beginning algebra

Beginning Algebra v 1.0 This is the book Beginning Algebra (v 1.0) This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/) license See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and make it available to everyone else under the same terms This book was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz (http://lardbucket.org) in an effort to preserve the availability of this book Normally, the author and publisher would be credited here However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed Additionally, per the publisher's request, their name has been removed in some passages More information is available on this project's attribution page (http://2012books.lardbucket.org/attribution.html?utm_source=header) For more information on the source of this book, or why it is available for free, please see the project's home page (http://2012books.lardbucket.org/) You can browse or download additional books there ii Table of Contents About the Author Acknowledgments Preface Chapter 1: Real Numbers and Their Operations Real Numbers and the Number Line Adding and Subtracting Integers 31 Multiplying and Dividing Integers 47 Fractions 67 Review of Decimals and Percents 99 Exponents and Square Roots 127 Order of Operations 149 Review Exercises and Sample Exam 167 Chapter 2: Linear Equations and Inequalities 184 Introduction to Algebra 185 Simplifying Algebraic Expressions 206 Solving Linear Equations: Part I 227 Solving Linear Equations: Part II 249 Applications of Linear Equations 270 Ratio and Proportion Applications 305 Introduction to Inequalities and Interval Notation 327 Linear Inequalities (One Variable) 346 Review Exercises and Sample Exam 372 Chapter 3: Graphing Lines 390 Rectangular Coordinate System 391 Graph by Plotting Points 420 Graph Using Intercepts 447 Graph Using the y-Intercept and Slope 470 Finding Linear Equations 502 Parallel and Perpendicular Lines 529 Introduction to Functions 548 Linear Inequalities (Two Variables) 578 Review Exercises and Sample Exam 597 iii Chapter 4: Solving Linear Systems 619 Solving Linear Systems by Graphing 620 Solving Linear Systems by Substitution 644 Solving Linear Systems by Elimination 664 Applications of Linear Systems 689 Solving Systems of Linear Inequalities (Two Variables) 715 Review Exercises and Sample Exam 732 Chapter 5: Polynomials and Their Operations 746 Rules of Exponents 747 Introduction to Polynomials 775 Adding and Subtracting Polynomials 797 Multiplying Polynomials 816 Dividing Polynomials 841 Negative Exponents 864 Review Exercises and Sample Exam 887 Chapter 6: Factoring and Solving by Factoring 903 Introduction to Factoring 904 Factoring Trinomials of the Form x^2 + bx + c 935 Factoring Trinomials of the Form ax^2 + bx + c 955 Factoring Special Binomials 978 General Guidelines for Factoring Polynomials 999 Solving Equations by Factoring 1013 Applications Involving Quadratic Equations 1038 Review Exercises and Sample Exam 1063 Chapter 7: Rational Expressions and Equations 1077 Simplifying Rational Expressions 1078 Multiplying and Dividing Rational Expressions 1107 Adding and Subtracting Rational Expressions 1127 Complex Rational Expressions 1152 Solving Rational Equations 1172 Applications of Rational Equations 1192 Variation 1219 Review Exercises and Sample Exam 1241 iv Chapter 8: Radical Expressions and Equations 1256 Radicals 1257 Simplifying Radical Expressions 1285 Adding and Subtracting Radical Expressions 1313 Multiplying and Dividing Radical Expressions 1330 Rational Exponents 1360 Solving Radical Equations 1385 Review Exercises and Sample Exam 1409 Chapter 9: Solving Quadratic Equations and Graphing Parabolas 1425 Extracting Square Roots 1426 Completing the Square 1449 Quadratic Formula 1469 Guidelines for Solving Quadratic Equations and Applications 1488 Graphing Parabolas 1514 Introduction to Complex Numbers and Complex Solutions 1553 Review Exercises and Sample Exam 1575 Appendix: Geometric Figures 1592 Plane 1593 Solid 1595 v About the Author John Redden earned his degrees at California State University–Northridge and Glendale Community College He is now a professor of mathematics at the College of the Sequoias, located in Visalia, California With over a decade of experience working with students to develop their algebra skills, he knows just where they struggle and how to present complex techniques in more understandable ways His student-friendly and commonsense approach carries over to his writing of Elementary Algebra and various other open-source learning resources Author site: http://edunettech.blogspot.com/ Acknowledgments I would like to thank the following reviewers whose feedback helped improve the final product: • • • • • • • • • • • • • • • • • • • Katherine Adams, Eastern Michigan University Sheri Berger, Los Angeles Valley College Seung Choi, Northern Virginia Community College Stephen DeLong, Colorado Mountain College Keith Eddy, College of the Sequoias Solomon Emeghara, William Patterson University Audrey Gillant, SUNY–Maritime Barbara Goldner, North Seattle Community College Joseph Grich, William Patterson University Caroll Hobbs, Pensacola State College Clark Ingham, Mott Community College Valerie LaVoice, NHTI, Concord’s Community College Sandra Martin, Brevard Schools Bethany Mueller, Pensacola State College Tracy Redden, College of the Sequoias James Riley, Northern Arizona University Bamdad Samii, California State University–Northridge Michael Scott, California State University–Monterey Bay Nora Wheeler, Santa Rosa Junior College I would also like to acknowledge Michael Boezi and Vanessa Gennarelli of Unnamed Publisher The success of this project is in large part due to their vision and expertise Finally, a special heartfelt thank-you is due to my wife, Tracy, who spent countless hours proofreading and editing these pages—all this while maintaining a tight schedule for our family Without her, this textbook would not have been possible Preface It is essential to lay a solid foundation in mathematics if a student is to be competitive in today’s global market The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines Traditionally, the study of algebra is separated into a two parts, elementary algebra and intermediate algebra This textbook, Elementary Algebra, is the first part, written in a clear and concise manner, making no assumption of prior algebra experience It carefully guides students from the basics to the more advanced techniques required to be successful in the next course This text is, by far, the best elementary algebra textbook offered under a Creative Commons license It is written in such a way as to maintain maximum flexibility and usability A modular format was carefully integrated into the design For example, certain topics, like functions, can be covered or omitted without compromising the overall flow of the text An introduction of square roots in Chapter is another example that allows for instructors wishing to include the quadratic formula early to so Topics such as these are carefully included to enhance the flexibility throughout This textbook will effectively enable traditional or nontraditional approaches to elementary algebra This, in addition to robust and diverse exercise sets, provides the base for an excellent individualized textbook instructors can use free of needless edition changes and excessive costs! A few other differences are highlighted below: • Equivalent mathematical notation using standard text found on a keyboard • A variety of applications and word problems included in most exercise sets • Clearly enumerated steps found in context within carefully chosen examples • Alternative methods and notation, modularly integrated, where appropriate • Video examples available, in context, within the online version of the textbook • Robust and diverse exercise sets with discussion board questions • Key words and key takeaways summarizing each section This text employs an early-and-often approach to real-world applications, laying the foundation for students to translate problems described in words into Preface mathematical equations It also clearly lays out the steps required to build the skills needed to solve these equations and interpret the results With robust and diverse exercise sets, students have the opportunity to solve plenty of practice problems In addition to embedded video examples and other online learning resources, the importance of practice with pencil and paper is stressed This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today In addition, textual notation is introduced as a means to communicate solutions electronically throughout the text While it is important to obtain the skills to solve problems correctly, it is just as important to communicate those solutions with others effectively in the modern era of instant communications Unnamed Publisher is the only publisher today willing to put in the resources that it takes to produce a quality, peer-reviewed textbook and allow it to be published under a Creative Commons license They have the system that implements the customizable, affordable, and open textbook of the twenty-first century In fact, this textbook was specifically designed and written to fully maximize the potential of the Unnamed Publisher system I feel that my partnership with Unnamed Publisher has produced a truly fine example in Elementary Algebra, which demonstrates what is possible in the future of publishing Chapter Real Numbers and Their Operations Chapter Solving Quadratic Equations and Graphing Parabolas h (t) = −16t + 48t, where t represents time in seconds What is the maximum height of the baseball? Introduction to Complex Numbers and Complex Solutions Rewrite in terms of i ⎯⎯⎯⎯⎯⎯⎯ 89 √ −36 ⎯⎯⎯⎯⎯⎯⎯ 90 √ −40 91 √ − ⎯⎯⎯⎯⎯⎯⎯8 ⎯ 25 92 −√ − ⎯⎯⎯⎯⎯1⎯ Perform the operations 93 (2 − 5i) + (3 + 4i) 95 (2 − 3i) (5 + i) 94 (6 96 − 7i) − (12 − 3i) 4−i 2−3i Solve 97 9x + 25 = 98 3x +1=0 99 y −y+5=0 100 y + 2y + 101 4x (x 9.7 Review Exercises and Sample Exam + 2) + = 8x 1582 Chapter Solving Quadratic Equations and Graphing Parabolas 102 (x 9.7 Review Exercises and Sample Exam + 2) (x + 3) = (x + 13) 1583 Chapter Solving Quadratic Equations and Graphing Parabolas SAMPLE EXAM Solve by extracting the roots 4x −9=0 + 1) − = (4x Solve by completing the square x + 10x + 19 = x −x−1=0 Solve using the quadratic formula −2x x +x+3=0 + 6x − 31 = Solve using any method (5x + 1) (x + 1) = x (x + 3) = −2 (x + 5) (x − 5) = 65 10 2(x − 2) − = 3x Set up an algebraic equation and solve 11 The length of a rectangle is twice its width If the diagonal measures ⎯⎯ 6√ centimeters, then find the dimensions of the rectangle 12 The height in feet reached by a model rocket launched from a platform is given by the function h(t) = −16t + 256t + , where t represents time in seconds after launch At what time will the rocket reach 451 feet? 9.7 Review Exercises and Sample Exam 1584 Chapter Solving Quadratic Equations and Graphing Parabolas Graph Find the vertex and the y-intercept In addition, find the x-intercepts if they exist 13 y = 2x − 4x − 14 y = −x + 4x − 15 y = 4x − 16 y = x + 2x − 17 Determine the maximum or minimum y-value: y = −3x + 12x − 15 18 Determine the x- and y-intercepts: y = x + x + 19 Determine the domain and range: y = 25x − 10x + 20 The height in feet reached by a model rocket launched from a platform is given by the function h(t) = −16t + 256t + , where t represents time in seconds after launch What is the maximum height attained by the rocket 21 A bicycle manufacturing company has determined that the weekly revenue in dollars can be modeled by the formula R = 200n − n , where n represents the number of bicycles produced and sold How many bicycles does the company have to produce and sell in order to maximize revenue? ⎯⎯⎯⎯⎯⎯⎯ 22 Rewrite in terms of i: √ −60 23 Divide: 4−2i 4+2i Solve 24 25x +3=0 25 −2x + 5x − = 9.7 Review Exercises and Sample Exam 1585 Chapter Solving Quadratic Equations and Graphing Parabolas REVIEW EXERCISES ANSWERS 1: ±16 ⎯⎯ 3: ±3√ 5: ± 5√3 7: 2, 9: 24±√3 ⎯⎯ 11: ±3√ 13: x −2=0 15: x − 6x + = (x − 3) 17: ⎯⎯⎯⎯ ± √ 35 19: ⎯⎯ ± 3√ 21: −5±√29 23: −3/2, 25: 8±3√6 27: −5±√23 29: −2 31: a 9.7 Review Exercises and Sample Exam ⎯⎯ ± 2√ = 1, b = −1 , and c = 1586 Chapter Solving Quadratic Equations and Graphing Parabolas 33: a = 1, b = 0, and c = −5 35: ⎯⎯ ± √3 37: 1±√13 39: ± 41: 6√5 5±√29 43: −12, 24 45: ⎯⎯ ± 3√ 47: Two real solutions 49: One real solution 51: −10, 53: ±1/5 55: 1±√13 57: 2±3√3 59: −4, 61: ⎯⎯ ± √5 63: Length: 6.14 inches; width: 4.07 inches 65: It is worth $22,000 new and when it is 24 years old 67: x-intercepts: (−3, 0), (1/2, 0); y-intercept: (0, −3) 9.7 Review Exercises and Sample Exam 1587 Chapter Solving Quadratic Equations and Graphing Parabolas 69: x-intercepts: none; y-intercept: (0, 2) 71: Vertex: (3, −8); line of symmetry: x =3 73: Vertex: (−3/2, −13/4); line of symmetry: x =− 75: 77: 79: 81: 9.7 Review Exercises and Sample Exam 1588 Chapter Solving Quadratic Equations and Graphing Parabolas 83: Minimum: y = −24 85: Maximum: y = 9/5 87: The car will have a minimum value 12 years after it is purchased 89: 6i 91: 2i√2 93: −i 95: 13 − 13i 97: ± 5i ± 99: 101: ± 9.7 Review Exercises and Sample Exam √19 i i√5 1589 Chapter Solving Quadratic Equations and Graphing Parabolas SAMPLE EXAM ANSWERS 1: ± 3: −5 ⎯⎯ ± √6 5: −1, 3/2 7: −6/5, 9: −2, −1 11: Length: 12 centimeters; width: centimeters 13: 15: 17: Maximum: y = −3 19: Domain: R; range: [0, ∞) 21: To maximize revenue, the company needs to produce and sell 100 bicycles a week 9.7 Review Exercises and Sample Exam 1590 Chapter Solving Quadratic Equations and Graphing Parabolas 23: 25: 5±√17 9.7 Review Exercises and Sample Exam − i 1591 Chapter 10 Appendix: Geometric Figures LEARNING OBJECTIVE 1592 Chapter 10 Appendix: Geometric Figures 10.1 Plane Area (A) is measured in square units, perimeter (P) is measured in units, and circumference (C) is measured in units Square Rectangle Parallelogram 1593 Chapter 10 Appendix: Geometric Figures Trapezoid Triangle Circle 10.1 Plane 1594 Chapter 10 Appendix: Geometric Figures 10.2 Solid Volume (V) is measured in cubic units and surface area (SA) is measured in square units Cube Rectangular Solid 1595 Chapter 10 Appendix: Geometric Figures Right Circular Cylinder Right Circular Cone Sphere 10.2 Solid 1596 ... parts, elementary algebra and intermediate algebra This textbook, Elementary Algebra, is the first part, written in a clear and concise manner, making no assumption of prior algebra experience... 167 Chapter 2: Linear Equations and Inequalities 184 Introduction to Algebra 185 Simplifying Algebraic Expressions 206 Solving Linear Equations: Part I ... importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines Traditionally, the study of algebra is separated