and weighing as much as80 tons, are still an enigma. (Source: Smithsonian, March 2002.)

Concept Check Answer true or false to each of the following.

7. The point lies on the graph of

8. The point lies on the graph of

9. The points and lie on the graph of

10. The graph of does not contain points in quadrant IV.

In Exercises 1116, the straight-line boundary has been drawn. Complete the graph by shadingthe correct region. See Examples 14.

11. 12. 13.

14. 15. 16.

17. Explain how to determine whether to use a dashed line or a solid line when graphing alinear inequality in two variables.

18. Explain why the point is not an appropriate choice for a test point when graphingan inequality whose boundary goes through the origin.

Graph each linear inequality. See Examples 14.

19. 20. 21.

22. 23. 24.

25. 26. 27.

28. 29. 30.

31. Explain why the graph of cannot lie in quadrant IV.

32. Explain why the graph of cannot lie in quadrant II.y 6 xy 7 x

y 2xy 4xy -3y 5x 7 1x 6 -2y 6 -3x + 1y 2x + 13x + 4y 6 122x + 3y 7 -6x + y 3x + y 5

10, 02

x 7 4y 6 -1x 3y

-3x + 4y 7 122x + y 5x + 2y 7

y 7 x3x - 2y 0.10, 0214, 12

3x - 4y 12.14, 023x - 4y 6 12.14, 02


x

y

01

1x

y

01 1

x

y

01

1

x

y

01

1x

y

01

1x

y

01

1

For the given information, (a) graph the inequality (here, and , so graph only thepart of the inequality in quadrant I) and (b) give two ordered pairs that satisfy the inequality.

33. A company will ship x units of merchandise to outlet I and y units of merchandise tooutlet II. The company must ship a total of at least 500 units to these two outlets. Thepreceding information can be expressed by writing

34. A toy manufacturer makes stuffed bears and geese. It takes 20 min to sew a bear and 30 min to sew a goose. There is a total of 480 min of sewing time available to make x bears and y geese. These restrictions lead to the inequality

20x + 30y 480.

x + y 500.

y 0x 0

SECTION 3.6 Introduction to Functions 229

Identifying Domains and Ranges of RelationsIdentify the domain and range of each relation.

(a)This relation has domain and

range The correspondence betweenthe elements of the domain and the elements ofthe range is shown in FIGURE 41.

(b)

This relation has domain and range NOW TRY{5, 6, 7, 8}.53613, 82613, 72,13, 62,513, 52,

51, 2, 5, 86. 50, 2, 3, 46510, 12, 12, 52, 13, 82, 14, 226

EXAMPLE 1

NOW TRY ANSWER1. domain: , 0, 2 ;

range: 3, 7, 8, 10656-25

Find the value of for each given value of x. See Section 1.3.

35. 36. 37.

38. 39. 40. -41- 53

4-103x2 + 8x + 5

PREVIEW EXERCISES

OBJECTIVES

Introduction to Functions3.6

1 Understand thedefinition of arelation.

2 Understand thedefinition of afunction.

3 Decide whether anequation defines afunction.

4 Find domains andranges.

5 Use functionnotation.

6 Apply the functionconcept in anapplication.

If gasoline costs $3.00 per gal and we buy 1 gal, then we must pay If we buy 2 gal, then the cost is If we buy 3 gal, then the cost is

and so on. Generalizing, if x represents the number of gallons,then the cost is $3.00x. If we let y represent the cost, then the equation

relates the number of gallons, x, to the cost in dollars, y. The ordered pairs thatsatisfy this equation form a relation.

OBJECTIVE 1 Understand the definition of a relation. In an ordered pairx and y are called the components of the ordered pair. Any set of ordered pairs

is called a relation. The set of all first components of the ordered pairs of a relationis

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