ALGEBRA COLLEGE STUDENTS for SEVENTH EDITION D. FRANKLIN WRIGHT Annotated Instructor’s Edition - Not for SaleLead Editor: Barbara Miller Editors: Allison Conger, Ian Craig, Jolie Even, S. Rebecca Johnson Creative Services Manager: Trudy Tronco Designers: Lizbeth Mendoza, Patrick Thompson, Joel Travis Cover Design: Caitlin Neville Design and Layout Assistance: U. Nagesh, E. Jeevan Kumar, D. Kanthi, K.V.S. Anil A division of Quant Systems, Inc. 546 Long Point Road, Mount Pleasant, SC 29464 Copyright © 2024, 2020, 2011, 2004, 2000, 1996, 1981 by Hawkes Learning / Quant Systems, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written consent of the publisher. Library of Congress Control Number: 2023937414 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-64277-608-9 AIE ISBN: 978-1-64277-609-6Table of Contents Ordering Operations ...................................................68 quations and Inequalities 2.1 Simplifying and Evaluating Algebraic Expressions ...........................................................70 Equations and Functions Expressions and Rational Equations Radicals, and Complex Numbers iv Table of Contents atic Functions unctions Sections ces, Series, and the Binomial Theorem 12.1 Sequences .........................................................902 12.2 Sigma Notation .................................................909 12.3 Arithmetic Sequences ......................................915 12.4 Geometric Sequences and Series ...................926 12.5 The Binomial Theorem ....................................940 CHAPTER 12 PROJECTS Constructing Fibonacci’s Legacy ...............................949 Ready Player One .......................................................950 Appendix A.1 Review of Fractions ............................................954 A.2 The Fundamental Counting Principle and Permutations ..............................................964 v Table of ContentsPreface Algebra for College Students: Purpose and Style The problem-solving and analytical skills of mathematics are essential in helping students excel during and beyond their college years. The purpose of Algebra for College Students is to provide students with a text that will prepare them for college mathematics. Its goal is to provide students with a learning tool that will help them 1. acquire learning skills necessary for succeeding in their coursework, 2. develop reasoning and problem-solving skills, 3. become familiar with algebraic notation, 4. expand basic algebra skills, 5. transition smoothly from prealgebra through algebra, and 6. achieve satisfaction in learning so that they will be encouraged to continue their education in mathematics. The writing style gives carefully worded, thorough explanations that are direct, easy to understand, and mathematically accurate. The use of color, boldface, subheadings, and shaded boxes helps students understand and reference important topics. Each topic is developed in a straightforward, step-by-step manner. Each section contains many detailed examples to lead students successfully through the exercises and help them develop an understanding of the related concepts. We have provided very little overlap with material covered in a beginning algebra course. While Chapters 1 and 2 provide a review of topics from beginning algebra, students will find that the review is comprehensive and that the pace of coverage is somewhat faster and in more depth than they have seen in previous courses. As with any text in mathematics, students should read the text carefully and thoroughly. Students are encouraged to use calculators when appropriate, and explicit directions and diagrams are provided as they relate to a simple four-function calculator, as well as to a TI-84 Plus graphing calculator. The NCTM and AMATYC curriculum standards have been taken into consideration in the development of the topics throughout the text. vi Preface Algebra for College Students: Purpose and StyleAlgebra for College Students: Content Highlights New Features Strategies for Academic Success Strategies for Academic Success lessons have been added to provide students with skills needed for success in a math course. These new lessons focus on stress management and staying organized. Other topics include time management, reading a math book and taking notes, effective study strategies, and reducing test anxiety. Strategies for Academic Success 0.3 Managing Your Time Effectively Have you ever made it to the end of a day and wondered where all your time went? Sometimes it feels like there aren't enough hours in the day. Finding time to balance work, school, and home life can be difficult. Some interruptions, like unexpected traffic or family emergencies, are simply outside of your control. However, other distractions are within your control, such as watching TV or scrolling through social media. It’s important to find a balance between activities you need to do (such as attending class and work) and activities you want to do (such as watching TV). Managing your time is important because you can never get time back. Here are three strategies for managing your time more effectively. Take Breaks When you are working on an important project or studying for a big exam, you may feel tempted to work as long as possible without taking a break. This is especially true when you’re working or studying at the last minute. While staying focused is important, working yourself for hours until you’re mentally drained will lower the quality of your work and force you to take even more time recovering. Think about the way that overworking can affect your body physically. If you’re weight-training, you must take frequent breaks both between individual sets and entire workout sessions. If you don’t let your muscles recover, you risk injuring yourself, which could leave you laid up for weeks. Just like taking breaks helps your physical body recover, it will also help your brain re-energize and refocus. During study sessions, you should plan to take a short study break at least once an hour. If you usually work indoors, take this time to get a breath of fresh air outside and clear your head. Study breaks and work breaks should usually last around five minutes. The longer the break, the harder it is to start working again. Instead of stopping for half an hour, take a five-minute break and reward yourself with some downtime when the task is complete. Similarly, if a course you are taking has a built-in break during the middle of the class period, use it to get up and move around. This little bit of physical movement can help you think more clearly. Avoid Multitasking Multitasking is working on more than one task at a time. When you have several assignments that need to be completed, you may be tempted to save time by working on two or three of them at once. While this strategy might seem like a time-saver, you will probably end up using more time than if you had completed each task individually. Not only will you have to switch your focus from one task to the next, but you will also make more mistakes that will need to be corrected later. People don’t multitask nearly as well as they think they do. For example, research studies have shown that multitasking while driving is similar to or even worse than driving while drunk. While multitasking on a project for school or work may not be dangerous, it can lead to wasted time and silly mistakes. Instead of trying to do two things at once, schedule yourself time to work on one task at a time. Multitasking can also become an excuse for distractions, especially electronic ones. Have you ever tried to complete a homework assignment, watch TV, and message friends all at the same time? You probably did one of these things well and two of these things badly. That’s because your brain can’t give its full attention to three tasks at once. To stay focused in class or while studying, try stashing your phone in your backpack or purse and staying logged out of your computer until you need it. viStrategies for Academic Success 0.3 Managing Your Time Effectively Chapter Projects This new feature promotes collaboration and shows students the practical side of mathematics through activities using real-world applications of the concepts taught in the chapter. Chapter 8 Project Rationally Increasing Precision in Population Problems An activity to demonstrate the use of rational exponents in real life. Sometimes we need to find the value of an exponential expression where the exponent is not an integer. This often happens when dealing with exponential growth. In the essay “Observations Concerning the Increase of Mankind, Peopling of Countries, etc.,” written in 1751, Benjamin Franklin projected that the human population in the thirteen US colonies was doubling in size every twenty-five years. For example, if one year the population was 300,000 people, then to estimate the number of people 75 years later, calculate 300,000 ⋅ 23 = 2,400,000. Note that this works because 75 25 =3, meaning the population would double 3 times in 75 years. Also notice that since 2 ⋅ 2 ⋅ 2 = 23 = 8, we multiply 300,000 by 8 to get the final population. If the timespan we want to estimate the future population for is not a multiple of 25, we can still calculate this value using rational exponents (with as much precision as we like). This investigation will suggest which types of numbers can be exponents, a topic which will be expanded upon in Chapter 10. In the following investigation, do not round the exponents. Round the answers to have 10 digits, if necessary. 1. Calculate 2 to each of the following powers. a. 3 b. 3.1 c. 3.14 d. 3.141 e. 3.1415 f. 3.14159 g. 3.141592 h. 3.1415926 i. 3.14159265 2. The sequence of exponents in Problem 1 is approaching which special number? p 3. In Problem 1, are the calculated powers of 2 increasing, decreasing, or is there no discernible pattern? If increasing (or decreasing), are they increasing (or decreasing) toward a particular value? 4. Consider raising the value 2 to the power of the special number found in Problem 2. a. If it is possible, state the result and compare it to the results of Problem 1. If this is not possible, explain why. b. How does this relate to your answer in Problem 3? If there is no relation, explain why. 5. Returning to Franklin’s population prediction, consider if someone wanted to know the population 77.5 years later, instead of 75 years later. a. Determine the decimal form of the exponent x = 77 5 25 . . b. Substitute the value of x found in part a. into the population equation and simplify: 300,000 ⋅ 2x. c. Explain what x stands for. Interpret the answer to part b. In this case, does it make sense to round or not? If rounding does make sense, what place would you round to and what is the result? d. If the population starts at 300,000, how many years have passed if 300,000 ⋅ 23.14 provides an estimate of the population size? 672Chapter 8 Roots, Radicals, and Complex Numbers Chapter 8 Project Let’s Get Radical! An activity to demonstrate the use of radical expressions in real life. There are many different situations in real life that require working with radicals, such as solving right-triangle problems, working with the laws of physics, calculating volumes, and solving investment problems. Let’s take a look at a simple investment problem to see how radicals are involved. The formula for computing compound interest for a principal P that is invested at an annual rate r and compounded annually is given by AP n 1 r , where A is the accumulated amount in the account after n years. 1. Let’s suppose that you have $5000 to invest for a term of 2 years. If you want to make $600 in interest, then at what interest rate should you invest the money? a. One way to approach this problem would be through trial and error, substituting various rates for r in the formula. This approach might take a while. Using the table below to organize your work, try substituting 3 values for r. Remember that rates are percentages and need to be converted to decimals before using the formula. Did you get close to $5600 for the accumulated amount in the account after 2 years? Annual Rate ( r )Principal ( P ) Number of Years ( n ) Amount, A = P (1 + r)n $50002 $50002 $50002 b. Let’s try a different approach. Substitute the value of 2 for n and solve this formula for r. Verify that you get the following result: r A P 1 (Hint: First solve for (1 + r)2 and then take the square root of both sides of the equation.) Notice that you now have a radical expression to work with. Substitute $5000 for P and $5600 for A (which is the principal plus $600 in interest) to see what your rate must be. Round your answer to the nearest percent. 2. Now, let’s suppose that you won’t need the money for 3 years. a. Use n = 3 years and solve the compound interest formula for r. b. What interest rate will you need to invest the principal of $5000 at in order to have at least $5600 at the end of 3 years? (To evaluate a cube root you may have to use the rational exponent of 1 3 on your calculator.) Round to the nearest percent. c. Compare the rates needed to earn at least $600 when n = 2 years and n = 3 years. What did you learn from this comparison? Write a complete sentence. 3. Using the above formulas for compound interest when n = 2 years and n = 3 years, write the general formula for r for any value of n. 4. Using the formula from Problem 3, compute the interest rate needed to earn at least $3000 in interest on a $5000 investment in 7 years. Round to the nearest percent. 5. Do an internet search on a local bank or financial institution to determine if the interest rate from Problem 4 is reasonable in the current economy. Using three to five sentences, briefly explain why or why not. 671 Chapter 8 Project Let’s Get Radical! CHAPTER 8 PROJECTS Connections Each chapter begins with a brief discussion related to a concept developed in the coming material and connects this concept to a real-life situation to improve student understanding and interest. Y Connections While they may not seem common, exponential and logarithmic functions are used in a variety of fields. Exponential functions can be used to describe the relationship between musical notes in music theory and to model population growth in ecology. Logarithmic functions can be used to describe the intensity of an earthquake in geology and to measure the acidity or alkalinity of solutions in chemistry. A common situation where you might find yourself working with exponential functions in your everyday life is in financial situations, such as calculating the amount of interest owed on your student loans or credit cards. When making large purchases, such as buying a car or a house, you’ll often have the option to choose between different financing options. Being comfortable with exponential functions can help you make the decision that is best for you and your circumstances. Suppose you’re buying a car that costs $15,000 and the dealership has two financing options available. If you put $5000 down, you can finance the rest at 2.5% interest for 6 years. Alternatively, you could put $0 down and finance the entire purchase at 5.5% for 4 years. What are some factors that might cause you to go with the first option? Under what circumstances might you go with the second option? Which option would result in you paying less overall for the car? Completion Examples Completion examples encourage students to practice the methods taught in the lesson by providing a partial solution to an example and guiding the student to fill in the missing parts. Answers for completion examples are located at the end of the instruction for that lesson. Example 1 Prime Factorizations Find the prime factorization of 60. Solution 60=6· 10 =2·3· 2 · 5 = Now work margin exercise 1. Completion Example Answers 1. 6 · 10 = 2 · 3 · 2 · 5 = 22 · 3 · 5 23 5 2 ⋅⋅ Margin Exercises Each example has a corresponding margin exercise to test students’ understanding of what was taught in the example. Answers for the margin exercises are located at the end of the instruction for that lesson. 1. Solve: 3x + 4 = 7 Margin Exercise Answers 1. x = 1 vii Algebra for College Students: Content Highlights PrefaceConcept Check Exercises to assess students’ conceptual understanding of topics and important definitions are included in every section. 1. The slope of a line is the ratio of rise to _______. Applications Real-world application problems have been added throughout the text to challenge students to apply the concepts taught in the lesson. 11. It takes Rosa, traveling at 30 mph, 30 minutes longer to go a certain distance than it takes Melody traveling at 50 mph. Find the distance traveled. Additional Features Objectives The objectives provide students with a clear and concise list of the main concepts and methods taught in each section, enabling students to focus their time and effort on the most important topics. Objectives have corresponding labels located in the section text where the topic is introduced for ease of reference. 2.5 Applications: Number Problems and Consecutive Integers A Number Problems In this section, we will learn how to read number problems and translate the sentences and phrases in the problem into a related equation. The solution of this equation will be the solution to the problem. Example 1 Solving Number Problems If a number is decreased by 36 and the result is 76 less than twice the number, what is the number? Solution Let n = the unknown number. A number is decreased by 36 The result is 76 less than twice the number nn nn n n n nn 36276 36276 3676 3676 40 7676 The number is 40. Now work margin exercise 1. Example 2 Solving Number Problems Three times the sum of a number and 5 is equal to twice the number plus 5. Find the number. Solution Let x = the unknown number. 3525 31525 31525 155 155 22 1515 xx xx xx x x x xx 110 3 times the sum of a number and 5 Is equal to Twice the number plus 5 The number is − 10. Now work margin exercise 2. Objectives A. Use linear equations to solve number problems. B. Use linear equations to solve word problems involving consecutive integers. C. Use linear equations to solve applications. 1. If a number is decreased by 28 and the result is 92 less than twice the number, what is the number? 2. Six times the sum of a number and 3 is equal to 3 times the number plus 3. Find the number. 107 2.5 Applications: Number Problems and Consecutive Integers Examples Examples are denoted with titled headers indicating the problem-solving skill being presented. Each section contains carefully explained examples with appropriate tables, diagrams, and graphs. Examples are presented in an easy-to-understand, step-by-step fashion and are annotated with notes for additional clarification. Example 1 Multiplying Fractions Multiply: 6 7 8 5 ⋅ Solution 6 7 8 5 68 75 48 35 Now work margin exercise 1. Notes Note boxes in the margin point out important information that will help deepen students’ understanding of the topics. Often these are helpful hints about subtle details in the definitions that many students do not notice upon first glance. Note Greek mathematician Euclid is often referred to as the ‘Father of Geometry’ for his revolutionary ideas and influential textbook called Elements that he wrote around the year 300 BC. Definition Boxes Straightforward definitions are presented in highly visible boxes for easy reference. Algebra Algebra is the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, and so on, in the description of such relations. DEFINITION viii Preface Algebra for College Students: Content HighlightsCommon Errors These hard-to-miss boxes highlight common mistakes and how to avoid them. Caution Don’t forget to carry the 1! Calculators For visual learners, key strokes and screenshots are provided when appropriate for visual reference. We also provide step-by-step instructions for using a simple four-function calculator for more basic operations, as well as a TI-84 Plus calculator for graphing skills. Ǟ CALCULATORS Performing a task with a calculator Press the keys . Then press . The display will read 1738. Exercises Each section includes a variety of exercises to give the students much-needed practice applying and reinforcing the skills they learned in the section. The exercises progress from relatively easy problems to more difficult problems. Writing & Thinking This feature gives students an opportunity to independently explore and expand on concepts presented in the chapter. These questions foster a better understanding of the concepts learned within each section. 61. Explain why an alternating sequence (one in which the terms alternate between positive and negative values) cannot be an arithmetic sequence. Collaborative Learning This feature encourages students to work with others to further explore and apply concepts learned in the chapter. These questions help students realize that they see many mathematical concepts in the world around them every day. 96. The class should be divided into teams of 2 or 3 students. Each team will need access to a digital camera, a printer, and a ruler. a. Take pictures of 8 things with a defined slope. (Suggestions: A roof, a stair railing, a beach umbrella, a crooked tree, etc. Be creative!) b. Print each picture. c. Use a ruler to draw a coordinate system on Answer Key Located in the back of the book, the answer key provides answers to all odd numbered exercises in each section. This allows students to check their work to ensure that they are accurately applying the methods and skills they have learned. Chapter 3: Linear Equations and Functions 3.1 Exercises Concept Check 1. quadrants 3. III 5. origin 7. True 9. True Practice 1. AB CD E ® °° ¯ ° ° ½ ¾ °° ¿ ° ° 5133 1112 22 ,,,, ,,,, , 3. A BC DE ® °° ¯ ° ° ½ ¾ °° ¿ ° ° 32 1313 0021 ,, ,,,, ,,, 5. AB CD E ® °° ¯ ° ° ½ ¾ °° ¿ ° ° 4434 0403 41 ,,,, ,,,, , 7. AB CD E ® °° ¯ ° ° ½ ¾ °° ¿ ° ° 3514 0131 60 ,,,, ,,,, , 9. AB CD E ® °° ¯ ° ° ½ ¾ °° ¿ ° ° 5022 1406 20 ,,,, ,,,, , 11. C(0, 5)E(1, 4) B(3, 2) A(4, 1) x y D(1, 1) 13. x y C(1, 2) A(1, 2) E(3, 2)B(0, 2) D(2, 2) 15. x y D(1 1) C(2 1) A(1 0) E(0 0) B(3 0) 17. x y A(4, 1) C(1, 2)D(2, 1) B(0, 3) E(4, 2) 19. x y C(0, 1) A(1, 4) B(1, 2) D(2, 7) E(2, 5) 21. x y § ©¨ · ¹¸ 4 3 4 , 1 2 4 § ©¨ · ¹¸ A(13) 22 1 2 § ©¨ · ¹¸ B C D 23. x y D(0, 2.3) A(1.6, 2) B(3, 2.5)C(1, 1.5) 25. (0, −4 ), (2, −2 ), (4, 0 ), (1, −3 ) 27. (0, 3 ), (2, 2 ), (6, 0 ), (−2, 4 ) 29. (0, −8 ), (1, −4 ), (2, 0 ), (3, 4 ) 31. (0, 2 ), § © ¨ · ¹ ¸1 8 3 ,, (3, 0 ), (6, −2 ) 33. 0 7 4 ,, § © ¨ · ¹ ¸ (1, −1 ), 7 3 0,, § © ¨ · ¹ ¸ 3 1 2 , § © ¨ · ¹ ¸ 35. x y 0013 2626 ,,,, ,,, 37. x y 0311 2733 ,,,, ,,, 39. x y 0930 1643 ,,,, ,,, 41. x y 0245 411 5 4 ,,,, ,,, § © ¨ · ¹ ¸ 43. x y 0 9 5 30 23 4 3 1 ,,,, ,,, § © ¨ · ¹ ¸ § © ¨ · ¹ ¸ 45. x y 0520 1 15 2 45 ,,,, ,,, § © ¨ · ¹ ¸ 47. x y 1 2 3 2 1 4 34 1432 4 3 1 2 0232 0 1920 83 5202 ,,.,, .,.,.,. 49. b, c, d 51. a, d 53. a, c, d 55. For example, and 33 0135 ,, ,,,. 57. For example, and 31 0031 ,, ,,,. 59. For example, and 23 0343 ,, ,,,. 61. For example, and 44 0342 ,, ,,,. 63. For example, and 33 2013 ,, ,,,. Applications 65. a. DE 100 85 200 170 300 255 400 340 500 425 b. 500 400 300 200 100 100200300400500 E D 67. a. td 116 264 3.5 196 4256 4.5 324 5400 b. 75 150 225 300 375 450 123 4 56 d t c. The time t is squared in the equation. 991Answer Key 3.1 Exercises ix Algebra for College Students: Content Highlights PrefaceNext >