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Charles P. McKeagueFor your course and learning solutions, visit academic.cengage.com

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Algebrawith Trigonometry for College StudentsFifth Edition

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Algebrawith Trigonometry for College StudentsFifth Edition

Charles P. McKeagueCuesta College

A u s t r a l i a • B r a z i l • J a p a n • K o r e a • M e x i c o • S i n g a p o r e • S p a i n • U n i t e d K i n g d o m • U n i t e d S t a t e s

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Algebra with Trigonometry for CollegeStudents, Fifth EditionCharles P. McKeague

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Equations and Inequalities in Two Variables 176

2.1 Paired Data and the Rectangular CoordinateSystem 178

2.2 The Slope of a Line 192

2.3 The Equation of a Line 202

2.4 Linear Inequalities in Two Variables 215

2.5 Introduction to Functions 223

2.6 Function Notation 236

2.7 Algebra With Functions 247

2.8 Variation 254

Summary 266

Review 269

Test 270

Cumulative Review 271

Projects 272

Systems of Linear Equationsand Inequalities 274

3.1 Systems of Linear Equations in Two Variables 276

3.2 Systems of Linear Equations in Three Variables 289

3.3 Introduction to Determinants 298

3.4 Cramer's Rule 306

3.5 Applications 312

3.6 Systems of Linear Inequalities 327

Summary 336

Review 339

Test 340


Projects 343

Contents

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Preface to the Instructor vi

Preface to the Student xiii

Basic Properties andDefinitions 1

R.1 Fundamental Definitions and Notation 3

R.2 Properties of Real Numbers 16

R.3 Arithmetic With Real Numbers 27

R.4 Exponents and Scientific Notation 38

R.5 Polynomials, Sums, Differences, and Products 49

R.6 Factoring 61

R.7 Special Factoring 70

R.8 Recognizing Patterns 79

Summary 90

Review 94

Test 96

Projects 97

Equations and Inequalities in One Variable 99

1.1 Linear and Quadratic Equations in One Variable 101

1.2 Formulas 112


1.4 Linear Inequalities in One Variable 143

1.5 Equations With Absolute Value 154

1.6 Inequalities Involving Absolute Value 159

Summary 167

Review 169

Test 171


Projects 173

RR

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Rational Expressions andRational Functions 345

4.1 Basic Properties and Reducing to Lower Terms 347

4.2 Division of Polynomials 360

4.3 Multiplication and Division of RationalExpressions 371

4.4 Addition and Subtraction of RationalExpressions 380

4.5 Complex Fractions 389

4.6 Equations Involving Rational Expressions 394


Summary 417

Review 419

Test 420


Projects 422

Rational Exponents and Roots 424

5.1 Rational Exponents 426

5.2 More Expressions Involving RationalExponents 438

5.3 Simplified Form for Radicals 445

5.4 Addition and Subtraction of RadicalExpressions 457

5.5 Multiplication and Division of RadicalExpressions 461

5.6 Equations With Radicals 468

5.7 Complex Numbers 478

Summary 486

Review 488

Test 489


Projects 491

Quadratic Functions 493

6.1 Completing the Square 494

6.2 The Quadratic Formula 505

6.3 Additional Items Involving Solutions toEquations 516

6.4 Equations Quadratic in Form 526

6.5 Graphing Parabolas 535

6.6 Quadratic Inequalities 549

Summary 560

Review 561

Test 562


Projects 564

Exponential and LogarithmicFunctions 566

7.1 Exponential Functions 567

7.2 The Inverse of a Function 578

7.3 Logarithms Are Exponents 588

7.4 Properties of Logarithms 597

7.5 Common Logarithms and Natural Logarithms 602

7.6 Exponential Equations and Change of Base 611

Summary 620

Review 621

Test 623


Projects 624

Sequences and Series 627

8.1 Sequences 628

8.2 Series 637

8.3 Arithmetic Sequences 644

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8.4 Geometric Sequences 651

8.5 The Binomial Expansion 661

Summary 669

Review 671

Test 672


Projects 674

Conic Sections 676

9.1 The Circle 677

9.2 Ellipses and Hyperbolas 686

9.3 Second-Degree Inequalities and NonlinearSystems 700

Summary 708

Review 709

Test 710


Projects 711

Introduction to Trigonometry 713

10.1 Degrees, Radians, and Special Triangles 714

10.2 Trigonometric Functions 724

10.3 Trigonometric Functions and Calculators 731

10.4 Graphing I: Amplitude and Period 743

10.5 Graphing II: Phase Shift 758

10.6 Finding an Equation from Its Graph 768

10.7 Inverse Trigonometric Functions 779

Summary 785

Review 788

Test 790


Projects 793

Trigonometric Identities and Equations 795

11.1 Introduction to Identities 796

11.2 Proving Identities 804

11.3 Sum and Difference Formulas 809

11.4 Double-Angle and Half-Angle Formulas 817

11.5 Solving Trigonometric Equations 823

Summary 830

Review 832

Test 833


Projects 835

Triangles 837

12.1 Definition II: Right Triangle Trigonometry 838

12.2 The Law of Sines 851

12.3 The Ambiguous Case 857

12.4 The Law of Cosines 862

Summary 868

Review 870

Test 871


Projects 873

Appendix A: Synthetic Division A–1

Appendix B: Matrix Solutions to Linear Systems B–1

Appendix C: Conditional Statements C–1

Appendix D: Answers to Odd-Numbered Problems,Chapter Reviews, Chapter Tests, andCumulative Reviews D–1

Index I–1

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T he first ten chapters of this book cover the topics usually found in a college-level intermediate algebra course. The last three chapters cover the essential

topics from trigonometry.

Same Approach, Slightly Higher LevelThough the basic content of the book has not changed, there are certain elementsnew to this edition that make it a bit more challenging.

• New, More Challenging Drill and Word ProblemsIn an effort to bolster the problem sets, over 1,500 new drill and word problemswere added to this edition. Though there is still a variety of drill problems in thebook, more than 20% of them are new and require a bit more effort from the stu-dent than the other drill problems. The new problems require the student either toemploy more steps or use other prerequisite knowledge to solve the problem.

Real-data application problems also were added to many of the sections of thebook. The information in these applications was derived from various sources, in-cluding the U.S. Census Bureau, Rand McNally, the Federal Bureau of Investiga-tion, and the Environmental Business Journal. These problems, some of which aremulti-step and multi-part, are meant to show the relevance of mathematics in thereal world.

• Increased Coverage of Functions and Function NotationThe concept of a function is used more throughout this edition than the previousedition. It is introduced in Chapter 2 and then used throughout the remainder of thebook, particularly in the chapters on Rational Functions, Quadratic Functions, andExponential and Logarithmic Functions. As a corollary to this, a concerted effortwas made to use function notation whenever possible and appropriate in this newedition. Also, a new section of the algebra and composition of functions has beenadded in Chapter 2.

• Streamlined Review of Elementary AlgebraIf you have a difficult time covering all the topics in intermediate algebra, you willlike the sequence of topics in this book: the review of elementary algebra has beencondensed considerably.

Chapter R reviews real numbers, exponents, and polynomials. It can be coveredin detail or simply left for reference, depending on the background of your students.

With polynomials and factoring covered in Chapter R, the sequence and varietyof topics in Chapter 1 become more interesting. For example, Section 1.1 coverslinear equations in one variable, as well as quadratic equations solvable by factor-ing. Covering these topics together is a pleasant deviation from the usual sequenceof intermediate algebra topics. It also allows for a wider variety of problems in thesections on applications and formulas that make up the next part of the chapter.

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You will find other interesting changes in both the scope and order of topics inthe rest of the book that result from the condensed review of elementary algebra. Ingeneral, spending less time on elementary algebra topics and more time on newtopics enhances both the level and content of the course.

Additional Features of the BookUsing Technology Scattered throughout the book is material that shows howgraphing calculators, spreadsheet programs, and computer graphing programs canbe used to enhance the topics being covered. This material is easy to find because itappears under the heading Using Technology.

Blueprint for Problem Solving The Blueprint for Problem Solving is a detailedoutline of the steps needed to successfully attempt application problems. Intended asa guide to problem solving in general, the Blueprint overlays the solution process toall the application problems in the first few chapters of the book. As students becomemore familiar with problem solving, the steps in the Blueprint are streamlined.

Increased Visualization of Topics This edition contains many more diagrams,charts, and graphs than the previous edition. The purpose of this is to give studentsadditional information, in visual form, to help them understand the topics we cover.

Facts From Geometry Many of the important facts from geometry are listedunder this heading. In most cases, an example or two accompanies each of the factsto give students a chance to see how topics from geometry are related to the algebrathey are learning.

New Number Sequences and Inductive Reasoning An introductory cover-age of number sequences and inductive reasoning is contained in Section R.8: Rec-ognizing Patterns. This material is carried into Chapter 1, and other parts of thebook. It is expanded on in Chapter 8. I find that there are many interesting topics Ican cover if students have some experience with number sequences. It is also theeasiest way to demonstrate inductive reasoning.

Unit Analysis Chapter 4 contains problems requiring students to convert fromone unit of measure to another. The method used to accomplish the conversions isthe method they will use if they take a chemistry class. Since this method is similarto the method we use to multiply rational expressions, unit analysis is covered inSection 4.7 as an application of multiplication and division of rational expressions.

New Getting Ready for Class Just before each problem set is a list of fourproblems under the heading Getting Ready for Class. These problems require writ-ten responses from students, and can be answered by reading the preceding section.They are to be done before students come to class.

New Chapter Openings Each chapter opens with an introduction in which areal-world application is used to stimulate interest in the chapter. These opening ap-

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plications are expanded on later in the chapter. Most of them are explained usingthe rule of four: in words, numerically, graphically, and algebraically.

Study Skills Found in the first six chapters, after the opening application, are alist of study skills intended to help students become organized and efficient withtheir time. The study skills point students in the direction of success.

New Conditional Statements An introductory look at the type of reasoningthat is the foundation of mathematics is contained in Appendix C. This appendixgives students experience with conditional statements and their associated forms:inverse, converse, and contrapositive.

Organization of the Problem Sets Six main ideas are incorporated into theproblem sets.

1. Drill There are enough problems in each set to ensure student proficiency inthe material.

2. Progressive difficulty The problems increase in difficulty as the problem setprogresses.

3. Odd-even similarities Each pair of consecutive problems is similar. Sincethe answers to the odd-numbered problems are listed in the back of the book,the similarity of the odd-even pairs allows your students to check their workon an odd-numbered problem and then to try the similar even-numberedproblem.

4. Applying the Concepts problems Students are always curious about how the al-gebra they are learning can be applied, but at the same time many of them areapprehensive about attempting application problems. I have found that they aremore likely to put some time and effort into trying application problems if theydo not have to work an overwhelming number of them at one time and if theywork on them every day. For these reasons, I have placed a few applicationproblems in almost every problem set in the book. Many of these applicationproblems are new to this edition, and I have made a considerable effort to usereal data whenever possible.

5. Review problems Each problem set, beginning with Chapter 1, contains a fewreview problems. Where appropriate, the review problems cover material thatwill be needed in the next section. Otherwise they cover material from the previ-ous chapter. That is, the review problems in Chapter 5 cover the important pointscovered in Chapter 4. Likewise, the review problems in Chapter 6 review the im-portant material from Chapter 5. If you give tests on two chapters at a time, youwill find this to be a timesaving feature. Your students will review one chapter asthey study the next chapter.

6. Extending the Concepts problems Many of the problem sets end with a fewproblems under this heading. These problems are more challenging than those inthe problem set are, or they are problems that extend some of the topics coveredin the section.

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End of Chapter Retrospective Each chapter ends with the following itemsthat together give a comprehensive reexamination of the chapter and some of theimportant problems from previous chapters.

Chapter Summary These list all main points from the chapter. In the mar-gin, next to each topic being reviewed, is an example that illustrates the type ofproblem associated with the topics being reviewed.

Chapter Review An extensive set of problems that review all the main top-ics in the chapter. The chapter reviews contain more problems than the chaptertests.

Chapter Test A set of problems representative of all the main points of thechapter.

New Cumulative Review New to this edition, each chapter after Chapter Rends with a set of problems that review material from all preceding chapters.The cumulative reviews keep students current with past topics and help themretain the information they study.

New Chapter Projects New to this edition are student projects that extendthe material found in the chapter. Each chapter ends with one group project forstudents to work on in class, and one research project for students to do outsideof class.

Changes to this EditionThis edition is more challenging than the previous edition. I have added more chal-lenging drill problems, and more application problems. In all, over 40% of theproblems are new to this edition. This edition also contains three new sections, newchapter projects, and new cumulative reviews. I have renumbered the chapters des-ignating the first chapter as Chapter R to support those instructors who leave thischapter for students to do on their own. Here are the details:

Chapter R This chapter was previously numbered Chapter 1. A new SectionR.8, Recognizing Patterns, has been added to this edition.

Chapter 1 New material on graphing with interval notation has been addedto the inequalities in this chapter.

Chapter 2 Section 2.7, Algebra with Functions, is new to this edition.

Chapter 3 Systems of equations has been moved up to Chapter 3. Section3.6, Systems of Linear Inequalities, is new to this edition. The section on ma-trix solutions to systems of equations has been moved to an appendix.

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Chapter 4 More examples and problems on difference quotients have beenadded to this chapter.

Chapter 7 Section 7.1 has been split into two sections: Section 7.1, Expo-nential Functions, and Section 7.2, The Inverse of a Function.

Supplements to the Textbook This fifth edition of Algebra with Trigonometry for College Students is accompa-nied by a number of useful supplements.

For the InstructorInstructor’s Manual The Instructor’s Manual contains complete annotated solu-tions to every problem in the problem sets, Chapter Reviews, Chapter Tests, andCumulative Reviews.

Printed Test Bank and Prepared Tests The test bank consists of multiple-choice and short-answer test items organized by chapter and section. It includes 15sample problems per section of the text for Chapters R through 9, labeled by chap-ter, section, and level of difficulty. The prepared tests comprise ten sets of ready-to-copy tests: one set for each chapter for Chapters R through 9. Each set comprisestwo multiple-choice and four show-your-work tests. Answers for every test item areprovided. The portion of test questions for Chapters 10 through 12 is organized bychapter and section. It contains short answer, true/false, and multiple-choice ques-tions, each of which are labeled for difficulty (1 = easy, 2 = medium, 3 = hard).Though there are not prepared tests for these chapters, there is an abundance ofquestions to choose from to create your own tests.

Computerized Test Bank A flexible, powerful computerized testing system, theComputerized Test Bank contains all the test bank questions and allows instructorsto prepare quizzes and examinations quickly and easily. It offers instructors theability to select, edit, and create not only test items but algorithms for test items aswell. Approximately 100 questions are taken from the printed test bank and madeinto algorithms in the computerized test bank from which over 300,000 differentquestions can be generated. Instructors can tailor tests according to a variety of cri-teria, scramble the order of test items, and administer tests on-line over a network.The Computerized Test Bank also includes full-function gradebook and graphingfeatures. Available in Macintosh or Windows format.

Excursions in Real-World Mathematics, by Gina Shankland and SaraWilliams of Mt. Hood Community College This exciting Web-based activityprogram will activate the minds of students as they solve problems that they mayencounter in their everyday lives. Real data obtained from actual business, industry,and other academic disciplines serve as the raw material used in solving problems.There are more than 80 application situations to increase students’ awareness of thepower of mathematics. The applications are organized by level, length, and concept.

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Professors access an easy-to-use, menu-based system to order exactly the problemsthey wish to teach. Professors receive an Instructor’s Manual with tips and solu-tions, and students receive the problems in the form of a workbook, which includesa guide to solving these applications and is available in the bookstore. Contact yoursales representative for details and passwords.

For StudentsStudent's Solutions Manual This manual contains complete annotated solu-tions to every other odd-numbered problem in problem sets and all Chapter Review,Chapter Test, and Cumulative Review Exercises.

Graphing Calculator Learning Resource The Graphing Calculator LearningResource contains programming ideas for various Texas Instruments graphing cal-culators and investigates the application of those programming operations to solveproblems.

Acknowledgments This revision began in May 2000 at an organizational meeting in San Luis Obispo,California, with Angus McDonald, former Mathematics Executive Editor, AnneScanlan-Rohrer, Two-Ravens Editorial Services, and myself. The vision for thebook became clear during that two-day meeting. The wonderful team of people thatturned our vision into the book you are looking at now begins and ends with JayCampbell, Developmental Editor at Harcourt College Publishing. Jay was involvedin every part of the revision process and he has done an exceptional job. To start,Jay and Anne took care of all the items I needed to get started on the manuscript.Jay then took my manuscript and put it into a form that could start the productionprocess. Donna King, of Progressive Publishing Alternatives, handled the produc-tion part of this revision. Julia Conover was responsible for the superb marketingcampaign. Kelley Tyner, my editor at Harcourt, has been helpful and encouragingin all parts of the revision process. Jess Collins (McLennan Community College),Sharon Edgmon (Bakersfield College), Ross Rueger (College of the Sequoias), andKay Haralson, Nancy Matthews, Maria Smith, and Loretta Griffy (Austin PeayState University), working under the guidance of Jay Campbell, suggested manynew applications and exercises for the problem sets. Their suggestions havestrengthened this revision. Ross Rueger, Mike Rosenborg, Sandra Beken, and AnnOstberg, did an excellent job of accuracy checking and proofreading. And finally,Mark Turner, Cuesta College, and Tammy Fisher-Vasta helped me with the pageproofs for this book and I doubt that I could have met my deadlines without them.My thanks to all these people: it was a pleasure working with such proficient andpleasant professionals.

Thanks also to Diane McKeague, for her encouragement with all my writingendeavors, and to Amy Jacobs for all her advice, encouragement, and ideas on mar-keting and sales for the book.

Finally, I am grateful to the following instructors for their suggestions andcomments on this revision. Some reviewed the entire manuscript, while others were

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asked to evaluate the development of specific topics, or the overall sequence of top-ics. My thanks go to the people listed below:

Connie Buller — Metropolitan Community CollegeNancy E. Carpenter — Johnson County Community CollegeJeremy Carr — Pensacola Junior CollegeSally Copeland — Johnson County Community CollegeJoe DiCostanzo — Johnson County Community CollegeSharon Edgmon — Bakersfield CollegeCeleste Hernandez — Richland CollegeCarrie Kyser — Cuyahoga Community CollegeNic LaHue — Penn Valley Community CollegeMohsen Manouchehri — Cleveland State UniversityMary Anne C. Petruska — Pensacola Junior CollegeFred Safier — City College of San FranciscoJim Stewart — Jefferson Community College

Charles P. McKeagueOctober 2001

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Many of my algebra students are apprehensive at first because they are worried thatthey will not understand the topics we cover. When I present a new topic that theydo not grasp completely, they think something is wrong with them for not under-standing it.

On the other hand, some students are excited about the course from the begin-ning. They are not worried about understanding algebra and, in fact, expect to findsome topics difficult.

What is the difference between these two types of students?Those who are excited about the course know from experience (as you do) that

a certain amount of confusion is associated with most new topics in mathematics.They don’t worry about it because they also know that the confusion gives way tounderstanding in the process of reading the textbook, working the problems, andgetting questions answered. If they find a topic they are having difficulty with, theywork as many problems as necessary to grasp the subject. They don’t wait for theunderstanding to come to them; they go out and get it by working lots of problems.In contrast, the students who lack confidence tend to give up when they becomeconfused. Instead of working more problems, they sometimes stop working prob-lems altogether and that of course guarantees that they will remain confused.

If you are worried about this course because you lack confidence in your abil-ity to understand algebra, and you want to change the way you feel about mathe-matics, then look forward to the first topic that causes you some confusion. As soonas that topic comes along, make it your goal to master it, in spite of your apprehen-sion. You will see that each and every topic covered in this course is one you caneventually master, even if your initial introduction to it is accompanied by someconfusion. As long as you have passed a college-level beginning algebra course (orits equivalent), you are ready to take this course.

If you have decided to do well in algebra, the following list will be importantto you.

How to Be Successful in Algebra 1. Attend all class sessions on time. You cannot know exactly what goes on in

class unless you are there. Missing class and then expecting to find out whatwent on from someone else is not the same as being there yourself.

2. Read the book. It is best to read the section that will be covered in class be-forehand. Reading in advance, even if you do not understand everything youread, is still better than going to class with no idea of what will be discussed.

3. Work problems every day and check your answers. The key to success in math-ematics is working problems. The more problems you work, the better you willbecome at working them. The answers to the odd-numbered problems are givenin the back of the book. When you have finished an assignment, be sure to com-

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pare your answers with those in the book. If you have made a mistake, find outwhat it is, and correct it.

4. Do it on your own. Don’t be misled into thinking someone else’s work is yourown. Having someone else show you how to work a problem is not the same asworking the same problem yourself. It is okay to get help when you are stuck.As a matter of fact, it is a good idea. Just be sure you do the work yourself.

5. Review every day. After you have finished the problems your instructor has as-signed, take another fifteen minutes and review a section you have already com-pleted. The more you review, the longer you will retain the material you havelearned.

6. Don’t expect to understand every new topic the first time you see it. Sometimesyou will understand everything you are doing, and sometimes you won’t. That’sjust the way things are in mathematics. Expecting to understand each new topicthe first time you see it can lead to disappointment and frustration. The processof understanding algebra takes time. It requires that you read the book, workproblems, and get your questions answered.

7. Spend as much time as it takes for you to master the material. No set formulaexists for the exact amount of time you need to spend on algebra to master it.You will find out as you go along what is or isn’t enough time for you. If youend up spending two or more hours on each section in order to master the mater-ial there, then that’s how much time it takes; trying to get by with less will notwork.

8. Relax. It’s probably not as difficult as you think.

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