Basic Math and Pre-Algebra D S ENISE ZECSEI Franklin Lakes NJ Copyright 2006 by Denise Szecsei All rights reserved under the Pan-American and International Copyright Conventions. This book may not be reproduced, in whole or in part, in any form or by any means electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system now known or hereafter invented, without written permission from the publisher, The Career Press. HOMEWORK HELPERS: BASIC MATHAND PRE-ALGEBRA EDITEDBY JODI L. BRANDON TYPESETBY EILEEN MUNSON Cover design by Lu Rossman/Digi Dog Design NYC Printed in the U.S.A. by Book-mart Press To order this title, please call toll-free 1-800-CAREER-1 (NJ and Canada: 201-848- 0310) to order using VISA or MasterCard, or for further information on books from Career Press. The Career Press, Inc., 3 Tice Road, PO Box 687, Franklin Lakes, NJ 07417 www.careerpress.com Library of Congress Cataloging-in-Publication Data Available upon request. Acknowledgments Writing a book is a labor-intensive process, not only for the author but for everyone else involved, directly and indirectly, with the project. This book would not have been completed without the help of a special group of people. I would like to thank Michael Pye, Kristen Parkes, and everyone else at Career Press who helped transform these mathematical ideas into the object you are now holding. I am most appreciative of the efforts of Jessica Faust, who coordinated things throughout the process. I am fortunate to be part of the faculty at Stetson University. It is because of the encouragement of the University and my department that I can take the time to write about my favorite subject and hopefully make it one of yours. Stetson University’s commitment to education is mani- fested by the support given to me throughout this process. Kendelyn Michaels played a pivotal role throughout this project. Be- cause of her thorough reading of the manuscript I was able to clarify some sections and improve on my explanations. This book would not have been the same without her influence. Alic Szecsei was in charge of finding all of the typographical errors in the manuscript and checking over my solutions. Writing this book was actually a ruse to get him to solve more math problems. Thanks to my family for their help throughout the writing and editing stage. The deadline for this manuscript was just before the holidays, and their presents will reflect the time and effort they put into helping me complete this project. This page intentionally left blank Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 1: Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Lesson 1-1: Symbols Used 14 Lesson 1-2: Addition 15 Lesson 1-3: Subtraction 17 Lesson 1-4: Multiplication 19 Lesson 1-5: Properties of 0 and 1 22 Lesson 1-6: Division 23 Lesson 1-7: Prime Numbers and Factoring 25 Lesson 1-8: Order of Operations 29 Lesson 1-9: Variables and the Distributive Property 31 Chapter 2: Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Lesson 2-1: The Number Line and Absolute Value 35 Lesson 2-2: Adding Integers 37 Lesson 2-3: Subtracting Integers 40 Lesson 2-4: Multiplying Integers 41 Lesson 2-5: Dividing Integers 43 Lesson 2-6: Integers and the Distributive Property 45 Chapter 3: Operations With Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . 49 Lesson 3-1: Multiplying Fractions 50 Lesson 3-2: Simplifying Fractions 52 Lesson 3-3: Dividing Fractions 57 Lesson 3-4: Adding Fractions 59 Lesson 3-5: Subtracting Fractions 61 Lesson 3-6: Fractions and the Distributive Property 64 Chapter 4: Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Lesson 4-1: Positive Integer Powers 67 Lesson 4-2: Rules for Exponents 71 Lesson 4-3: Negative Integer Powers 74 Lesson 4-4: When the Power Is Zero 76 Lesson 4-5: Powers of Quotients and Products 77 Lesson 4-6: Square Roots and Irrational Numbers 79 Chapter 5: Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Lesson 5-1: The Decimal System 83 Lesson 5-2: Adding and Subtracting Decimal Numbers 88 Lesson 5-3: Multiplying Decimal Numbers 91 Lesson 5-4: Dividing Decimal Numbers 94 Lesson 5-5: Rounding and Estimating 96 Lesson 5-6: Comparing Decimals and Fractions 99 Chapter 6: Expressions and Equations. . . . . . . . . . . . . . . . . . . . . . . . . 105 Lesson 6-1: Evaluating Expressions 106 Lesson 6-2: Equations 108 Lesson 6-3: Equality 109 Lesson 6-4: Algebraic Properties of Equality 110 Lesson 6-5: Solving Linear Equations in 1 Step 111 Lesson 6-6: Solving Linear Equations in 2 Steps 116 Chapter 7: Ratios, Proportions, and Percents. . . . . . . . . . . . . . . . . . . 121 Lesson 7-1: Ratios and Unit Rates 121 Lesson 7-2: Proportions 124 Lesson 7-3: Proportions and Percents 128 Lesson 7-4: Percents and Equations 131 Lesson 7-5: Percent Change 133 Lesson 7-6: Markup and Discount 136 Chapter 8: Inequalities and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Lesson 8-1: Inequalities 143 Lesson 8-2: Properties of Inequalities 145 Lesson 8-3: Solving Inequalities in 1 Step 146 Lesson 8-4: Solving Inequalities in 2 Steps 150 Lesson 8-5: Graphing Inequalities Using the Number Line 152 Chapter 9: Linear Functions and Graphs. . . . . . . . . . . . . . . . . . . . . . . 159 Lesson 9-1: Relations and Functions 159 Lesson 9-2: Linear Functions and Equations 162 Lesson 9-3: The Cartesian Coordinate System 163 Lesson 9-4: Slope and Intercepts 166 Lesson 9-5: Graphing Linear Equations 172 Chapter 10: Algebra and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Lesson 10-1: Polygons 179 Lesson 10-2: Midpoint 181 Lesson 10-3: Perimeter 183 Lesson 10-4: Right Triangles and the Pythagorean Theorem 186 Lesson 10-5: Special Right Triangles 189 Lesson 10-6: The Distance Formula 192 Lesson 10-7: Area 197 Lesson 10-8: Circles 200 Chapter 11: Monomials and Polynomials. . . . . . . . . . . . . . . . . . . . . . . 205 Lesson 11-1: Adding and Subtracting Monomials 205 Lesson 11-2: Multiplying and Dividing Monomials 207 Lesson 11-3: Adding and Subtracting Polynomials 210 Lesson 11-4: Multiplying Polynomials 213 Chapter 12: Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Lesson 12-1: How to Approach Word Problems 217 Lesson 12-2: Finding Integers 218 Lesson 12-3: Rate Problems 220 Lesson 12-4: Money Problems 222 Lesson 12-5: Spatial Problems 225 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Preface Welcome to Homework Helpers: Basic Math and Pre-Algebra! Math. One word that, when spoken, can invoke extreme reactions of dread and terror. Most people would rather fight an alien invasion than work out a math problem. In this day and age, working out math prob- lems is more common than fighting alien invasions, so, although it is advantageous to be prepared for an alien attack, you will derive more benefit from developing your math skills. When children first start learning about numbers, it is fun. Counting things such as the number of candles on a birthday cake or the number of presents under the tree is exciting. There is a feeling of accomplishment in being able to count higher and faster than anyone else. But then, some- thing changes. Math stops being a game and starts to become serious. Getting the right answer becomes important, especially when it involves receiving the correct change from a twenty. Balancing a checkbook, calculating the finance charges on a credit card balance, figuring out the price of an item on sale, and budgeting your monthly expenses can take all of the fun out of working with numbers, but you can’t let that stop you from developing the skills necessary to function in our number-centered world. 9