Algebra i essentials for dummies

Algebra I Essentials For Dummies® Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com Copyright © 2019 by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada 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, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 7486011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc and may not be used without written permission John Wiley & Sons, Inc is not associated with any product or vendor mentioned in this book LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ For general information on our other products and services, please contact our Customer Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993, or fax 317-572-4002 For technical support, please visit https://hub.wiley.com/community/support/dummies Wiley publishes in a variety of print and electronic formats and by printon-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Library of Congress Control Number: 2019932871 ISBN: 978-1-119-59096-5 (pbk); ISBN: 978-1-119-59098-9 (ePDF); ISBN: 978-1-119-59095-8 (ePub) Algebra I Essentials For Dummies® To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Algebra I Essentials For Dummies Cheat Sheet” in the Search box Table of Contents Cover Introduction About This Book Conventions Used in This Book Foolish Assumptions Icons Used in This Book Where to Go from Here Beyond the Book Chapter 1: Setting the Scene for Actions in Algebra Making Numbers Count Giving Meaning to Words and Symbols Operating with Signed Numbers Dealing with Decimals and Fractions Chapter 2: Examining Powers and Roots Expanding and Contracting with Exponents Exhibiting Exponent Products Taking Division to Exponents Taking on the Power of Zero Taking on the Negativity of Exponents Putting Powers to Work Circling around Square Roots Chapter 3: Ordering and Distributing: The Business of Algebra Taking Orders for Operations Dealing with Distributing Making Numbers and Variables Cooperate Making Distributions over More than One Term Chapter 4: Factoring in the First and Second Degrees Making Factoring Work Getting at the Basic Quadratic Expression Following Up on FOIL and unFOIL Making UnFOIL and the GCF Work Together Getting the Best of Binomials Chapter 5: Broadening the Factoring Horizon Grabbing onto Grouping Tackling Multiple Factoring Methods Knowing When Enough Is Enough Recruiting the Remainder Theorem Factoring Rational Expressions Chapter 6: Solving Linear Equations Playing by the Rules Solving Equations with Two Terms Taking on Three Terms Breaking Up the Groups Focusing on Fractions Changing Formulas by Solving for Variables Chapter 7: Tackling Second-Degree Quadratic Equations Recognizing Quadratic Equations Finding Solutions for Quadratic Equations Applying Factorizations Solving Three-Term Quadratics Applying Quadratic Solutions Calling on the Quadratic Formula Ignoring Reality with Imaginary Numbers Chapter 8: Expanding the Equation Horizon Queuing Up to Cubic Equations Using Synthetic Division Working Quadratic-Like Equations Rooting Out Radicals Chapter 9: Reconciling Inequalities Introducing Interval Notation Performing Operations on Inequalities Finding Solutions for Linear Inequalities Expanding to More than Two Expressions Taking on Quadratic and Rational Inequalities Chapter 10: Absolute-Value Equations and Inequalities Acting on Absolute-Value Equations Working Absolute-Value Inequalities Chapter 11: Making Algebra Tell a Story Making Plans to Solve Story Problems Finding Money and Interest Interesting Formulating Distance Problems Stirring Things Up with Mixtures Chapter 12: Putting Geometry into Story Problems Triangulating a Problem with the Pythagorean Theorem Being Particular about Perimeter Making Room for Area Problems Validating with Volume Chapter 13: Grappling with Graphing Preparing to Graph a Line Incorporating Intercepts Sliding the Slippery Slope Making Parallel and Perpendicular Lines Toe the Line Criss-Crossing Lines Turning the Curve with Curves Chapter 14: Ten Warning Signs of Algebraic Pitfalls Including the Middle Term Keeping Distributions Fair Creating Two Fractions from One Restructuring Radicals Including the Negative (or Not) Making Exponents Fractional Keeping Bases the Same Powering Up a Power Making Reasonable Reductions Catching All the Negative Exponents Index About the Author Advertisement Page Connect with Dummies End User License Agreement List of Illustrations Chapter FIGURE 9-1: A graph of the inequality FIGURE 9-2: A graph of the interval FIGURE 9-3: A number line helps you find the signs of the factors and their prod FIGURE 9-4: The sign changes at each critical number in this problem FIGURE 9-5: The and –1 are included in the solution Chapter 11 FIGURE 11-1: Visualizing containers can help with mixture problems Chapter 12 FIGURE 12-1: Triangulating the “right” way FIGURE 12-2: A shape for rooms, posters, and corrals FIGURE 12-3: Triangles come in all shapes and sizes Chapter 13 FIGURE 13-1: Pick a line — see its slope FIGURE 13-2: The y-intercept is located; use run and rise to find another point FIGURE 13-3: The simplest parabola FIGURE 13-4: Parabolas galore Introduction One of the most commonly asked questions in a mathematics classroom is, “What will I ever use this for?” Some teachers can give a good, convincing answer Others hem and haw and stare at the floor My favorite answer is, “Algebra gives you power.” Algebra gives you the power to move on to bigger and better things in mathematics Algebra gives you the power of knowing that you know something that your neighbor doesn’t know Algebra gives you the power to be able to help someone else with an algebra task or to explain to your child these logical mathematical processes Algebra is a system of symbols and rules that is universally understood, no matter what the spoken language Algebra provides a clear, methodical process that can be followed from beginning to end What power! About This Book What could be more essential than Algebra I Essentials For Dummies? In this book, you find the main points, the nitty-gritty (made spiffy-jiffy), and a format that lets you find what you need about an algebraic topic as you need it I keep the same type of organization that you find in Algebra I For Dummies, 2nd Edition, but I keep the details neat, sweet, and don’t repeat The fundamentals are here for your quick reference or, if you prefer, a more thorough perusal The choice is yours This book isn’t like a mystery novel; you don’t have to read it from beginning to end I divide the book into some general topics — from the beginning vocabulary and processes and operations to the important tool of factoring to equations and applications So you can dip into the book wherever you want, to find the information you need Conventions Used in This Book I don’t use many conventions in this book, but you should be aware of the rules for, 22–23 symbol for (√), 9, 10, 22 rational expression, factoring, 54–55 rational inequality, 113–114 rational number, real number, reciprocal, 62 rectangle, 135–138 reductions, reasonable, 160 relatively prime coefficient, 35 remainder theorem, 51–54 repeating decimal, 15 right angle, 151 roots See also radical expression double root, 72 expression rules, 22–23 in quadratic equations, 72–73 symbol for (radical), 9, 10, 22 S S rule for adding signed numbers, 11 second-degree expression See entries beginning with quadratic signed numbers, 10–14 slope, 143, 146–151 slope-intercept form, 149–151 square, 135–138 square root, 9, 22–23, 72–73 story problem distance, 126–129 geometry in, 133–142 mixture, 129–131 money and interest, 123–126 planning out, 121–122 substitution for graphing line intersect, 153 subtraction exponents, 17–18 numbers in inequalities, 107 signed numbers, 12 symbol for (−), sum, symbols || (absolute value), ≈ (approximately equal), { } (braces), 10 [ ] (brackets), 10 ÷ (division), — (fraction line), 10 ≥ (greater than or equal to), 104 > (greater than), 104 grouping, 8, 10 inequality, 104 interval notation, 104–105 ≤ (less than or equal to), 104 < (less than), 104 − (minus or negative number), * (multiplication), · (multiplication), × (multiplication), operations, 8–9 ( ) (parentheses grouping), 10 π (pi), 9, 136–137, 139, 141 + (plus or positive number), √ (square root or radical), synthetic division, 52–54, 93–95 T term, terminating decimal, 15 three-term linear equation, 62–64 three-term quadratic equation, 76–80 triangle, 134–135, 138–139 trinomial, 62–64, 76–80 two-term linear equation, 58–62 See also binomial U unFOIL, 37–41, 46–47 V variables in cubic equation factoring, 89–90 defined, 5, in distribution, 28–31 formulas, changing, 69–70 in linear equations, 63–64 vertex, 155–156 vocabulary of algebra, 7–8 volume, 140–142 W whole number, X x-axis, 144, 145 Y y-axis, 144, 145 Z zero in exponential expressions, 20 in inequalities, 108 multiplication property of, 74 About the Author Mary Jane Sterling has been an educator since graduating from college Teaching at the junior high, high school, and college levels, she has had the full span of experiences and opportunities to determine how best to explain how mathematics works She has been teaching at Bradley University in Peoria, Illinois, for the past 30 years She is also the author of Algebra I For Dummies, 2nd Edition; Algebra II For Dummies; Trigonometry For Dummies; Math Word Problems For Dummies; Business Math For Dummies; and Linear Algebra For Dummies Publisher’s Acknowledgments Project Editor: Elizabeth Kuball Senior Acquisitions Editor: Lindsay Sandman Lefevere Copy Editor: Elizabeth Kuball Assistant Editor: Erin Calligan Mooney Editorial Program Coordinator: Joe Niesen Technical Editors: Tony Bedenikovic, Michael McAsey Senior Editorial Manager: Jennifer Ehrlich Editorial Supervisor and Reprint Editor: Carmen Krikorian Editorial Assistants: Rachelle Amick, Jennette ElNaggar Senior Editorial Assistant: David Lutton Production Editor: Siddique Shaik Cover Photos: © Snvv18870020330/Shutterstock Take Dummies with you everywhere you go! Go to our Website Like us on Facebook Follow us on Twitter Watch us on YouTube Join us on LinkedIn Pin us on Pinterest Subscribe to our newsletter Create your own Dummies book cover WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... Radicals Chapter 9: Reconciling Inequalities Introducing Interval Notation Performing Operations on Inequalities Finding Solutions for Linear Inequalities Expanding to More than Two Expressions... Quadratic Formula Ignoring Reality with Imaginary Numbers Chapter 8: Expanding the Equation Horizon Queuing Up to Cubic Equations Using Synthetic Division Working Quadratic-Like Equations Rooting... applying the distribution process twice Following is an example with the steps telling you how to distribute a binomial over some polynomial Multiply using distribution: Break the binomial into its