PREPARATION COLLEGE MATHEMATICS for THIRD EDITION D. FRANKLIN WRIGHTLead 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, 2019, 2018, 2017 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: 2023936426 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-64277-626-3 AIE ISBN: 978-1-64277-627-0Table of Contents Strategies for Academic Success ............................ xiv 1.1 Introduction to Whole Numbers .........................2 1.2 Addition and Subtraction with Whole Numbers .............................................................12 1.3 Multiplication with Whole Numbers................29 1.4 Division with Whole Numbers..........................42 1.5 Rounding and Estimating with Whole Numbers .............................................................52 1.6 Problem Solving with Whole Numbers ...........67 1.7 Solving Equations with Whole Numbers: x + b = c and ax = c .............................................82 1.8 Exponents and Order of Operations ...............90 1.9 Tests for Divisibility..........................................100 1.10 Prime Numbers and Prime Factorizations .................................................106 CHAPTER 1 PROJECTS Aspiring to New Heights! ........................................118 Just Between You and Me ......................................119 2.1 Introduction to Integers ..................................122 2.2 Addition with Integers .....................................131 2.3 Subtraction with Integers ...............................140 2.4 Multiplication, Division, and Order of Operations with Integers ................................149 2.5 Simplifying and Evaluating Expressions........161 2.6 Translating English Phrases and Algebraic Expressions ......................................................170 2.7 Solving Equations with Integers: ax + b = c ...178 CHAPTER 2 PROJECTS Going to Extremes! .................................................188 Ordering Operations ..............................................189 d Proportions 3.1 Introduction to Fractions and Mixed Numbers ...........................................................192 3.2 Multiplication with Fractions .......................... 207 3.3 Division with Fractions ....................................220 3.4 Multiplication and Division with Mixed Numbers ...........................................................228 3.5 Least Common Multiple (LCM) .......................238 3.6 Addition and Subtraction with Fractions ......251 3.7 Addition and Subtraction with Mixed Numbers ...........................................................267 3.8 Comparisons and Order of Operations with Fractions ...........................................................281 3.9 Solving Equations with Fractions ...................294 3.10 Ratios and Unit Rates ....................................303 3.11 Proportions ....................................................317 3.12 Probability ......................................................330 CHAPTER 3 PROJECTS On a Budget .............................................................337 What’s Cookin’, Good Lookin’? ...............................3394.1 Introduction to Decimal Numbers .................342 4.2 Addition and Subtraction with Decimal Numbers ...........................................................353 4.3 Multiplication and Division with Decimal Numbers ...........................................................364 4.4 Estimating and Order of Operations with Decimal Numbers ............................................377 4.5 Statistics: Mean, Median, Mode, and Range .........................................................387 4.6 Decimal Numbers and Fractions ...................398 4.7 Solving Equations with Decimal Numbers ...410 CHAPTER 4 PROJECTS What Would You Weigh on the Moon? ................417 A Trip to The Grocery Store ...................................419 5.1 Basics of Percent..............................................422 5.2 Solving Percent Problems Using Proportions ......................................................436 5.3 Solving Percent Problems Using Equations ..........................................................444 5.4 Applications of Percent ...................................452 5.5 Simple and Compound Interest .....................467 5.6 Reading Graphs ...............................................485 CHAPTER 5 PROJECTS Take Me Out to the Ball Game! .............................498 How Much Will This Cell Phone Cost? ..................499 6.1 US Measurements ...........................................502 6.2 The Metric System: Length and Area ............510 6.3 The Metric System: Capacity and Weight .....521 6.4 US and Metric Equivalents ..............................530 6.5 Angles and Triangles .......................................540 6.6 Perimeter ..........................................................559 6.7 Area ...................................................................573 6.8 Volume and Surface Area ...............................587 6.9 Similar and Congruent Triangles ...................599 6.10 Square Roots and the Pythagorean Theorem .........................................................615 CHAPTER 6 PROJECTS Metric Cooking ........................................................628 This Mixtape Is Fire! ................................................629 and Inequalities 7.1 Properties of Real Numbers ...........................632 7.2 Solving Linear Equations: x + b = c and ax = c ..................................................................638 7.3 Solving Linear Equations: ax + b = c...............650 7.4 Solving Linear Equations: ax + b = cx + d ......659 7.5 Working with Formulas ...................................669 7.6 Applications: Number Problems and Consecutive Integers ....................................... 681 7.7 Applications: Distance-Rate-Time, Interest, Average, and Cost ............................................691 7.8 Solving Linear Inequalities in One Variable ..705 7.9 Compound Inequalities ..................................720 7.10 Absolute Value Equations .............................729 7.11 Absolute Value Inequalities ..........................733 CHAPTER 7 PROJECTS A Linear Vacation ....................................................740 Breaking Even ..........................................................741 iv Table of Contents g Linear Equations and Inequalities 8.1 The Cartesian Coordinate System .................744 8.2 Graphing Linear Equations in Two Variables ...................................................761 8.3 Slope-Intercept Form ......................................772 8.4 Point-Slope Form .............................................788 8.5 Introduction to Functions and Function Notation ............................................................800 8.6 Graphing Linear Inequalities in Two Variables ...................................................819 CHAPTER 8 PROJECTS What’s Your Car Worth? .........................................828 Demand and It Shall Be Supplied .........................830 9.1 Systems of Linear Equations: Solutions by Graphing ...........................................................832 9.2 Systems of Linear Equations: Solutions by Substitution ......................................................845 9.3 Systems of Linear Equations: Solutions by Addition.............................................................854 9.4 Applications: Distance-Rate-Time, Number Problems, Amounts, and Cost .......................864 9.5 Applications: Interest and Mixture ................875 9.6 Systems of Linear Equations: Three Variables ................................................883 9.7 Matrices and Gaussian Elimination ...............894 9.8 Systems of Linear Inequalities .......................907 CHAPTER 9 PROJECTS Don’t Put All Your Eggs in One Basket! .................913 Limitations of Money-Making Opportunities ......915 10.1 Rules for Exponents ......................................918 10.2 Power Rules for Exponents ..........................930 10.3 Applications: Scientific Notation ..................939 10.4 Introduction to Polynomials .........................946 10.5 Addition and Subtraction with Polynomials ....................................................956 10.6 Multiplication with Polynomials ...................964 10.7 Special Products of Binomials ......................974 10.8 Division with Polynomials .............................981 10.9 Synthetic Division and the Remainder Theorem .........................................................990 CHAPTER 10 PROJECTS Math in a Box ..........................................................997 Small but Mighty .....................................................998 g Polynomials 11.1 Greatest Common Factor (GCF) and Factoring by Grouping ................................1002 11.2 Factoring Trinomials: x2 + bx + c .................1014 11.3 Factoring Trinomials: ax2 + bx + c ..............1022 11.4 Special Factoring Techniques .....................1032 11.5 Review of Factoring Techniques ................1040 11.6 Solving Quadratic Equations by Factoring .......................................................1043 11.7 Applications: Quadratic Equations ............1054 CHAPTER 11 PROJECTS Building a Dog Pen................................................1065 Planting Seeds .......................................................1066 12.1 Introduction to Rational Expressions ........1070 12.2 Multiplication and Division with Rational Expressions ..................................................1078 12.3 Least Common Multiple of Polynomials ...1086 12.4 Addition and Subtraction with Rational Expressions ..................................................1092 12.5 Simplifying Complex Fractions ...................1104 12.6 Solving Rational Equations .........................1112 12.7 Applications: Rational Expressions ............1126 12.8 Applications: Variation ................................1137 CHAPTER 12 PROJECTS Let’s Be Rational Here! .........................................1149 You Are What You Eat ...........................................1151 v Table of Contents13.1 Evaluating Radicals ......................................1154 13.2 Simplifying Radicals .....................................1164 13.3 Rational Exponents .....................................1171 13.4 Addition, Subtraction, and Multiplication with Radicals ................................................1183 13.5 Rationalizing Denominators .......................1191 13.6 Solving Radical Equations ...........................1200 13.7 Functions with Radicals ..............................1210 13.8 Introduction to Complex Numbers ...........1221 13.9 Multiplication and Division with Complex Numbers .......................................................1228 CHAPTER 13 PROJECTS Let’s Get Radical! ...................................................1235 Rationally Increasing Precision in Population Problems ................................................................1236 tic Equations 14.1 Quadratic Equations: The Square Root Method .........................................................1238 14.2 Quadratic Equations: Completing the Square ...........................................................1248 14.3 Quadratic Equations: The Quadratic Formula .........................................................1257 14.4 More Applications of Quadratic Equations ......................................................1266 14.5 Equations in Quadratic Form .....................1278 14.6 Graphing Quadratic Functions ...................1285 14.7 More on Graphing Quadratic Functions and Applications ..........................................1294 14.8 Solving Polynomial and Rational Inequalities ...................................................1305 CHAPTER 14 PROJECTS Gateway to the West ............................................1320 Determining Product Pricing to Maximize Revenue .................................................................1321 unctions 15.1 Algebra of Functions ...................................1324 15.2 Composition of Functions and Inverse Functions ......................................................1337 15.3 Exponential Functions.................................1353 15.4 Logarithmic Functions ................................1366 15.5 Properties of Logarithms ............................1373 15.6 Common Logarithms and Natural Logarithms ...................................................1383 15.7 Logarithmic and Exponential Equations and Change-of-Base ....................................1390 15.8 Applications: Exponential and Logarithmic Functions ................................1399 CHAPTER 15 PROJECTS The Ups and Downs of Population Change .......1406 Twice as Loud but Way More than Twice as Intense ....................................................................1407 Sections 16.1 Translations and Reflections ......................1410 16.2 Parabolas as Conic Sections .......................1423 16.3 Distance Formula, Midpoint Formula, and Circles ....................................................1432 16.4 Ellipses and Hyperbolas ..............................1444 16.5 Nonlinear Systems of Equations ...............1455 CHAPTER 16 PROJECTS What’s in a Logo? ..................................................1462 Conic Sections in Medicine ..................................1463 Answer Key ................................................................. 1465 Index ............................................................................... 1525 Formula Pages ........................................................... 1534 vi Table of Contents Preface Preparation for College Mathematics: 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 Preparation for College Mathematics is to provide students with an all-in-one 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. review basic arithmetic skills, 3. develop reasoning and problem-solving skills, 4. become familiar with algebraic notation, 5. understand the connections between arithmetic and algebra, 6. develop basic algebra skills, 7. provide a smooth transition from arithmetic through prealgebra to algebra, and 8. 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. Algebra skills and topics from geometry are integrated within the discussions and problems. In particular, Chapters 1 through 5 introduce basic arithmetic skills while providing students with an introduction to solving equations. Chapter 6 provides an in-depth study of measurement and geometrical concepts (perimeter, area, volume, and so on). From Chapter 7 on, the text concentrates on developing useful algebraic skills and concepts. 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. vii Preparation for College Mathematics: Purpose and Style PrefacePreparation for College Mathematics: Content Highlights New Features Strategies for Academic Success The Strategies for Academic Success lessons have been updated to provide more of a focus on time management and stress management along with success in a math course. Other topics include 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 New projects promote collaboration and show students the practical side of mathematics through activities using real-world applications of the concepts taught in the chapter. 945 Chapter 11 Project Limitations of Money-Making Opportunities Chapter 11 Project Limitations of Money-Making Opportunities An activity to demonstrate the use of solving systems of linear equations and inequalities in real life. In this project, you will investigate how limitations (known as constraints) on the production of goods can lead to limitations on how much profit a company can make. Suppose the Soaring Eagle Book Store wishes to produce two types of coffee mugs: Type A and Type B. Each Type A mug will result in a profit of $3, and each Type B mug will result in a profit of $2.75. Manufacturing one Type A mug requires 6 minutes on Machine I and 3 minutes on Machine II. Manufacturing one Type B mug requires 4 minutes on Machine I and 6 minutes on Machine II. According to the schedule, Machine I has 12 hours available and Machine II has 10 hours available to make the mugs. How many of each type of mug should the bookstore make to maximize its profit? 1. Using the variables x and y, identify the two unknown quantities. 2. Consider the information provided about the time available on each machine. This information allows us to write two linear inequalities in the standard form Ax + By ≤ C. These are called constraints. a. Explain what each constraint represents. b. Explain why the constraints should be of the less- than-or-equal-to type. c. Write the two constraints. (Note: Be mindful of how the times are expressed, minutes versus hours. You’ll need to convert to make units consistent throughout.) 3. Regardless of maximizing profit, there are two more “common sense” constraints. Consider whether there is a minimum or maximum number of mugs. Using this information, write two more constraints and explain why they are constraints. 4. Graph the system of linear inequalities resulting from the four constraints. (Hint: You may wish to first graph the two inequalities from Problem 3, shading lightly. Once you’ve identified the intersection, erase extra shading. Then add the other two constraints found in Problem 2 to the graph and, once you’ve identified the new (smaller) intersection, erase extra shading again.) 5. The final result of graphing the four constraints should be a four-cornered region. Use systems of equations to solve for the (x, y) coordinates of the four corners. 6. Recall the profit that the bookstore can make from each type of mug. Write an equation of the form P = Ax + By, with A and B filled in but not P. 7. Consider the following six values of P: −400, −200, 0, 200, 400, and 600. Using your answer to Problem 6, lightly graph six lines for these six different values of P over the shaded four-cornered region from Problem 5. (Hint: These six lines will be parallel.) 8. Would all of the lines created in Problem 7 indicate a possible value of P? Explain why or why not. 9. Is there a maximum value of P or can it keep growing? Why or why not? 10. Let’s determine the maximum profit that Soaring Eagle Book Store can make from selling mugs. a. If there is a maximum value of P, at what point does it seem to occur? b. Is there anything special about the point found in part a.? c. Substitute this point into your P = Ax + By equation from Problem 6 to calculate the maximum profit. Express your final answer in words; that is, state how many mugs of each type should be made to produce the maximum profit and what that maximum profit is. Chapter 15 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 APr n 1, 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. 1235 Chapter 15 Project Let’s Get Radical! CHAPTER 15 PROJECTS 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. 521 7.2 Reading Graphs 7.2 Reading Graphs A Introduction to Graphs Graphs are pictures of numerical information. Graphs appear almost daily in newspapers and magazines and frequently in textbooks and corporate reports. Well-drawn graphs can organize and communicate information accurately, effectively, and fast. Most computers can be programmed to draw graphs, and anyone whose work involves a computer in any way will probably be expected to understand graphs and even to create graphs. There are many different types of graphs, each type particularly well-suited to the display and clarification of certain types of information. There have been various graphs used throughout the text. In this section, we will discuss in more detail the uses of bar graphs, circle graphs, line graphs, and histograms. Four Types of Graphs and Their Purposes 1. Bar Graphs: To emphasize comparative amounts 2. Circle Graphs: To help in understanding percents or parts of a whole (Circle graphs are also called pie charts.) 3. Line Graphs: To indicate tendencies or trends over a period of time 4. Histograms: To indicate data in classes (a range or interval of numbers) DEFINITION A common characteristic of all graphs is that they are intended to communicate information about numerical data quickly and easily. With this in mind, note the following three properties of all graphs. Properties of Graphs Every graph should 1. be clearly labeled, 2. be easy to read, and 3. have an appropriate title. PROPERTIES Objectives A. Learn the purposes and properties of graphs. B. Read bar graphs. C. Read circle graphs. D. Read line graphs. E. Read histograms. @ Math Tip Not all graphs are created equal. Since the purpose of a graph is to communicate information in a concise manner, be sure that the type of graph chosen presents the information clearly. For example, if a bar graph or a circle graph might work, consider whether you want to compare total amounts (bar graph) or if you want to compare how the data is made up as a whole (circle graph). 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 The Swiss psychologist Jean Piaget once said, “Logic and mathematics are nothing other than specialized linguistic structures.” Variables are used as place holders for numerical values in algebraic expressions and equations. Expressions and equations can be seen as tools that allow us to record, reuse, and communicate important relationships using a universally understood language. For example, in the United States, we commonly measure temperature in degrees Fahrenheit (°F). In this temperature scale, 32 °F is the freezing temperature of water while 212 °F is the boiling point. Most of the rest of the world uses the Celsius scale (°C), where water freezes at 0 °C and boils at 100 °C. The two scales are related by the following formula. CF 5 9 32 According to the National Oceanic and Atmospheric Administration, the ocean water temperature at Conimicut Lighthouse in Rhode Island was 39.9 °F on February 2, 2020. How can this formula be used to determine the water temperature in degree Celsius? viii Preface Preparation for College Mathematics: Content HighlightsExamples 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. 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 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 ix Preparation for College Mathematics: Content Highlights PrefaceNext >