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# Beginning algebra

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### Beginning algebra

1. 1. 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
3. 3. 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
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. 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
8. 8. 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
9. 9. 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
10. 10. 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
11. 11. 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
12. 12. 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.
13. 13. 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.