miL28711_fm_i-xxvi 12:12:06 10:07am Page i BA CONFIRMING PAGES Beginning ALGEBRA Second Edition Julie Miller Daytona Beach Community College Molly O’Neill Daytona Beach Community College Nancy Hyde Formerly of Broward Community College With Contributions by Mitchel Levy
miL28711_fm_i-xxvi 11:16:2006 12:21 Page iii BA CONFIRMING PAGES Contents Preface xi Chapter R Reference 1 R.1 Study Tips 2 R.2 Fractions 6 R.3 Decimals and Percents 22 R.4 Introduction to Geometry 32 Chapter 1 The Set of Real Numbers 49 1.1 Sets of Numbers and the Real Number Line 50 1.2 Order of Operations 61 1.3 Addition of Real Numbers 71 1.4 Subtraction of Real Numbers 80 Problem Recognition Exercises—Addition and Subtraction of Signed Numbers 88 1.5 Multiplication and Division of Real Numbers 89 1.6 Properties of Real Numbers and Simplifying Expressions 100 Chapter 1 Summary 113 Chapter 1 Review Exercises 117 Chapter 1 Test 119 Chapter 2 Linear Equations and Inequalities 121 2.1 Addition, Subtraction, Multiplication, and Division Properties of Equality 122 2.2 Solving Linear Equations 133 2.3 Linear Equations: Clearing Fractions and Decimals 142 2.4 Applications of Linear Equations: Introduction to Problem Solving 149 2.5 Applications Involving Percents 164 2.6 Formulas and Applications of Geometry 171 2.7 Linear Inequalities 182 Chapter 2 Summary 197 Chapter 2 Review Exercises 203 Chapter 2 Test 206 Cumulative Review Exercises, Chapters 1–2 207 iii
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page iv BA CONFIRMING PAGES iv Contents Chapter 3 Graphing Linear Equations in Two Variables 209 3.1 Rectangular Coordinate System 210 3.2 Linear Equations in Two Variables 221 3.3 Slope of a Line 238 3.4 Slope-Intercept Form of a Line 253 3.5 Point-Slope Formula 264 3.6 Applications of Linear Equations 273 Chapter 3 Summary 284 Chapter 3 Review Exercises 288 Chapter 3 Test 292 Cumulative Review Exercises, Chapters 1–3 294 Chapter 4 Systems of Linear Equations and Inequalities in Two Variables 297 4.1 Solving Systems of Equations by the Graphing Method 298 4.2 Solving Systems of Equations by the Substitution Method 310 4.3 Solving Systems of Equations by the Addition Method 320 4.4 Applications of Linear Equations in Two Variables 331 4.5 Linear Inequalities in Two Variables 341 4.6 Systems of Linear Inequalities in Two Variables 349 Chapter 4 Summary 357 Chapter 4 Review Exercises 362 Chapter 4 Test 366 Cumulative Review Exercises, Chapters 1–4 368 Chapter 5 Polynomials and Properties of Exponents 371 5.1 Exponents: Multiplying and Dividing Common Bases 372 5.2 More Properties of Exponents 382 5.3 Definitions of b0 and b؊n 387 5.4 Scientific Notation 395 Problem Recognition Exercises—Properties of Exponents 404 5.5 Addition and Subtraction of Polynomials 404 5.6 Multiplication of Polynomials 416 5.7 Division of Polynomials 426 Problem Recognition Exercises—Operations on Polynomials 434 Chapter 5 Summary 435 Chapter 5 Review Exercises 438 Chapter 5 Test 441 Cumulative Review Exercises, Chapters 1–5 442
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page v BA CONFIRMING PAGES Contents v Chapter 6 Factoring Polynomials 445 6.1 Greatest Common Factor and Factoring by Grouping 446 6.2 Factoring Trinomials of the Form x2 ϩ bx ϩ c (Optional) 456 6.3 Factoring Trinomials: Trial-and-Error Method 462 6.4 Factoring Trinomials: AC-Method 473 6.5 Factoring Binomials 482 6.6 General Factoring Summary 490 6.7 Solving Equations Using the Zero Product Rule 495 Chapter 6 Summary 509 Chapter 6 Review Exercises 513 Chapter 6 Test 515 Cumulative Review Exercises, Chapters 1–6 516 Chapter 7 Rational Expressions 517 7.1 Introduction to Rational Expressions 518 7.2 Multiplication and Division of Rational Expressions 528 7.3 Least Common Denominator 534 7.4 Addition and Subtraction of Rational Expressions 542 Problem Recognition Exercises—Operations on Rational Expressions 552 7.5 Complex Fractions 553 7.6 Rational Equations 561 Problem Recognition Exercises—Comparing Rational Equations and Rational Expressions 571 7.7 Applications of Rational Equations and Proportions 572 7.8 Variation (Optional) 584 Chapter 7 Summary 594 Chapter 7 Review Exercises 599 Chapter 7 Test 602 Cumulative Review Exercises, Chapters 1–7 603 Chapter 8 Radicals 605 8.1 Introduction to Roots and Radicals 606 8.2 Simplifying Radicals 619 8.3 Addition and Subtraction of Radicals 629 8.4 Multiplication of Radicals 636 8.5 Rationalization 643 Problem Recognition Exercises—Operations on Radicals 650 8.6 Radical Equations 651 8.7 Rational Exponents 658
miL28711_fm_i-xxvi 10/6/06 12:54 AM Page vi BA CONFIRMING PAGES vi Contents Chapter 8 Summary 666 Chapter 8 Review Exercises 670 Chapter 8 Test 673 Cumulative Review Exercises, Chapters 1–8 674 Chapter 9 Functions, Complex Numbers, and Quadratic Equations 677 9.1 Introduction to Functions 678 9.2 Complex Numbers 693 9.3 The Square Root Property and Completing the Square 702 9.4 Quadratic Formula 711 9.5 Graphing Quadratic Functions 721 Chapter 9 Summary 734 Chapter 9 Review Exercises 737 Chapter 9 Test 741 Cumulative Review Exercises, Chapters 1–9 742 Student Answer Appendix SA–1
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page vii BA CONFIRMING PAGES Dedication To our students . . . —Julie Miller —Molly O’Neill —Nancy Hyde
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page viii BA CONFIRMING PAGES About the Authors Julie Miller Julie Miller has been on the faculty of the Mathematics Department at Daytona Beach Community College for 18 years, where she has taught developmental and upper-level courses. Prior to her work at DBCC, she worked as a software engineer for General Electric in the area of flight and radar simulation. Julie earned a bachelor of science in applied mathematics from Union College in Schenectady, New York, and a master of science in mathematics from the Uni- versity of Florida. In addition to this textbook, she has authored several course supplements for college algebra, trigonometry, and precalculus, as well as several short works of fiction and nonfiction for young readers. “My father is a medical researcher, and I got hooked on math and science when I was young and would visit his laboratory. I can remember using graph paper to plot data points for his experiments and doing simple calculations. He would then tell me what the peaks and features in the graph meant in the context of his experiment. I think that applications and hands-on experience made math come alive for me and I’d like to see math come alive for my students.” —Julie Miller Molly O’Neill Molly O’Neill is also from Daytona Beach Community College, where she has taught for 20 years in the Mathematics Department. She has taught a variety of courses from developmental mathematics to calculus. Before she came to Florida, Molly taught as an adjunct instructor at the University of Michigan–Dearborn, Eastern Michigan University, Wayne State University, and Oakland Community College. Molly earned a bachelor of science in mathematics and a master of arts and teaching from Western Michigan Uni- versity in Kalamazoo, Michigan. Besides this textbook, she has authored several course supplements for college algebra, trigonometry, and precalculus and has reviewed texts for devel- opmental mathematics. “I differ from many of my colleagues in that math was not always easy for me. But in seventh grade I had a teacher who taught me that if I follow the rules of mathematics, even I could solve math problems. Once I un- derstood this, I enjoyed math to the point of choosing it for my career. I now have the greatest job because I get to do math everyday and I have the oppor- tunity to influence my students just as I was influenced. Authoring these texts has given me another avenue to reach even more students.” —Molly O’Neill viii
miL28711_fm_i-xxvi 10/6/06 04:31 AM Page ix BA CONFIRMING PAGES About the Authors ix Nancy Hyde Nancy Hyde served as a full-time faculty member of the Mathematics Department at Broward Community College for 24 years. During this time she taught the full spectrum of courses from developmental math through dif- ferential equations. She received a bachelor of science degree in math education from Florida State University and a mas- ter’s degree in math education from Florida Atlantic Univer- sity. She has conducted workshops and seminars for both students and teachers on the use of technology in the class- room. In addition to this textbook, she has authored a graph- ing calculator supplement for College Algebra. “I grew up in Brevard County, Florida, with my father working at Cape Canaveral. I was always excited by mathematics and physics in relation to the space program. As I studied higher levels of mathematics I became more intrigued by its abstract nature and infinite possibilities. It is enjoyable and rewarding to convey this perspective to students while helping them to under- stand mathematics.” —Nancy Hyde Mitchel Levy Mitchel Levy of Broward Community College joined the team as the exercise consultant for the Miller/O’Neill/Hyde paperback series. Mitchel received his BA in mathematics in 1983 from the State University of New York at Albany and his MA in mathematical statistics from the University of Maryland, College Park in 1988. With over 17 years of teaching and extensive reviewing experience, Mitchel knows what makes exercise sets work for students. In 1987 he received the first annual “Excellence in Teaching” award for graduate teaching assistants at the University of Maryland. Mitchel was honored as the Broward Community College Professor of the year in 1994, and has co-coached the Broward math team to 3 state championships over 7 years. “I love teaching all level of mathematics from Elementary Algebra through Calculus and Statistics.” —Mitchel Levy
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page x BA CONFIRMING PAGES x Introducing . . . The Miller/O’Neill/Hyde Series in Developmental Mathematics Miller/O’Neill/Hyde casebound series Beginning Algebra, 2/e Intermediate Algebra, 2/e Beginning and Intermediate Algebra, 2/e Miller/O’Neill/Hyde worktext series Basic College Mathematics Introductory Algebra Intermediate Algebra x
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page xi BA CONFIRMING PAGES Preface From the Authors First and foremost, we would like to thank the students and colleagues who have helped us prepare this text. The content and organization are based on a wealth of resources. Aside from an accumulation of our own notes and experiences as teachers, we recognize the influence of colleagues at Daytona Beach Community College as well as fellow presenters and attendees of national mathematics con- ferences and meetings. Perhaps our single greatest source of inspiration has been our students, who ask good, probing questions every day and challenge us to find new and better ways to convey mathematical concepts. We gratefully acknowl- edge the part that each has played in the writing of this book. In designing the framework for this text, the time we have spent with our students has proved especially valuable. Over the years we have observed that students struggle consistently with certain topics. We have also come to know the influence of forces beyond the math, particularly motivational issues. An awareness of the various pitfalls has enabled us to tailor pedagogy and tech- niques that directly address students’ needs and promote their success. These techniques and pedagogy are outlined here. Active Classroom First, we believe students retain more of what they learn when they are actively engaged in the classroom. Consequently, as we wrote each section of text, we also wrote accompanying worksheets called Classroom Activities to foster ac- countability and to encourage classroom participation. Classroom Activities re- semble the examples that students encounter in the textbook. The activities can be assigned to individual students or to pairs or groups of students. Most of the activities have been tested in the classroom with our own students. In one class in particular, the introduction of Classroom Activities transformed a group of “clock watchers” into students who literally had to be ushered out of the class- room so that the next class could come in. The activities can be found in the Instructor’s Resource Manual, which is available through MathZone. Conceptual Support While we believe students must practice basic skills to be successful in any math- ematics class, we also believe concepts are important. To this end, we have in- cluded numerous writing questions and homework exercises that ask students to “interpret the meaning in the context of the problem.” These questions make stu- dents stop and think, so they can process what they learn. In this way, students will learn underlying concepts. They will also form an understanding of what their answers mean in the contexts of the problems they solve. Writing Style Many students believe that reading a mathematics text is an exercise in futility. However, students who take the time to read the text and features may cast that notion aside. In particular, the Tips and Avoiding Mistakes boxes should prove xi
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page xii BA CONFIRMING PAGES xii Preface especially enlightening. They offer the types of insights and hints that are usually only revealed during classroom lecture. On the whole, students should be very comfortable with the reading level, as the language and tone are consistent with those used daily within our own developmental mathematics classes. Real-World Applications Another critical component of the text is the inclusion of contemporary real- world examples and applications. We based examples and applications on in- formation that students encounter daily when they turn on the news, read a magazine, or surf the Internet. We incorporated data for students to answer math- ematical questions based on information in tables and graphs. When students en- counter facts or information that is meaningful to them, they will relate better to the material and remember more of what they learn. Study Skills Many students in this course lack the basic study skills needed to be successful. Therefore, at the beginning of the homework exercises, we included a set of Study Skills Exercises. The exercises focus on one of nine areas: learning about the course, using the text, taking notes, completing homework assignments, test taking, time management, learning styles, preparing for a final exam, and defining key terms. Through completion of these exercises, students will be in a better posi- tion to pass the class and adopt techniques that will benefit them throughout their academic careers. Language of Mathematics Finally, for students to succeed in mathematics, they must be able to understand its language and notation. We place special emphasis on the skill of translating mathematical notation to English expressions and vice versa through Translat- ing Expressions Exercises. These appear intermittently throughout the text. We also include key terms in the homework exercises and ask students to define these terms. What Sets This Book Apart? We believe that the thoughtfully designed pedagogy and contents of this text- book offer any willing student the opportunity to achieve success, opening the door to a wider world of possibilities. While this textbook offers complete coverage of the beginning algebra cur- riculum, there are several concepts that receive special emphasis. Problem Recognition Problem recognition is an important theme carried throughout this edition and is integrated into the textbook in a number of different ways. First, we developed Problem Recognition Exercises that appear in selected chapters. The purpose of the Problem Recognition Exercises is to present a collection of problems that may look very similar to students, upon first glance, but are actually quite different in the manner of their individual solutions. By completing these exercises, students gain a greater awareness in problem recognition—that is, identifying a particular problem type upon inspection and applying the appropriate method to solve it.
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page xiii BA CONFIRMING PAGES Preface xiii We have carefully selected the content areas that seem to present the greatest challenge for students in terms of their ability to differentiate among various problem types. For these topics (listed below) we developed Problem Recognition Exercises. Addition and Subtraction of Signed Numbers (page 88) Properties of Exponents (page 404) Operations on Polynomials (page 434) Operations on Rational Expressions (page 552) Comparing Rational Equations and Rational Expressions (page 571) Operations on Radicals (page 650) The problem recognition theme is also threaded throughout the book within the exercise sets. In particular, Section 6.6, “General Factoring Summary,” is worth special mention. While not formally labeled “Problem Recognition,” each factor- ing exercise requires students to label the type of factoring problem presented before attempting to factor the polynomial (see page 493, directions for problems 7–74). The concept of problem recognition is also applied in every chapter, on a more micro level. We looked for opportunities within the section-ending Practice Ex- ercises to include exercises that involve comparison between problem types. See, for example, page 422, exercises 2–13 page 628, exercises 91 and 93 as well as 103 and 105 page 642, exercises 74–79 (contrast parts a and b) page 649, exercises 51–58 page 649, exercises 59–62 We also included Mixed Exercises that appear in many of the Practice Ex- ercise sets to give students opportunities to practice exercises that are not lumped together by problem type. We firmly believe that students who effectively learn how to distinguish between various problems and the methods to solve them will be better prepared for Intermediate Algebra and courses beyond. Design While the content of a textbook is obviously critical, we believe that the design of the page is equally vital. We have often heard instructors describe the impor- tance of “white space” in the design of developmental math textbooks and the need for simplicity, to prevent distractions. All of these comments have factored heavily in the page layout of this textbook. For example, we left ample space between exercises in the section-ending Practice Exercise sets to make it easier for students to read and complete homework assignments. Similarly, we developed design treatments within sections to present the content, including examples, definitions, and summary boxes, in an organized and reader-friendly way. We also limited the number of colors and photos in use, in an effort to avoid distraction on the pages. We believe these considerations di- rectly reflect the needs of our students and will enable them to navigate through the content successfully. Chapter R Chapter R is a reference chapter. We designed it to help students reacquaint them- selves with the fundamentals of fractions, decimals, percents, and geometry. This chapter also addresses study skills and helpful hints to use the resources provided in the text and its supplements.
miL28711_fm_i-xxvi 11:16:2006 12:21 Page xiv BA CONFIRMING PAGES xiv Preface Factoring Many years ago, we experimented in the classroom with our approach to factoring. We began factoring trinomials with the general case first, that is, with leading co- efficient not equal to 1. This gave the students one rule for all cases, and it pro- vided us with an extra class day for practice and group work. Most importantly, this approach forces students always to consider the leading coefficient, whether it be 1 or some other number. Thus, when students take the product of the inner terms and the product of the outer terms, the factors of the leading coefficient always come into play. While we recommend presenting trinomials with leading coefficient other than 1 first, we want to afford flexibility to the instructors using this textbook. Therefore, we have structured our chapter on Factoring, Chapter 6, to offer in- structors the option of covering trinomials with leading coefficient of 1 first (Section 6.2), or to cover trinomials with leading coefficient other than 1 first (Sections 6.3 and 6.4). For those instructors who have never tried the method of presenting trino- mials with leading coefficient other than 1 first, we suggest giving it a try. We have heard other instructors, who were at first resistant to this approach, remark how well it has worked for their students. In fact, one instructor told us, “I was skep- tical, but I had the best exam results I have ever seen in factoring as a result of this approach! It has a big thumbs up from me and students alike.” Calculator Usage The use of a scientific or a graphing calculator often inspires great debate among faculty who teach developmental mathematics. Our Calculator Connections boxes offer screen shots and some keystrokes to support applications where a calcula- tor might enhance learning. Our approach is to use a calculator as a verification tool after analytical methods have been applied. The Calculator Connections boxes are self-contained units with accompanying exercises and can be employed or easily omitted at the recommendation of the instructor. Calculator Exercises appear within the section-ending Practice Exercise sets where appropriate, but they are clearly noted as such and can be assigned or omitted at the instructor’s discretion. Listening to Students’ and Instructors’ Concerns Our editorial staff has amassed the results of reviewer questionnaires, user diaries, focus groups, and symposia. We have consulted with a seven-member panel of beginning algebra instructors and their students on the development of this book. In addition, we have read hundreds of pages of reviews from instructors across the country. At McGraw-Hill symposia, faculty from across the United States gath- ered to discuss issues and trends in developmental mathematics. These efforts have involved hundreds of faculty and have explored issues such as content, read- ability, and even the aesthetics of page layout. In our continuing efforts to improve our products, we invite you to contact us directly with your comments. Julie Miller Molly O’Neill Nancy Hyde email@example.com firstname.lastname@example.org email@example.com
miL28711_fm_i-xxvi 11:16:2006 12:21 Page xv BA CONFIRMING PAGES Preface xv Acknowledgments and Reviewers The development of this textbook would never have been possible without the cre- ative ideas and constructive feedback offered by many reviewers. We are especially thankful to the following instructors for their valuable feedback and careful re- view of the manuscript. Board of Advisors Valarie Beaman-Hackle, Macon State College Michael Everett, Santa Ana College Teresa Hasenauer, Indian River Community College Lori Holdren, Manatee Community College–Bradenton Becky Hubiak, Tidewater Community College Barbara Hughes, San Jacinto College–Pasadena Mike Kirby, Tidewater Community College Manuscript Reviewers Cedric Atkins, Charles Stewart Mott Community College David Bell, Florida Community College Emilie Berglund, Utah Valley State College Kirby Bunas, Santa Rosa Junior College Annette Burden, Youngstown State University Susan Caldiero, Cosumnes River College Edie Carter, Amarillo College Marcial Echenique, Broward Community College Karen Estes, St. Petersburg College Renu Gupta, Louisiana State University–Alexandria Rodger Hergert, Rock Valley College Linda Ho, El Camino College Michelle Hollis, Bowling Green Community College Laura Hoye, Trident Technical College Randa Kress, Idaho State University–Pocatello Donna Krichiver, Johnson County Community College Karl Kruczek, Northeastern State University Kathryn Lavelle, Westchester Community College Jane Loftus, Utah Valley State College Gary McCracken, Shelton State Community College Karen Pagel, Dona Ana Branch Community College Peter Remus, Chicago State University Daniel Richbart, Erie Community College–City Campus Special thanks go to Doris McClellan-Lewis for preparing the Instructor’s Solutions Manual and the Student’s Solution Manual and to Lauri Semarne for her work ensuring accuracy. Many, many thanks to Yolanda Davis,Andrea Hendricks, Patricia Jayne, and Cynthia Cruz for their work in the video series and to Kelly Jackson, an awesome teacher, for preparing the Instructor Notes. To Pat Steele, we appre- ciate your watchful eye over our manuscript. Finally, we are forever grateful to the many people behind the scenes at McGraw-Hill, our publishing family. To Erin Brown, our lifeline on this project, without whom we’d be lost. To Liz Haefele for her passion for excellence and
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page xvi BA CONFIRMING PAGES xvi Preface constant inspiration. To David Millage and Barb Owca for their countless hours of support and creative ideas promoting all of our efforts. To Jeff Huettman and Amber Huebner for the awesome technology so critical to student success, and fi- nally to Vicki Krug for keeping the train on the track during production. Most importantly, we give special thanks to all the students and instructors who use Beginning Algebra second edition in their classes. Julie Miller Molly O’Neill Nancy Hyde
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page xvii BA CONFIRMING PAGES Preface xvii A COMMITMENT TO ACCURACY You have a right to expect an accurate textbook, and McGraw-Hill invests consid- erable time and effort to make sure that we deliver one. Listed below are the many steps we take to make sure this happens. OUR ACCURACY VERIFICATION PROCESS 1st Round: First Round Author’s Manuscript Step 1: Numerous college math instructors review the manuscript and report on any errors that they may find, and the authors make these corrections in their final man- uscript. ✓ Multiple Rounds of Review by College Second Round Math Instructors Step 2: Once the manuscript has been typeset, the authors check their manuscript against the first page proofs to ensure that all illustrations, graphs, examples, exer- cises, solutions, and answers have been correctly laid out on the pages, and that all 2nd Round: notation is correctly used. Typeset Pages Step 3: An outside, professional mathematician works through every example and exercise in the page proofs to verify the accuracy of the answers. Accuracy Checks by: Step 4: A proofreader adds a triple layer of accuracy assurance in the first pages by ✓ Authors hunting for errors, then a second, corrected round of page proofs is produced. ✓ Professional Mathematician ✓ 1st Proofreader Third Round Step 5: The author team reviews the second round of page proofs for two reasons: (1) to make certain that any previous corrections were properly made, and (2) to 3rd Round: look for any errors they might have missed on the first round. Typeset Pages Step 6: A second proofreader is added to the project to examine the new round of page proofs to double check the author team’s work and to lend a fresh, critical eye to the book before the third round of paging. Accuracy Checks by: ✓ Authors Fourth Round ✓ 2nd Proofreader Step 7: A third proofreader inspects the third round of page proofs to verify that all previous corrections have been properly made and that there are no new or remaining errors. 4th Round: Step 8: Meanwhile, in partnership with independent mathematicians, the text accu- Typeset Pages racy is verified from a variety of fresh perspectives: • The test bank author checks for consistency and accuracy as they prepare the computerized test item file. Accuracy Checks by: • The solutions manual author works every single exercise and verifies their an- ✓ 3rd Proofreader swers, reporting any errors to the publisher. ✓ Test Bank Author • A consulting group of mathematicians, who write material for the text’s ✓ Solutions Manual Author MathZone site, notifies the publisher of any errors they encounter in the page ✓ Consulting Mathematicians for MathZone site proofs. ✓ Math Instructors for text’s video series • A video production company employing expert math instructors for the text’s videos will alert the publisher of any errors they might find in the page proofs. Final Round Final Round: Step 9: The project manager, who has overseen the book from the beginning, per- Printing forms a fourth proofread of the textbook during the printing process, providing a fi- nal accuracy review. ⇒ What results is a mathematics textbook that is as accurate and error-free as is ✓ Accuracy Check by humanly possible, and our authors and publishing staff are confident that our 4th Proofreader many layers of quality assurance have produced textbooks that are the leaders of the industry for their integrity and correctness.
miL28711_fm_i-xxvi 11:16:2006 12:21 Page xviii BA CONFIRMING PAGES Guided Tour Chapter Opener Each chapter opens with a puzzle relating to the content of the chapter. Section titles are clearly listed for easy reference. Linear Equations and Inequalities 2.1 Addition, Subtraction, Multiplication, and Division 2 Properties of Equality 2.2 Solving Linear Equations 2.3 Linear Equations: Clearing Fractions and Decimals 2.4 Applications of Linear Equations: Introduction to Problem Solving 2.5 Applications Involving Percents 2.6 Formulas and Applications of Geometry 2.7 Linear Inequalities In this chapter we learn how to solve linear equations and inequalities in one variable. This important category of equations and inequalities is used in a variety of applications. The list of words contains key terms used in this chapter. Search for them in the puzzle and in the text throughout the chapter. By the end of this chapter, you should be familiar with all of Z C these terms. E L Q A U M P R N I E L Key Terms C D K I W E O P N V M X C O N T R A D I C T I O N D compound identity notation L M S S O C L T K E T K O E conditional inequality percent I P L N N Y I I Z G U I I C consecutive integer property N O A O O T U D V E C N T I contradiction interest set-builder E U R I I I B N Y R E T A M decimal interval solution A N E T T T T O A Q S E U A equation linear R D T U A N E C U H N R Q L fractions literal N I L T E S A A N O E E B L O O D L Z Y R C S A N S N I L Q G E F T I N T E R V A L R Q Y T R E P O R P T N E C R E P A P L M Q 121 Section 3.1 Rectangular Coordinate System Concepts 1. Interpreting Graphs Concepts 1. Interpreting Graphs Mathematics is a powerful tool used by scientists and has directly contributed 2. Plotting Points in a to the highly technical world we live in. Applications of mathematics have led Rectangular Coordinate System to advances in the sciences, business, computer technology, and medicine. A list of important learning concepts is provided One fundamental application of mathematics is the graphical representation of 3. Applications of Plotting and Identifying Points numerical information (or data). For example, Table 3-1 represents the number of clients admitted to a drug and alcohol rehabilitation program over a 12-month at the beginning of each section. Each concept period. corresponds to a heading within the section and Table 3-1 Number of within the exercises, making it easy for students Month Clients Jan. 1 55 to locate topics as they study or as they work Feb. 2 62 March 3 64 through homework exercises. April 4 60 May 5 70 June 6 73 July 7 77 Aug. 8 80 Sept. 9 80 Oct. 10 74 Nov. 11 85 Dec. 12 90 In table form, the information is difficult to picture and interpret. It appears that on a monthly basis, the number of clients fluctuates. However, when the data are represented in a graph, an upward trend is clear (Figure 3-1). y Number of Clients Admitted to Program 100 Number of Clients 80 60 40 20 0 x 0 1 2 3 4 5 6 7 8 9 10 11 12 Month (x ϭ 1 corresponds to Jan.) Figure 3-1 xviii
miL28711_fm_i-xxvi 10/5/06 01:03 AM Page xix BA CONFIRMING PAGES Guided Tour xix Worked Examples Example 5 Solving Linear Equations Solve the equations: Examples are set off in boxes and organized a. 2 ϩ 7x Ϫ 5 ϭ 61x ϩ 32 ϩ 2x so that students can easily follow the solutions. b. 9 Ϫ 1z Ϫ 32 ϩ 4z ϭ 4z Ϫ 51z ϩ 22 Ϫ 6 Explanations appear beside each step and color- Solution: a. 2 ϩ 7x Ϫ 5 ϭ 61x ϩ 32 ϩ 2x coding is used, where appropriate. For additional Ϫ3 ϩ 7x ϭ 6x ϩ 18 ϩ 2x Step 1: Add like terms on the left. step-by-step instruction, students can run the Clear parentheses on the right. “e-Professors” in MathZone. The e-Professors are Ϫ3 ϩ 7x ϭ 8x ϩ 18 Combine like terms. based on worked examples from the text and use Ϫ3 ϩ 7x Ϫ 7x ϭ 8x Ϫ 7x ϩ 18 Step 2: Subtract 7x from both sides. Ϫ3 ϭ x ϩ 18 Simplify. the solution methodologies presented in the text. Ϫ3 Ϫ 18 ϭ x ϩ 18 Ϫ 18 Step 3: Subtract 18 from both sides. Ϫ21 ϭ x Step 4: Because the coefficient of the x term is already 1, there is no x ϭ Ϫ21 need to apply the multiplication or division property of equality. Step 5: The check is left to the reader. Skill Practice Exercises b. 9 Ϫ 1z Ϫ 32 ϩ 4z ϭ 4z Ϫ 51z ϩ 22 Ϫ 6 9 Ϫ z ϩ 3 ϩ 4z ϭ 4z Ϫ 5z Ϫ 10 Ϫ 6 Step 1: Clear parentheses. Every worked example is followed by one or 12 ϩ 3z ϭ Ϫz Ϫ 16 Combine like terms. 12 ϩ 3z ϩ z ϭ Ϫz ϩ z Ϫ 16 Step 2: Add z to both sides. more Skill Practice exercises. These exercises 12 ϩ 4z ϭ Ϫ16 offer students an immediate opportunity to work 12 Ϫ 12 ϩ 4z ϭ Ϫ16 Ϫ 12 Step 3: Subtract 12 from both sides. problems that mirror the examples. Students can 4z ϭ Ϫ28 4z Ϫ28 then check their work by referring to the 4 ϭ 4 Step 4: Divide both sides by 4. answers at the bottom of the page. z ϭ Ϫ7 Step 5: The check is left for the reader. Skill Practice Solve the equations. 6. 6y ϩ 3 Ϫ y ϭ 412y Ϫ 12 7. 313a Ϫ 42 Ϫ 415a ϩ 22 ϭ 7a Ϫ 2 Avoiding Mistakes: c. a w2 Ϫ wb ϩ a w2 ϩ w Ϫ b 1 4 2 3 3 4 1 6 1 2 Avoiding Mistakes Example 3(c) is an expression, not an equation. Therefore, we cannot 1 3 2 1 1 ϭ w2 ϩ w2 Ϫ w ϩ w Ϫ Clear parentheses, and group like clear fractions. 4 4 3 6 2 1 3 4 1 1 terms. Through notes labeled Avoiding Mistakes ϭ w2 ϩ w2 Ϫ w ϩ w Ϫ Get common denominators for like 4 4 6 6 2 terms. students are alerted to common errors and are 4 3 1 ϭ w2 Ϫ w Ϫ 4 6 2 Add like terms. shown methods to avoid them. 1 1 ϭ w2 Ϫ w Ϫ Simplify. 2 2 Skill Practice Add the polynomials. 3. 17q 2 Ϫ 2q ϩ 42 ϩ 15q 2 ϩ 6q Ϫ 92 Tips 4. a t 3 Ϫ 1b ϩ a t 3 ϩ b 1 2 1 2 3 5 5. 15x 2 ϩ x Ϫ 22 ϩ 1x ϩ 42 ϩ 13x 2 ϩ 4x2 Tip boxes appear throughout the text and offer helpful hints and insight. 4. Subtraction of Polynomials The opposite (or additive inverse) of a real number a is Ϫa. Similarly, if A is a polynomial, then ϪA is its opposite. Example 4 Finding the Opposite of a Polynomial Find the opposite of the polynomials. a. 5x b. 3a Ϫ 4b Ϫ c c. 5.5y4 Ϫ 2.4y3 ϩ 1.1y Ϫ 3 Solution: TIP: Notice that the a. The opposite of 5x is Ϫ15x2, or Ϫ5x. sign of each term is changed when finding the b. The opposite of 3a Ϫ 4b Ϫ c is Ϫ13a Ϫ 4b Ϫ c2 or equivalently, opposite of a polynomial. Ϫ3a ϩ 4b ϩ c. c. The opposite of 5.5y4 Ϫ 2.4y3 ϩ 1.1y Ϫ 3 is Ϫ15.5y4 Ϫ 2.4y3 ϩ 1.1y Ϫ 32, or equivalently, Ϫ5.5y4 ϩ 2.4y3 Ϫ 1.1y ϩ 3. Chapter 5 Problem Recognition Exercises— Properties of Exponents Simplify completely. Assume that all variables 21. 125bϪ3 2 Ϫ3 22. 13Ϫ2y3 2 Ϫ2 represent nonzero real numbers. 3x Ϫ4 6c Ϫ2 1. t t3 5 3 5 2. 2 2 23. a b 24. a b 2y 5d3 y7 p9 3. 4. 25. 13ab 2 1a b2 2 2 3 26. 14x2y3 2 3 1xy2 2 y2 p3 xy2 4 a3b 5 5. 1r2s4 2 2 6. 1ab3c2 2 3 27. a b 28. a b Problem Recognition Exercises w4 7. Ϫ2 8. mϪ14 x3y 1tϪ2 2 3 a5b3 1p3 2 Ϫ4 w m2 29. 30. tϪ4 pϪ5 These exercises appear in selected chapters and 9. yϪ7x4 zϪ3 10. a3bϪ6 cϪ8 31. a 2w2x3 3 b 32. a 5a0b4 2 b 3y0 4c3 cover topics that are inherently difficult for 11. 12.5 ϫ 10Ϫ3 215.0 ϫ 105 2 qr3 Ϫ2 nϪ3m2 33. 34. students. Problem recognition exercises give 12. 13.1 ϫ 106 214.0 ϫ 10Ϫ2 2 sϪ1t 5 pϪ3qϪ1 4.8 ϫ 107 5.4 ϫ 10Ϫ2 1y 2 1y 2 Ϫ3 2 5 1w2 2 Ϫ4 1wϪ2 2 students the opportunity to practice the 13. 6.0 ϫ 10Ϫ2 14. 9.0 ϫ 106 35. 1yϪ3 2 Ϫ4 36. 1w5 2 Ϫ4 important, yet often overlooked, skill of 1 15. Ϫ6 Ϫ8 Ϫ1 6 8 16. p p p 37. a Ϫ2a2bϪ3 Ϫ3 b 38. a Ϫ3xϪ4y3 b Ϫ2 p p p aϪ4bϪ5 2x5yϪ2 distinguishing among different problem types. 2 0 3 2 4 3 5 4 0 2 5