501Algebra Questions

501Algebra Questions2nd Edition NEW YORK

Copyright 2006 LearningExpress, LLC.All rights reserved under International and Pan-American Copyright Conventions.Published in the United States by LearningExpress, LLC, New York.Library of Congress Cataloging-in-Publication Data:501 algebra questions.—2nd ed.p. cm.Rev. ed. of: 501 algebra questions / [William Recco]. 1st ed. 2002.ISBN 1-57685-552-X1. Algebra—Problems, exercises, etc. I. Recco, William. 501 algebraquestions. II. LearningExpress (Organization). III. Title: Five hundred onealgebra questions. IV. Title: Five hundred and one algebra questions.QA157.A15 2006512—dc222006040834Printed in the United States of America9 8 7 6 5 4 3 21Second EditionISBN 1-57685-552-XFor more information or to place an order, contact LearningExpress at:55 Broadway8th FloorNew York, NY 10006Or visit us

The LearningExpress Skill Builder in Focus Writing Team iscomprised of experts in test preparation, as well as educators andteachers who specialize in language arts and math.LearningExpress Skill Builder in Focus Writing TeamBrigit DermottFreelance WriterEnglish Tutor, New York CaresNew York, New YorkSandy GadeProject EditorLearningExpressNew York, New YorkKerry McLeanProject EditorMath TutorShirley, New YorkWilliam ReccoMiddle School Math Teacher, Grade 8New York Shoreham/Wading River School DistrictMath TutorSt. James, New YorkColleen SchultzMiddle School Math Teacher, Grade 8Vestal Central School DistrictMath TutorVestal, New York

ContentsIntroductionix1Working with Integers12Working with Algebraic Expressions123Combining Like Terms244Solving Basic Equations415Solving Multi-Step Equations496Solving Equations with Variables on Both Sidesof an Equation587Using Formulas to Solve Equations728Graphing Linear Equations819Solving Inequalities11010 Graphing Inequalities11911 Graphing Systems of Linear Equationsand Inequalities14212 Solving Systems of Equations Algebraically17213 Working with Exponents186

Contents14 Multiplying Polynomials19415 Factoring Polynomials20616 Using Factoring21517 Solving Quadratic Equations22918 Simplifying Radicals24219 Solving Radical Equations25020 Solving Equations with the Quadratic Formula261viii

IntroductionThis book is designed to provide you with review and practice for algebrasuccess! It is not intended to teach common algebra topics. Instead, it provides501 problems so you can flex your muscles and practice a variety of mathematical and algebraic skills. 501 Algebra Questions is designed for many audiences. It’s for anyone who has ever taken a course in algebra and needs torefresh and revive forgotten skills. It can be used to supplement current instruction in a math class. Or, it can be used by teachers and tutors who need to reinforce student skills. If, at some point, you feel you need further explanationabout some of the algebra topics highlighted in this book, you can find them inthe LearningExpress publication Algebra Success in 20 Minutes a Day.How to Use This BookFirst, look at the table of contents to see the types of algebra topics covered inthis book. The book is organized into 20 chapters with a variety of arithmetic,algebra, and word problems. The structure follows a common sequence of concepts introduced in basic algebra courses. You may want to follow the sequence,as each succeeding chapter builds on skills taught in previous chapters. But if

501 Algebra Questionsyour skills are just rusty, or if you are using this book to supplement topics you arecurrently learning, you may want to jump around from topic to topic.Chapters are arranged using the same method. Each chapter has an introductiondescribing the mathematical concepts covered in the chapter. Second, there arehelpful tips on how to practice the problems in each chapter. Last, you are presented with a variety of problems that generally range from easier to more difficultproblems and their answer explanations. In many books, you are given one modelproblem and then asked to do many problems following that model. In this book,every problem has a complete step-by-step explanation for the solutions. If you findyourself getting stuck solving a problem, you can look at the answer explanation anduse it to help you understand the problem-solving process.As you are solving problems, it is important to be as organized and sequential inyour written steps as possible. The purpose of drills and practice is to make you proficient at solving problems. Like an athlete preparing for the next season or a musician warming up for a concert, you become skillful with practice. If, aftercompleting all the problems in a section, you feel that you need more practice, dothe problems over. It’s not the answer that matters most—it’s the process and thereasoning skills that you want to master.You will probably want to have a calculator handy as you work through some ofthe sections. It’s always a good idea to use it to check your calculations. If you havedifficulty factoring numbers, the multiplication chart on the next page may helpyou. If you are unfamiliar with prime numbers, use the list on the next page so youwon’t waste time trying to factor numbers that can’t be factored. And don’t forgetto keep lots of scrap paper on hand.Make a CommitmentSuccess does not come without effort. Make the commitment to improve your algebra skills. Work for understanding. Why you do a math operation is as importantas how you do it. If you truly want to be successful, make a commitment to spendthe time you need to do a good job. You can do it! When you achieve algebra success, you have laid the foundation for future challenges and success. So sharpenyour pencil and practice!x

501 Algebra Questions 617677751823883967Multiplication 556677889960728496108Commonly 1732292813494094635416016597338098639411,013

501Algebra Questions

1Working withIntegersFor some people, it is helpful to try to simplify expressions containingsigned numbers as much as possible. When you find signed numbers withaddition and subtraction operations, you can simplify the task by changingall subtraction to addition. Subtracting a number is the same as adding itsopposite. For example, subtracting a three is the same as adding a negativethree. Or subtracting a negative 14 is the same as adding a positive 14. Asyou go through the step-by-step answer explanations, you will begin to seehow this process of using only addition can help simplify your understanding of operations with signed numbers. As you begin to gain confidence,you may be able to eliminate some of the steps by doing them in your headand not having to write them down. After all, that’s the point of practice!You work at the problems until the process becomes automatic. Then youown that process and you are ready to use it in other situations.The Tips for Working with Integers section that follows gives yousome simple rules to follow as you solve problems with integers. Refer tothem each time you do a problem until you don’t need to look at them.That’s when you can consider them yours.You will also want to review the rules for Order of Operations withnumerical expressions. You can use a memory device called a mnemonicto help you remember a set of instructions. Try remembering the wordPEMDAS. This nonsense word helps you remember to:

501 Algebra QuestionsPEMDASdo operations inside Parenthesesevaluate terms with Exponentsdo Multiplication and Division in order from left to rightAdd and Subtract terms in order from left to rightTips for Working with IntegersAdditionSigned numbers the same? Find the SUM and use the same sign. Signed numbers different? Find the DIFFERENCE and use the sign of the larger number. (The larger number is the one whose value without a positive or negativesign is greatest.)Addition is commutative. That is, you can add numbers in any order andthe result is the same. As an example, 3 5 5 3, or –2 –1 –1 –2.SubtractionChange the operation sign to addition, change the sign of the number following the operation, then follow the rules for addition.Multiplication/DivisionSigns the same? Multiply or divide and give the result a positive sign. Signsdifferent? Multiply or divide and give the result a negative sign.Multiplication is commutative. You can multiply terms in any order and theresult will be the same. For example: (2 5 7) (2 7 5) (5 2 7) (5 7 2) and so on.Evaluate the following expressions.1. 27 52. 18 20 163. 15 74. 33 165. 8 4 126. 38 2 97. 258. 5· 3 15 · 5· 9 · 29. 24 · 8 210. 2 · 3 · 711. 15 5 112

501 Algebra Questions12. (49 7) (48 4)13. 3 7 14 514. (5· 3) (12 4)15. ( 18 2) (6 · 3)16. 23 (64 16)17. 23 ( 4)218. (3 5)3 (18 6)219. 21 (11 8)320. (32 6) ( 24 8)21. A scuba diver descends 80 feet, rises 25 feet, descends 12 feet, and thenrises 52 feet where he will do a safety stop for five minutes beforesurfacing. At what depth did he do his safety stop?22. A digital thermometer records the daily high and low temperatures. Thehigh for the day was 5 C. The low was 12 C. What was the differencebetween the day’s high and low temperatures?23. A checkbook balance sheet shows an initial balance for the month of 300.During the month, checks were written in the amounts of 25, 82, 213,and 97. Deposits were made into the account in the amounts of 84 and 116. What was the balance at the end of the month?24. A gambler begins playing a slot machine with 10 in quarters in her coinbucket. She plays 15 quarters before winning a jackpot of 50 quarters. Shethen plays 20 more quarters in the same machine before walking away.How many quarters does she now have in her coin bucket?25. A glider is towed to an altitude of 2,000 feet above the ground beforebeing released by the tow plane. The glider loses 450 feet of altitudebefore finding an updraft that lifts it 1,750 feet. What is the glider’saltitude now?3

501 Algebra QuestionsAnswersNumerical expressions in parentheses like this [ ] are operations performed on only part of the originalexpression. The operations performed within these symbols are intended to show how to evaluate thevarious terms that make up the entire expression.Expressions with parentheses that look like this ( ) contain either numerical substitutions or expressions that are part of a numerical expression. Once a single number appears within these parentheses,the parentheses are no longer needed and need not be used the next time the entire expression iswritten.When two pair of parentheses appear side by side like this ( )( ), it means that the expressions withinare to be multiplied.Sometimes parentheses appear within other parentheses in numerical or algebraic expressions.Regardless of what symbol is used, ( ), { }, or [ ], perform operations in the innermost parentheses firstand work outward.Underlined expressions show the original algebraic expression as an equation with the expressionequal to its simplified result.1. The signs of the terms are different, so find the differenceof the values.The sign of the larger term is positive, so the sign ofthe result is positive.2. Change the subtraction sign to addition bychanging the sign of the number that follows it.Since all the signs are negative, add theabsolute value of the numbers.Since the signs were negative, the resultis negative.The simplified result of the numeric expressionis as follows:3. Change the subtraction sign to addition bychanging the sign of the number that follows it.Signs different? Subtract the absolute value ofthe numbers.Give the result the sign of the larger term.The simplified expression is as follows:4. Signs different? Subtract the value of the numbers.Give the result the sign of the larger term.4[27 5 22]27 5 22 18 20 ( 16)[18 20 16 54] 18 20 16 54 18 20 16 54 15 7[15 7 8] 15 7 8 15 7 8[33 16 17]33 16 17

501 Algebra Questions5. Change the subtraction sign to addition by changing8 4 128 ( 4 12)the sign of the number that follows it.With three terms, first group like terms and add.Signs the same? Add the value of the terms andgive the result the same sign.Substitute the result into the first expression.Signs different? Subtract the value of the numbers.Give the result the sign of the larger term.The simplified result of the numeric expressionis as follows:6. First divide. Signs different? Divide and give theresult the negative sign.Substitute the result into the expression.Signs different? Subtract the value of the numbers.Give the result the sign of the term with thelarger value.The simplified result of the numeric expressionis as follows:[( 4 12) 16]8 ( 16)[16 8 8]8 ( 16) 88 4 12 8[(38 2) 19]( 19) 9[19 9 10]( 19) 9 1038 2 9 107. First perform the multiplications.Signs the same? Multiply the terms and givethe result a positive sign.Signs different? Multiply the terms and givethe result a negative sign.Now substitute the results into the originalexpression.Signs different? Subtract the value of thenumbers.The simplified result of the numericexpression is as follows:[ 25 · 3 75][15 · 5 75]( 75) (

23.08.2017 · the LearningExpress publication Algebra Success in 20 Minutes a Day. How to Use This Book First, look at the table of contents to see the types of algebra topics covered in this book. The book is organized into 20 chapters with a variety of arithmetic, algebra, and word problems. The structure follows a common sequence of con-File Size: 959KBPage Count: 285