Elayn MartinGay's developmental math textbooks and video resources are motivated by her firm belief that every stu
963 131 36MB
English Pages vi, 936 pages: colour illustrations; 28 cm [945] Year 2013;2014
Table of contents :
Cover......Page 1
Table of Contents......Page 4
1. Review of Real Numbers......Page 10
1. Tips for Success in Mathematics......Page 11
2. Symbols and Sets of Numbers......Page 16
3. Fractions and Mixed Numbers......Page 25
4. Exponents, Order of Operations, Variable Expressions, and Equations......Page 34
5. Adding Real Numbers......Page 44
6. Subtracting Real Numbers......Page 52
Integrated Review—Operations on Real Numbers......Page 59
7. Multiplying and Dividing Real Numbers......Page 60
8. Properties of Real Numbers......Page 70
Chapter Highlights......Page 77
Chapter Review......Page 81
Chapter Test......Page 84
2. Equations, Inequalities, and Problem Solving......Page 91
1. Simplifying Algebraic Expressions......Page 92
2. The Addition and Multiplication Properties of Equality......Page 100
3. Solving Linear Equations......Page 110
Integrated Review—Solving Linear Equations......Page 118
4. An Introduction to Problem Solving......Page 119
5. Formulas and Problem Solving......Page 130
6. Percent and Mixture Problem Solving......Page 141
7. Further Problem Solving......Page 153
8. Solving Linear Equations......Page 160
Chapter Highlights......Page 172
Chapter Review......Page 177
Chapter Test......Page 180
3. Graphing......Page 187
1. Reading Graphs and the Rectangular Coordinate System......Page 188
2. Graphing Linear Equations......Page 203
3. Intercepts......Page 213
4. Slope and Rate Change......Page 221
Integrated Review—Summary on Slope and Graphing Linear Equations......Page 235
5. Equations of Lines......Page 236
6. Functions......Page 245
Chapter Highlights......Page 257
Chapter Review......Page 261
Chapter Test......Page 264
4. Solving Systems of Linear Equations......Page 274
1. Solving Systems of Linear Equations by Graphing......Page 275
2. Solving Systems of Linear Equations by Substitution......Page 283
3. Solving Systems of Linear Equations by Addition......Page 290
Integrated Review—Solving Systems of Equations......Page 297
4. Solving Systems of Linear Equations in Three Variables......Page 298
5. Systems of Linear Equations and Problem Solving......Page 305
Chapter Highlights......Page 323
Chapter Review......Page 326
Chapter Test......Page 328
5. Exponents and Polynomials......Page 335
1. Exponents......Page 336
2. Polynomial Functions and Adding and Subtracting Polynomials......Page 347
3. Multiplying Polynomials......Page 359
4. Special Products......Page 366
5. Negative Exponents and Scientific Notation......Page 373
6. Dividing Polynomials......Page 382
7. Synthetic Division and the Remainder Theorem......Page 389
Chapter Vocabulary Check......Page 393
Chapter Highlights......Page 394
Chapter Review......Page 396
Chapter Test......Page 399
6. Factoring Polynomials......Page 406
1. The Greatest Common Factor and Factoring by Grouping......Page 407
2. Factoring Trinomials of the Form x2 + bx + c......Page 415
3. Factoring Trinomials of the Form ax2 + bx + c and Perfect Square Trinomials......Page 422
4. Factoring Trinomials of the Form ax2 + bx + c by Grouping......Page 430
5. Factoring Binomials......Page 435
Integrated Review Choosing a Factoring Strategy......Page 442
6. Solving Quadratic Equations by Factoring......Page 445
7. Quadratic Equations and Problem Solving......Page 454
Chapter Vocabulary Check......Page 463
Chapter Highlights......Page 464
Chapter Review......Page 467
Chapter Test......Page 469
7. Rational Expressions......Page 474
1. Rational Functions and Simplifying Rational Expressions......Page 475
2. Multiplying and Dividing Rational Expressions......Page 486
3. Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator......Page 495
4. Adding and Subtracting Rational Expressions with Unlike Denominators......Page 503
5. Solving Equations Containing Rational Expressions......Page 509
Integrated Review—Summary on Rational Expressions......Page 516
6. Proportion and Problem Solving with Rational Equations......Page 517
7. Simplifying Complex Fractions......Page 530
Chapter Vocabulary Check......Page 536
Chapter Highlights......Page 537
Chapter Review......Page 540
Chapter Test......Page 543
8. More on Functions and Graphs......Page 549
1. Graphing and Writing Linear Functions......Page 550
2. Reviewing Function Notation and Graphing Nonlinear Functions......Page 558
Integrated Review—Summary on Functions and Equations of Lines......Page 566
3. Graphing PiecewiseDefined Functions and Shifting and Reflecting Graphs of Functions......Page 567
4. Variation and Problem Solving......Page 575
Chapter Vocabulary Check......Page 583
Chapter Highlights......Page 584
Chapter Review......Page 586
Chapter Test......Page 587
9. Inequalities and Absolute Value......Page 594
1. Compound Inequalities......Page 595
2. Absolute Value Equations......Page 602
3. Absolute Value Inequalities......Page 607
4. Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities......Page 613
Chapter Vocabulary Check......Page 622
Chapter Highlights......Page 623
Chapter Review......Page 625
Chapter Test......Page 626
10. Rational Exponents, Radicals, and Complex Numbers......Page 631
1. Radicals and Radical Functions......Page 632
2. Rational Exponents......Page 641
3. Simplifying Radical Expressions......Page 648
4. Adding, Subtracting, and Multiplying Radical Expressions......Page 656
5. Rationalizing Denominators and Numerators of Radical Expressions......Page 662
Integrated Review—Radicals and Rational Exponents......Page 668
6. Radical Equations and Problem Solving......Page 669
7. Complex Numbers......Page 679
Chapter Highlights......Page 686
Chapter Review......Page 690
Chapter Test......Page 692
11. Quadratic Equations and Functions......Page 698
1. Solving Quadratic Equations by Completing the Square......Page 699
2. Solving Quadratic Equations by the Quadratic Formula......Page 709
3. Solving Equations by Using Quadratic Methods......Page 719
Integrated Review—Summary on Solving Quadratic Equations......Page 728
4. Nonlinear Inequalities in One Variable......Page 729
5. Quadratic Functions and Their Graphs......Page 736
6. Further Graphing of Quadratic Functions......Page 744
Chapter Highlights......Page 752
Chapter Review......Page 755
Chapter Test......Page 756
12. Exponential and Logarithmic Functions......Page 765
1. The Algebra of Function; Composite Functions......Page 766
2. Inverse Functions......Page 771
3. Exponential Functions......Page 782
4. Exponential Growth and Decay Functions......Page 791
5. Logarithmic Functions......Page 795
6. Properties of Logarithms......Page 803
Integrated Review—Functions and Properties of Logarithms......Page 809
7. Common Logarithms, Natural Logarithms, and Change of Base......Page 810
8. Exponential and Logarithmic Equations and Problem Solving......Page 816
Chapter Vocabulary Check......Page 822
Chapter Highlights......Page 823
Chapter Review......Page 826
Chapter Test......Page 828
13. Conic Sections......Page 836
1. The Parabola and the Circle......Page 837
2. The Ellipse and the Hyperbola......Page 846
Integrated Review—Graphing Conic Sections......Page 853
3. Solving Nonlinear Systems of Equations......Page 854
4. Nonlinear Inequalities and Systems of Inequalities......Page 859
Chapter Highlights......Page 863
Chapter Review......Page 866
Chapter Test......Page 867
14. Appendix: Operations on Decimals/Table of Percent, Decimal, and Fraction Equivalents......Page 873
15. Appendix: Review of Algebra Topics......Page 878
1. Equations (Linear and Quadratic Solved by Factoring)......Page 879
2. Problem Solving......Page 882
3. Graphing......Page 891
4. Polynomials and Factoring......Page 895
5. Rational Expressions......Page 899
16. Appendix: An Introduction to Using a Graphing Utility......Page 905
17. Appendix: Solving Systems of Equations by Matrices......Page 910
18. Appendix: Solving Systems of Equations by Determinants......Page 917
19. Appendix: Mean, Median, and Mode......Page 926
20. Appendix: Review of Angles, Lines, and Special Triangles......Page 930
C......Page 938
F......Page 939
L......Page 940
N......Page 941
Q......Page 942
S......Page 943
Y......Page 944
Z......Page 945
Beginning and Intermediate Algebra MartinGay Fifth Edition
ISBN 9781292022871
9 781292 022871
Beginning and Intermediate Algebra Elayn MartinGay Fifth Edition
Beginning and Intermediate Algebra Elayn MartinGay Fifth Edition
Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 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 either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any afﬁliation with or endorsement of this book by such owners.
ISBN 10: 1292022876 ISBN 13: 9781292022871
British Library CataloguinginPublication Data A catalogue record for this book is available from the British Library Printed in the United States of America
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Table of Contents 1. Review of Real Numbers Elayn MartinGay 1. Tips for Success in Mathematics
1 2
2. Symbols and Sets of Numbers
7
3. Fractions and Mixed Numbers
16
4. Exponents, Order of Operations, Variable Expressions, and Equations
25
5. Adding Real Numbers
35
6. Subtracting Real Numbers
43
Integrated Review—Operations on Real Numbers
50
7. Multiplying and Dividing Real Numbers
51
8. Properties of Real Numbers
61
Chapter Vocabulary Check
68
Chapter Highlights
68
Chapter Review
72
Chapter Test
75
2. Equations, Inequalities, and Problem Solving Elayn MartinGay 1. Simplifying Algebraic Expressions
81 83
2. The Addition and Multiplication Properties of Equality
91
3. Solving Linear Equations
101
Integrated Review—Solving Linear Equations
109
4. An Introduction to Problem Solving
110
5. Formulas and Problem Solving
121
6. Percent and Mixture Problem Solving
132
I
7. Further Problem Solving
144
8. Solving Linear Equations
151
Chapter Vocabulary Check
163
Chapter Highlights
163
Chapter Review
168
Chapter Test
171
3. Graphing Elayn MartinGay 1. Reading Graphs and the Rectangular Coordinate System
177 179
2. Graphing Linear Equations
194
3. Intercepts
204
4. Slope and Rate Change
212
Integrated Review—Summary on Slope and Graphing Linear Equations
226
5. Equations of Lines
227
6. Functions
236
Chapter Vocabulary Check
248
Chapter Highlights
248
Chapter Review
252
Chapter Test
255
4. Solving Systems of Linear Equations Elayn MartinGay 1. Solving Systems of Linear Equations by Graphing
265 266
2. Solving Systems of Linear Equations by Substitution
274
3. Solving Systems of Linear Equations by Addition
281
Integrated Review—Solving Systems of Equations
288
4. Solving Systems of Linear Equations in Three Variables
289
5. Systems of Linear Equations and Problem Solving
296
Chapter Vocabulary Check
314
Chapter Highlights
314
Chapter Review
317
Chapter Test
319
5. Exponents and Polynomials Elayn MartinGay 1. Exponents
II
325 327
2. Polynomial Functions and Adding and Subtracting Polynomials
338
3. Multiplying Polynomials
350
4. Special Products
357
Integrated Review—Exponents and Operations on Polynomials
364
5. Negative Exponents and Scientific Notation
364
6. Dividing Polynomials
373
7. Synthetic Division and the Remainder Theorem
380
Chapter Vocabulary Check
384
Chapter Highlights
385
Chapter Review
387
Chapter Test
390
6. Factoring Polynomials Elayn MartinGay 1. The Greatest Common Factor and Factoring by Grouping
397 398
2. Factoring Trinomials of the Form x2 + bx + c
406
3. Factoring Trinomials of the Form ax2 + bx + c and Perfect Square Trinomials
413
4. Factoring Trinomials of the Form ax2 + bx + c by Grouping
421
5. Factoring Binomials
426
Integrated Review Choosing a Factoring Strategy
433
6. Solving Quadratic Equations by Factoring
436
7. Quadratic Equations and Problem Solving
445
Chapter Vocabulary Check
454
Chapter Highlights
455
Chapter Review
458
Chapter Test
460
7. Rational Expressions Elayn MartinGay 1. Rational Functions and Simplifying Rational Expressions
465 466
2. Multiplying and Dividing Rational Expressions
477
3. Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator
486
4. Adding and Subtracting Rational Expressions with Unlike Denominators
494
5. Solving Equations Containing Rational Expressions
500
Integrated Review—Summary on Rational Expressions
507
6. Proportion and Problem Solving with Rational Equations
508
III
7. Simplifying Complex Fractions
521
Chapter Vocabulary Check
527
Chapter Highlights
528
Chapter Review
531
Chapter Test
534
8. More on Functions and Graphs Elayn MartinGay 1. Graphing and Writing Linear Functions
539 541
2. Reviewing Function Notation and Graphing Nonlinear Functions
549
Integrated Review—Summary on Functions and Equations of Lines
557
3. Graphing PiecewiseDefined Functions and Shifting and Reflecting Graphs of Functions
558
4. Variation and Problem Solving
566
Chapter Vocabulary Check
574
Chapter Highlights
575
Chapter Review
577
Chapter Test
578
9. Inequalities and Absolute Value Elayn MartinGay 1. Compound Inequalities
585 586
2. Absolute Value Equations
593
3. Absolute Value Inequalities
598
Integrated Review—Solving Compound Inequalities and Absolute Value Equations and Inequalities
604
4. Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities
604
Chapter Vocabulary Check
613
Chapter Highlights
614
Chapter Review
616
Chapter Test
617
10. Rational Exponents, Radicals, and Complex Numbers Elayn MartinGay
IV
1. Radicals and Radical Functions
621 623
2. Rational Exponents
632
3. Simplifying Radical Expressions
639
4. Adding, Subtracting, and Multiplying Radical Expressions
647
5. Rationalizing Denominators and Numerators of Radical Expressions
653
Integrated Review—Radicals and Rational Exponents
659
6. Radical Equations and Problem Solving
660
7. Complex Numbers
670
Chapter Vocabulary Check
677
Chapter Highlights
677
Chapter Review
681
Chapter Test
683
11. Quadratic Equations and Functions Elayn MartinGay 1. Solving Quadratic Equations by Completing the Square
689 690
2. Solving Quadratic Equations by the Quadratic Formula
700
3. Solving Equations by Using Quadratic Methods
710
Integrated Review—Summary on Solving Quadratic Equations
719
4. Nonlinear Inequalities in One Variable
720
5. Quadratic Functions and Their Graphs
727
6. Further Graphing of Quadratic Functions
735
Chapter Vocabulary Check
743
Chapter Highlights
743
Chapter Review
746
Chapter Test
747
12. Exponential and Logarithmic Functions Elayn MartinGay 1. The Algebra of Function; Composite Functions
755 757
2. Inverse Functions
762
3. Exponential Functions
773
4. Exponential Growth and Decay Functions
782
5. Logarithmic Functions
786
6. Properties of Logarithms
794
Integrated Review—Functions and Properties of Logarithms
800
7. Common Logarithms, Natural Logarithms, and Change of Base
801
8. Exponential and Logarithmic Equations and Problem Solving
807
Chapter Vocabulary Check
813
Chapter Highlights
814
Chapter Review
817
V
Chapter Test
819
13. Conic Sections Elayn MartinGay 1. The Parabola and the Circle
827 828
2. The Ellipse and the Hyperbola
837
Integrated Review—Graphing Conic Sections
844
3. Solving Nonlinear Systems of Equations
845
4. Nonlinear Inequalities and Systems of Inequalities
850
Chapter Vocabulary Check
854
Chapter Highlights
854
Chapter Review
857
Chapter Test
858
14. Appendix: Operations on Decimals/Table of Percent, Decimal, and Fraction Equivalents Elayn MartinGay
863
15. Appendix: Review of Algebra Topics Elayn MartinGay 1. Equations (Linear and Quadratic Solved by Factoring)
869 870
2. Problem Solving
873
3. Graphing
882
4. Polynomials and Factoring
886
5. Rational Expressions
890
16. Appendix: An Introduction to Using a Graphing Utility Elayn MartinGay
895
17. Appendix: Solving Systems of Equations by Matrices Elayn MartinGay
901
18. Appendix: Solving Systems of Equations by Determinants Elayn MartinGay
907
19. Appendix: Mean, Median, and Mode Elayn MartinGay
917
20. Appendix: Review of Angles, Lines, and Special Triangles
VI
Elayn MartinGay
921
Index
929
Review of Real Numbers National Park Service Regions Midwest
1
Tips for Success in Mathematics
2
Symbols and Sets of Numbers
3
Fractions and Mixed Numbers
4
Exponents, Order of Operations, Variable Expressions, and Equations
5
Adding Real Numbers
6
Subtracting Real Numbers
Northeast
Pacific West National Capital Southeast
Intermountain Alaska
The National Park Service (NPS) is a federal agency that manages all national parks and many other historic monuments and properties. The NPS has 21,989 employees and had a total of 281 million recreational visitors in 2010 alone. The map above shows the different geographic regions of the NPS, and the bar graph below shows the number of recreational visits per year shown. In Section 2, Exercises 71 through 76, we shall study this bar graph further.
Number of Visitors (in millions)
Total Annual Recreational Visitors to U.S. National Park System 300 290 280
286 280
277
270
281
280
277 273
273
2005
2006
Integrated Review– Operations on Real Numbers
7
Multiplying and Dividing Real Numbers
8
Properties of Real Numbers
In this chapter, we review the basic symbols and words—the language— of arithmetic and introduce using variables in place of numbers. This is our starting place in the study of algebra.
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266
260 250 2001
2002
2003
2004
2007
2008
2009
2010
Year
From Chapter 1 of Beginning and Intermediate Algebra, Fifth Edition. Elayn MartinGay. Copyright © 2013 by Pearson Education, Inc. All rights reserved.
1
Review of Real Numbers
1
Tips for Success in Mathematics
OBJECTIVES 1 Get Ready for This Course. 2 Understand Some General Tips for Success.
3 Get Help as Soon as You Need It.
4 Learn How to Prepare for and Take an Exam.
5 Develop Good Time Management.
Before reading this section, remember that your instructor is your best source of information. Please see your instructor for any additional help or information. OBJECTIVE
Getting Ready for This Course 1 Now that you have decided to take this course, remember that a positive attitude will make all the difference in the world. Your belief that you can succeed is just as important as your commitment to this course. Make sure you are ready for this course by having the time and positive attitude that it takes to succeed. Next, make sure that you have scheduled your math course at a time that will give you the best chance for success. For example, if you are also working, you may want to check with your employer to make sure that your work hours will not conflict with your course schedule. On the day of your first class period, doublecheck your schedule and allow yourself extra time to arrive on time in case of traffic problems or difficulty locating your classroom. Make sure that you bring at least your textbook, paper, and a writing instrument. Are you required to have a lab manual, graph paper, calculator, or some other supplies besides this text? If so, also bring this material with you. OBJECTIVE
2
Helpful Hint MyMathLab® and MathXL® If you are doing your homework online, you can work and rework those exercises that you struggle with until you master them. Try working through all the assigned exercises twice before the due date.
Helpful Hint MyMathLab® and MathXL® If you are completing your homework online, it’s important to work each exercise on paper before submitting the answer. That way, you can check your work and follow your steps to find and correct any mistakes.
2
General Tips for Success
Following are some general tips that will increase your chance for success in a mathematics class. Many of these tips will also help you in other courses you may be taking. Exchange names and phone numbers or email addresses with at least one other person in class. This contact person can be a great help if you miss an assignment or want to discuss math concepts or exercises that you find difficult. Choose to attend all class periods. If possible, sit near the front of the classroom. This way, you will see and hear the presentation better. It may also be easier for you to participate in classroom activities. Do your homework. You’ve probably heard the phrase “practice makes perfect” in relation to music and sports. It also applies to mathematics. You will find that the more time you spend solving mathematics exercises, the easier the process becomes. Be sure to schedule enough time to complete your assignments before the next due date assigned by your instructor. Check your work. Review the steps you made while working a problem. Learn to check your answers in the original problems. You may also compare your answers with the “Answers to Selected Exercises” section in the back of the book. If you have made a mistake, try to figure out what went wrong. Then correct your mistake. If you can’t find what went wrong, don’t erase your work or throw it away. Bring your work to your instructor, a tutor in a math lab, or a classmate. It is easier for someone to find where you had trouble if he or she looks at your original work. Learn from your mistakes and be patient with yourself. Everyone, even your instructor, makes mistakes. (That definitely includes me—Elayn MartinGay.) Use your errors to learn and to become a better math student. The key is finding and understanding your errors. Was your mistake a careless one, or did you make it because you can’t read your own math writing? If so, try to work more slowly or write more neatly and make a conscious effort to check your work carefully. Did you make a mistake because you don’t understand a concept? Take the time to review the concept or ask questions to understand it better. Did you skip too many steps? Skipping steps or trying to do too many steps mentally may lead to preventable mistakes. Know how to get help if you need it. It’s all right to ask for help. In fact, it’s a good idea to ask for help whenever there is something that you don’t understand. Make sure you know when your instructor has office hours and how to find his or her office. Find
Review of Real Numbers
Helpful Hint MyMathLab® and MathXL® When assignments are turned in online, keep a hard copy of your complete written work. You will need to refer to your written work to be able to ask questions and to study for tests later.
Helpful Hint MyMathLab® and MathXL® Be aware of assignments and due dates set by your instructor. Don’t wait until the last minute to submit work online. Allow 6–8 hours before the deadline in case you have technology trouble.
Helpful Hint MyMathLab® and MathXL® • Use the Help Me Solve This button to get stepbystep help for the exercise you are working. You will need to work an additional exercise of the same type before you can get credit for having worked it correctly. • Use the Video button to view a video clip of the author working a similar exercise.
out whether math tutoring services are available on your campus. Check on the hours, location, and requirements of the tutoring service. Organize your class materials, including homework assignments, graded quizzes and tests, and notes from your class or lab. All of these items will make valuable references throughout your course and when studying for upcoming tests and the final exam. Make sure that you can locate these materials when you need them. Read your textbook before class. Reading a mathematics textbook is unlike reading a novel or a newspaper. Your pace will be much slower. It is helpful to have paper and a pencil with you when you read. Try to work out examples on your own as you encounter them in your text. You should also write down any questions that you want to ask in class. When you read a mathematics textbook, sometimes some of the information in a section will be unclear. But after you hear a lecture or watch a lecture video on that section, you will understand it much more easily than if you had not read your text beforehand. Don’t be afraid to ask questions. You are not the only person in class with questions. Other students are normally grateful that someone has spoken up. Turn in assignments on time. This way, you can be sure that you will not lose points for being late. Show every step of a problem and be neat and organized. Also be sure that you understand which problems are assigned for homework. If allowed, you can always doublecheck the assignment with another student in your class.
OBJECTIVE
3 Getting Help If you have trouble completing assignments or understanding the mathematics, get help as soon as you need it! This tip is presented as an objective on its own because it is so important. In mathematics, usually the material presented in one section builds on your understanding of the previous section. This means that if you don’t understand the concepts covered during a class period, there is a good chance that you will not understand the concepts covered during the next class period. If this happens to you, get help as soon as you can. Where can you get help? Many suggestions have been made in this section on where to get help, and now it is up to you to get it. Try your instructor, a tutoring center, or a math lab, or you may want to form a study group with fellow classmates. If you do decide to see your instructor or go to a tutoring center, make sure that you have a neat notebook and are ready with your questions.
3
Review of Real Numbers OBJECTIVE
Helpful Hint MyMathLab® and MathXL® Review your written work for previous assignments. Then, go back and rework previous assignments. Open a previous assignment and click Similar Exercise to generate new exercises. Rework the exercises until you fully understand them and can work them without help features.
4 Preparing for and Taking an Exam Make sure that you allow yourself plenty of time to prepare for a test. If you think that you are a little “math anxious,” it may be that you are not preparing for a test in a way that will ensure success. The way that you prepare for a test in mathematics is important. To prepare for a test: 1. Review your previous homework assignments. 2. Review any notes from class and sectionlevel quizzes you have taken. (If this is a final exam, also review chapter tests you have taken.) 3. Review concepts and definitions by reading the Chapter Highlights at the end of this chapter. 4. Practice working out exercises by completing the Chapter Review found at the end of this chapter. 5. It is important to place yourself in conditions similar to test conditions to find out how you will perform. In other words, as soon as you feel that you know the material, get a few blank sheets of paper and take a sample test. A Chapter Test is available at the end of this chapter, or you can work selected problems from the Chapter Review. Your instructor may also provide you with a review sheet. During this sample test, do not use your notes or your textbook. Then check your sample test. If you are not satisfied with the results, study the areas that you are weak in and try again. 6. On the day of the test, allow yourself plenty of time to arrive where you will be taking your exam. When taking your test: 1. Read the directions on the test carefully. 2. Read each problem carefully as you take the test. Make sure that you answer the question asked. 3. Watch your time and pace yourself so that you can attempt each problem on your test. 4. If you have time, check your work and answers. 5. Do not turn your test in early. If you have extra time, spend it doublechecking your work.
4
Review of Real Numbers OBJECTIVE
5
Managing Your Time
As a college student, you know the demands that classes, homework, work, and family place on your time. Some days, you probably wonder how you’ll ever get everything done. One key to managing your time is developing a schedule. Here are some hints for making a schedule:
Stylephotographs/Dreamstime
1
1. Make a list of all your weekly commitments for the term. Include classes, work, regular meetings, extracurricular activities, etc. You may also find it helpful to list such things as laundry, regular workouts, grocery shopping, etc. 2. Next, estimate the time needed for each item on the list. Also make a note of how often you will need to do each item. Don’t forget to include time estimates for the reading, studying, and homework you do outside of your classes. You may want to ask your instructor for help estimating the time needed. 3. In the exercise set that follows, you are asked to block out a typical week on the schedule grid given. Start with items with fixed time slots like classes and work. 4. Next, include the items on your list with flexible time slots. Think carefully about how best to schedule items such as study time. 5. Don’t fill up every time slot on the schedule. Remember that you need to allow time for eating, sleeping, and relaxing! You should also allow a little extra time in case some items take longer than planned. 6. If you find that your weekly schedule is too full for you to handle, you may need to make some changes in your workload, classload, or other areas of your life. You may want to talk to your advisor, manager or supervisor at work, or someone in your college’s academic counseling center for help with such decisions.
Exercise Set
1. What is your instructor’s name? 2. What are your instructor’s office location and office hours? 3. What is the best way to contact your instructor? 4. Do you have the name and contact information of at least one other student in class? 5. Will your instructor allow you to use a calculator in this class? 6. Why is it important that you write stepbystep solutions to homework exercises and keep a hard copy of all work submitted?
7. Is a tutoring service available on campus? If so, what are its hours? What services are available? 8. Have you attempted this course before? If so, write down ways that you might improve your chances of success during this second attempt. 9. List some steps that you can take if you begin having trouble understanding the material or completing an assignment. If you are completing your homework in MyMathLab® and MathXL®, list the resources you can use for help.
5
Review of Real Numbers 10. How many hours of studying does your instructor advise for each hour of instruction? 11. How many times is it suggested that you work through the homework exercises in MathXL® before the submission deadline? 12. How far in advance of the assigned due date is it suggested that homework be submitted online? Why? 13. Chapter Highlights are found at the end of this chapter. Find the Chapter Highlights and explain how you might use it and how it might be helpful.
Monday
Tuesday
14. Chapter Reviews are found at the end of this chapter. Find the Chapter Review and explain how you might use it and how it might be useful. 15. Chapter Tests are at the end of this chapter. Find the Chapter Test and explain how you might use it and how it might be helpful when preparing for an exam on this chapter. If you are working in MyMathLab® and MathXL®, how can you use previous homework assignments to study? 16. Read or reread objective 5 and fill out the schedule grid below.
Wednesday
Thursday
Friday
Saturday
Sunday
1:00 a.m. 2:00 a.m. 3:00 a.m. 4:00 a.m. 5:00 a.m. 6:00 a.m. 7:00 a.m. 8:00 a.m. 9:00 a.m. 10:00 a.m. 11:00 a.m. Noon 1:00 p.m. 2:00 p.m. 3:00 p.m. 4:00 p.m. 5:00 p.m. 6:00 p.m. 7:00 p.m. 8:00 p.m. 9:00 p.m. 10:00 p.m. 11:00 p.m. Midnight
6 M01_MART5121_05_SE_C01_PCL.indd 6
28/03/12 2:34 AM
Review of Real Numbers
2
Symbols and Sets of Numbers OBJECTIVE
OBJECTIVES 1 Use a Number Line to Order Numbers.
2 Translate Sentences into
1
Using a Number Line to Order Numbers
We begin with a review of the set of natural numbers and the set of whole numbers and how we use symbols to compare these numbers. A set is a collection of objects, each of which is called a member or element of the set. A pair of brace symbols 5 6 encloses the list of elements and is translated as “the set of” or “the set containing.”
Mathematical Statements.
3 Identify Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers.
4 Find the Absolute Value of a Real Number.
0 1
2
3 4
5
A number line
Natural Numbers The set of natural numbers is 5 1, 2, 3, 4, 5, 6, c6 . Whole Numbers The set of whole numbers is 5 0, 1, 2, 3, 4, c6 .
The three dots (an ellipsis) means that the list continues in the same manner indefinitely.
These numbers can be pictured on a number line. We will use number lines often to help us visualize distance and relationships between numbers. To draw a number line, first draw a line. Choose a point on the line and label it 0. To the right of 0, label any other point 1. Being careful to use the same distance as from 0 to 1, mark off equally spaced distances. Label these points 2, 3, 4, 5, and so on. Since e the whole numbers continue indefinitely, it is not possible to show every whole number on this number line. The arrow at the right end of the line indicates that the pattern continues indefinitely. Picturing whole numbers on a number line helps us see the order of the numbers. Symbols can be used to describe concisely in writing the order that we see. The equal symbol = means ;is equal to.< The symbol ⬆ means ;is not equal to.< These symbols may be used to form a mathematical statement. The statement might be true or it might be false. The two statements below are both true. 2 = 2 states that ;two is equal to two.< 2 ⬆ 6 states that ;two is not equal to six.
s total
T
number of days T
T
score each day T
=
3
#
( 7)
=
#
= 21 Thus, the golfer’s total score was 21, or 21 strokes under par. PRACTICE
11
A card player had a score of 13 for each of four games. Find the total score.
Lunamarina/Fotolia
Graphing Calculator Explorations Entering Negative Numbers on a Scientific Calculator To enter a negative number on a scientific calculator, find a key marked +/ . (On some calculators, this key is marked CHS for “change sign.”) To enter 8, for example, press the keys 8 +/ . The display will read 8 .
57
Review of Real Numbers Entering Negative Numbers on a Graphing Calculator To enter a negative number on a graphing calculator, find a key marked 1  2 . Do not confuse this key with the key  , which is used for subtraction. To enter 8, for example, press the keys 1  2 8 . The display will read 8 . Operations with Real Numbers To evaluate 217  92  20 on a calculator, press the keys 2 +/
*
12 2
1 7
1 7  9 2  9 2

 2 0 = or 2 0 ENTER 217  92  20 . 16
The display will read 16 or
Use a calculator to simplify each expression. 1. 38126  272 3. 134 + 25168  912 5012942 5. 175  265
2. 591 82 + 1726 4. 451322  812182 444  444.8 6. 181  324
7. 95  4550
8. 58  6259
9. 1 1252 2 (Be careful.)
10. 1252 (Be careful.)
Vocabulary & Readiness Check Use the choices below to fill in each blank. positive 1.
0
negative
undefined
If n is a real number, then n # 0 =
0 2. If n is a real number, but not 0, then = n
and we say
. n is 0
.
3. The product of two negative numbers is a number. number. 4. The quotient of two negative numbers is a 5. The quotient of a positive number and a negative number is a
number.
6. The product of a positive number and a negative number is a
number.
7. The reciprocal of a positive number is a 8. The opposite of a positive number is a
58
and 0 # n =
number. number.
Review of Real Numbers
7
Exercise Set Divide. See Examples 6 through 8.
Multiply. See Examples 1 through 3.
63.
18 2
64.
20 10
65.
16 4
6. 61  112
66.
18 6
67.
48 12
68.
2 # 8 6 # 0
60 5
69.
0 4
70.
0 9
71. 
24 8
73.
5 0
74.
3 0
76.
45 9
77.
30 2
1. 6142
2. 8152
3. 21 12
4. 71 42
5. 51 102 3 # 4 7 # 0
7. 9.
8. 10.
15 3
11. 21 92
12. 31 52
1 3 13.  a  b 2 5
1 1 14.  a  b 8 3
72. 
3 8 15.  a  b 4 9
5 3 16.  a  b 6 10
75.
 12 4
17. 51 1.42
18. 61 2.52
78.
14 2
79.
80.
1 4 , a b 5 2
81. 
5 3 , a b 9 4
83. 
4 4 , 9 9
19. 0.21 0.72
20. 0.51 0.32
21. 101802
22. 201602
23. 41 72
24. 51 92
25. 1 521 52
26. 1 721  72
27.
4 2 a b 3 9
28.
29.  111112 31. 
2 2 a b 7 11
30. 121122
20 5 a b 25 16
32. 
33. 1 121221 3)1 52
1 8 , a b 10 11 5 5 84. , 12 12
82. 
MIXED PRACTICE
25 6 a b 36 15
Simplify. See Examples 1 through 9.
34. 1  221 3)1 421 22
85.
35. 122152  1112132
36. 81 32  41  52
37. 1 62112122  152
38. 20  1 421321  22
Decide whether each statement is true or false. 39. The product of three negative integers is negative.
91.
40. The product of three positive integers is positive. 41. The product of four negative integers is negative.
93.
42. The product of four positive integers is positive. 95. Evaluate. See Example 4. 45. 1
44. 24 46. 1 12
97. 5
47. 1 52 2
48. 52
49. 72
50. 1 72 2
99.
Find each reciprocal or multiplicative inverse. See Example 5. 51. 9
2 3
52. 100
53.
1 7
55. 14
56. 8
3 57. 11
6 58. 13
60. 1.5
61.
54.
1 6.3
62 + 4 2 8 + 1  42 2 4  12 22 + 1321 22 5  2 3  5 21 72
59. 0.2 1 8.9
61 32
15 88. 1  4 90. 92. 94.
2
6  21 32 4  31 22 3  21 92 15  31 42
0 5  9 0 + 0 10  15 0 101. 0 21 32 0
96. 98. 100. 102.
32 + 4 5 6 + 1 22 2 4  9  20 + 1 42132 1  5 2  42 31 62 8  31 22 2  51 42 4  81 22 9  21 32
0 3 + 6 0 + 0 2 + 7 0 0 2 # 2 0
If x = 5 and y = 3, evaluate each expression. See Example 10. 103. 3x + 2y
104. 4x + 5y
105. 2x 2  y 2
106. x 2  2y 2
3
108. y 3 + 3x 2y  12 110. x  4
107. x + 3y
62.
86.
4
12 87. 9  12 89.
5
91 32 6
Perform the indicated operations. See Example 2.
43. 1 22 4
6 1 , a b 7 3
2x  5 y  2 3  y 111. x  4
109.
112.
4  2x y + 3
59
Review of Real Numbers 123. A deepsea diver must move up or down in the water in short steps to keep from getting a physical condition called the “bends.” Suppose a diver moves down from the surface in five steps of 20 feet each. Represent his total movement as a product of signed numbers and find the total depth.
TRANSLATING Translate each phrase to an expression. Use x to represent “a number.” See Example 11. 113. The product of 71 and a number 114. The quotient of 8 and a number
124. A weather forecaster predicts that the temperature will drop five degrees each hour for the next six hours. Represent this drop as a product of signed numbers and find the total drop in temperature.
115. Subtract a number from  16. 116. The sum of a number and 12 117. 29 increased by a number 118. The difference of a number and 10
Decide whether the given number is a solution of the given equation.
119. Divide a number by 33 .
125. Is 7 a solution of 5x = 35?
120. Multiply a number by 17.
126. Is 4 a solution of 2x = x  1? x 127. Is 20 a solution of = 2? 10
Solve. See Example 11. 121. A football team lost four yards on each of three consecutive plays. Represent the total loss as a product of signed numbers and find the total loss.
128. Is 3 a solution of
122. Joe Norstrom lost $400 on each of seven consecutive days in the stock market. Represent his total loss as a product of signed numbers and find his total loss.
45 = 15? x
129. Is 5 a solution of 3x  5 = 20? 130. Is 4 a solution of 2x + 4 = x + 8?
CONCEPT EXTENSIONS Study the bar graph below showing the average surface temperatures of planets. Use Exercises 131 and 132 to complete the planet temperatures on the graph. (Pluto is now classified as a dwarf planet.) 135. Find any real numbers that are their own reciprocal.
Average Surface Temperature of Planets* 1200
136. Explain why 0 has no reciprocal.
867
800
If q is a negative number, r is a negative number, and t is a positive number, determine whether each expression simplifies to a positive or negative number. If it is not possible to determine, state so.
600 400
330
200
59
0
137.
q r#t
138. 139. 140. 141. 142.
q2 # r # t q + t t + r t1q + r2 r1q  t2
ⴚ323 Neptune
Jupiter
Mars
Earth
600
Venus
400
Saturn
ⴚ218
Uranus
ⴚ81
200
Mercury
Degrees Fahrenheit
1000
*For some planets, the temperature given is the temperature where the atmosphere pressure equals 1 Earth atmosphere; data from The World Almanac
Write each of the following as an expression and evaluate. 131. The surface temperature of Jupiter is twice the temperature of Mars. 143. Find this temperature. 144. 132. The surface temperature of Neptune is equal to the temperature of 145. Mercury divided by 1. Find this temperature. 146. 133. Explain why the product of an even number of negative numbers is a positive number. 134. If a and b are any real numbers, is the statement a # b = b # a always true? Why or why not?
60
The sum of 2 and the quotient of 15 and 3 The sum of 1 and the product of 8 and 5 Twice the sum of 5 and 3 7 subtracted from the quotient of 0 and 5
Review of Real Numbers
8
Properties of Real Numbers OBJECTIVE
OBJECTIVES 1 Use the Commutative and Associative Properties.
2 Use the Distributive Property.
3 Use the Identity and Inverse Properties.
Using the Commutative and Associative Properties 1 In this section, we give names to properties of real numbers with which we are already familiar. Throughout this section, the variables a, b, and c represent real numbers. We know that order does not matter when adding numbers. For example, we know that 7 + 5 is the same as 5 + 7. This property is given a special name—the commutative property of addition. We also know that order does not matter when multiplying numbers. For example, we know that 5162 = 61 52. This property means that multiplication is commutative also and is called the commutative property of multiplication. Commutative Properties a + b = b + a a#b = b#a
Addition: Multiplication:
These properties state that the order in which any two real numbers are added or multiplied does not change their sum or product. For example, if we let a = 3 and b = 5, then the commutative properties guarantee that 3 + 5 = 5 + 3
and
3#5 = 5#3
Helpful Hint Is subtraction also commutative? Try an example. Does 3  2 = 2  3? No! The left side of this statement equals 1; the right side equals 1. There is no commutative property of subtraction. Similarly, there is no commutative property for division. For example, 10 , 2 does not equal 2 , 10.
EXAMPLE 1
Use a commutative property to complete each statement. b. 3 # x =
a. x + 5 =
Solution a. x + 5 = 5 + x b. 3 # x = x # 3
By the commutative property of addition By the commutative property of multiplication
PRACTICE
1 a.
Use a commutative property to complete each statement.
x#8 =
b. x + 17 =
CONCEPT CHECK Which of the following pairs of actions are commutative? a. b. c. d.
“raking the leaves” and “bagging the leaves” “putting on your left glove” and “putting on your right glove” “putting on your coat” and “putting on your shirt” “reading a novel” and “reading a newspaper”
Answers to Concept Check: b, d
Let’s now discuss grouping numbers. We know that when we add three numbers, the way in which they are grouped or associated does not change their sum. For example, we know that 2 + 13 + 42 = 2 + 7 = 9. This result is the same if we group the numbers differently. In other words, 12 + 32 + 4 = 5 + 4 = 9 also. Thus, 2 + 13 + 42 = 12 + 32 + 4. This property is called the associative property of addition. We also know that changing the grouping of numbers when multiplying does not change their product. For example, 2 # 13 # 42 = 12 # 32 # 4 (check it). This is the associative property of multiplication.
61
Review of Real Numbers
Associative Properties Addition: Multiplication:
1a + b2 + c = a + 1b + c2 1a # b2 # c = a # 1b # c2
These properties state that the way in which three numbers are grouped does not change their sum or their product.
EXAMPLE 2
Use an associative property to complete each statement. b. 1 1 # 22 # 5 =
a. 5 + 14 + 62 =
Solution a. 5 + 14 + 62 = 15 + 42 + 6 b. 1 1 # 22 # 5 = 1 # 12 # 52
By the associative property of addition By the associative property of multiplication
PRACTICE
2
Use an associative property to complete each statement. b. 4 # 12 # 72 =
a. 12 + 92 + 7 =
Helpful Hint Remember the difference between the commutative properties and the associative properties. The commutative properties have to do with the order of numbers, and the associative properties have to do with the grouping of numbers.
Let’s now illustrate how these properties can help us simplify expressions.
EXAMPLE 3 a. 10 + 1x + 122
Simplify each expression. b. 317x2
Solution a. 10 + 1x + 122 = 10 + 112 + x2 = 110 + 122 + x = 22 + x b. 317x2 = 1 3 # 72x = 21x
By the commutative property of addition By the associative property of addition Add. By the associative property of multiplication Multiply.
PRACTICE
3
Simplify each expression.
a. 15 + x2 + 9
b. 51 6x2
OBJECTIVE
2 Using the Distributive Property The distributive property of multiplication over addition is used repeatedly throughout algebra. It is useful because it allows us to write a product as a sum or a sum as a product. We know that 712 + 42 = 7162 = 42. Compare that with 7122 + 7142 = 14 + 28 = 42. Since both original expressions equal 42, they must equal each other, or 7(2+4)=7(2)+7(4) This is an example of the distributive property. The product on the left side of the equal sign is equal to the sum on the right side. We can think of the 7 as being distributed to each number inside the parentheses.
62
Review of Real Numbers
Distributive Property of Multiplication Over Addition a(b+c)=ab+ac Since multiplication is commutative, this property can also be written as (b+c)a=ba+ca The distributive property can also be extended to more than two numbers inside the parentheses. For example, 3(x+y+z)=3(x)+3(y)+3(z) = 3x + 3y + 3z Since we define subtraction in terms of addition, the distributive property is also true for subtraction. For example 2(xy)=2(x)2(y) = 2x  2y
E X A M P L E 4 Use the distributive property to write each expression without parentheses. Then simplify if possible. a. 21x + y2
b. 51 3 + 2z2
d. 112  y2
e. 13 + x  w2
c. 51x + 3y  z2 1 f. 16x + 142 + 10 2
Solution a. 2(x+y)=2 # x+2 # y = 2x + 2y b. –5(–3+2z)=–5(–3)+(–5)(2z) = 15  10z c. 5(x+3yz)=5(x)+5(3y)5(z) = 5x + 15y  5z d. 112  y2 = 1 12122  1 121y2 Helpful Hint = 2 + y Notice in part (e) that 13 + x  w2 is first rewritten as 113 + x  w2. e. 13 + x  w2 = 113 + x  w2 = 1 12132 + 1 121x2  1 121w2 = 3  x + w 1 1 1 16x2 + 1142 + 10 Apply the distributive property. f. 16x + 142 + 10 = 2 2 2 Multiply. = 3x + 7 + 10 = 3x + 17
Add.
63
Review of Real Numbers PRACTICE
4 Use the distributive property to write each expression without parentheses. Then simplify if possible. a. 51x  y2
b. 614 + 2t2 d. 13  y2 # 1 12 1 f. 12x + 42 + 9 2
c. 213x  4y  z2 e. 1x  7 + 2s2
We can use the distributive property in reverse to write a sum as a product.
EXAMPLE 5
Use the distributive property to write each sum as a product.
a. 8 # 2 + 8 # x
b. 7s + 7t
Solution a. 8 # 2 + 8 # x = 812 + x2
b. 7s + 7t = 71s + t2
PRACTICE
5 a.
Use the distributive property to write each sum as a product.
5#w + 5#3
b. 9w + 9z
OBJECTIVE
3 Using the Identity and Inverse Properties Next, we look at the identity properties. The number 0 is called the identity for addition because when 0 is added to any real number, the result is the same real number. In other words, the identity of the real number is not changed. The number 1 is called the identity for multiplication because when a real number is multiplied by 1, the result is the same real number. In other words, the identity of the real number is not changed.
Identities for Addition and Multiplication 0 is the identity element for addition. a + 0 = a
and
0 + a = a
1 is the identity element for multiplication. a#1 = a
and
1#a = a
Notice that 0 is the only number that can be added to any real number with the result that the sum is the same real number. Also, 1 is the only number that can be multiplied by any real number with the result that the product is the same real number. Additive inverses or opposites were introduced in Section 5. Two numbers are called additive inverses or opposites if their sum is 0. The additive inverse or opposite of 6 is 6 because 6 + 1 62 = 0. The additive inverse or opposite of 5 is 5 because 5 + 5 = 0.
64
Review of Real Numbers Reciprocals or multiplicative inverses were introduced in Section 3. Two nonzero numbers are called reciprocals or multiplicative inverses if their product is 1. The recipro2 3 2 3 cal or multiplicative inverse of is because # = 1. Likewise, the reciprocal of 5 is 3 2 3 2 1 1  because 5a  b = 1. 5 5
CONCEPT CHECK Which of the following, 1, a. opposite of 
3 ? 10
10 3 3 10 , , 0, ,  , is the 3 10 3 10 3 b. reciprocal of  ? 10
Additive or Multiplicative Inverses The numbers a and a are additive inverses or opposites of each other because their sum is 0; that is, a + 1 a2 = 0 1 The numbers b and (for b ⬆ 0) are reciprocals or multiplicative inverses of each b other because their product is 1; that is, 1 b# = 1 b
EXAMPLE 6
Name the property or properties illustrated by each true statement.
Solution a. 3 # y = y # 3 b. 1x + 72 + 9 = x + 17 + 92
Commutative property of multiplication (order changed)
c. 1b + 02 + 3 = b + 3 d. 0.2 # 1z # 52 = 0.2 # 15 # z2 1 e. 2 # a  b = 1 2
Identity element for addition
f. 2 + 2 = 0 g. 6 # 1y # 22 = 1 6 # 22 # y
Additive inverse property
Associative property of addition (grouping changed) Commutative property of multiplication (order changed) Multiplicative inverse property
Commutative and associative properties of multiplication (order and grouping changed)
PRACTICE
6
Name the property or properties illustrated by each true statement.
Commutative property of multiplication a. 17 # 3x2 # 4 = 13x # 72 # 4 b. 6 + 13 + y2 = 16 + 32 + y Associative property of addition Identity element for addition c. 8 + 1t + 02 = 8 + t 3 4 d.  # a  b = 1 Multiplicative inverse property 4 3
Answers to Concept Check: 3 10 a. b. 10 3
e. 12 + x2 + 5 = 5 + 12 + x2 f. 3 + 1 32 = 0 g. 1 3b2 # 7 = 1 3 # 72 # b
Commutative property of addition Additive inverse property Commutative and associative properties of multiplication
65
Review of Real Numbers
Vocabulary & Readiness Check Use the choices below to fill in each blank. distributive property opposites or additive inverses reciprocals or multiplicative inverses
associative property of multiplication associative property of addition commutative property of multiplication
1. x + 5 = 5 + x is a true statement by the 2. 3. 4.
.
x # 5 = 5 # x is a true statement by the 31y + 62 = 3 # y + 3 # 6 is true by the 2 # 1x # y2 = 12 # x2 # y is a true statement by the
. . .
5. x + 17 + y2 = 1x + 72 + y is a true statement by the 6. The numbers 
2 3 and  are called 3 2
7. The numbers 
2 2 and are called 3 3
8
. . .
Exercise Set
Use a commutative property to complete each statement. See Example 1.
Use the commutative and associative properties to simplify each expression. See Example 3.
1. x + 16 =
2. 4 + y =
17. 8 + 19 + b2
18. 1r + 32 + 11
3. 4 # y =
4. 2 # x =
19. 4(6y)
20. 2(42x)
5. xy =
6. ab =
7. 2x + 13 =
8. 19 + 3y =
1 15y2 21. 5
22.
23. 113 + a2 + 13
24. 7 + 1x + 42
25. 918x2 3 4 a sb 27. 4 3
26.  3112y2 2 7 28. a rb 7 2
Use an associative property to complete each statement. See Example 2. 9. 1xy2 # z = 11. 2 + 1a + b2 = 13.
4 # 1ab2
=
15. 1a + b2 + c = 16. 6 + 1r + s2 =
66
commutative property of addition
10. 3 # 1xy2 = 12. 1y + 42 + z = 14. 1 3y2 # z =
29.
4 2 + a + xb 3 3
30.
1 18z2 8
7 2 + a + yb 9 9
Review of Real Numbers
Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. 31. 33. 35. 37. 39. 41. 43.
41x + y2 91x  62 213x + 52 714x  32 316 + x2  21y  z2 713y + 52
44. 45. 46. 47. 48. 49. 50. 51. 52.
512r + 112 51x + 4m + 22 813y + z  62 411  2m + n2 414 + 2p + 5q2 15x + 22 19r + 52 1r  3  7p2 1q  2 + 6r2
32. 34. 36. 38. 40. 42.
71a + b2 111y  42 517 + 8y2 318x  12 21x + 52 31z  y2
0 + 6 = 6 1a + 92 + 6 = a + 19 + 62 41y + 72 = 4 # y + 1 42 # 7 111 + r2 + 8 = 1r + 112 + 8 4 # 18 # 32 = 18 # 42 # 3 r + 0 = r
CONCEPT EXTENSIONS Fill in the table with the opposite (additive inverse), and the reciprocal (multiplicative inverse). Assume that the value of each expression is not 0. Expression
Opposite
Reciprocal
8
85. 86.
1 53. 16x + 82 2 1 54. 14x  22 4 1 55.  13x  9y2 3

2 3
87.
x
88.
4y 1 2x
89. 90.
1 56.  110a  25b2 5 57. 58. 59. 60. 61. 62.
79. 80. 81. 82. 83. 84.
7x
Decide whether each statement is true or false. See the second Concept Check in this section.
312r + 52  7 1014s + 62  40 914x + 82 + 2 1115x + 32 + 10 414x + 52  5 612x + 12  1
a 2 is  . 2 a a a 92. The reciprocal of  is . 2 2
91. The opposite of 
Use the distributive property to write each sum as a product. See Example 5. 63. 4 # 1 + 4 # y
64. 14 # z + 14 # 5
65. 11x + 11y
66. 9a + 9b
67. 1 12 # 5 + 1 12 # x
68. 1 32a + 1 32b
69. 30a + 30b
70. 25x + 25y
Determine which pairs of actions are commutative. See the first Concept Check in this section. 93. “taking a test” and “studying for the test” 94. “putting on your shoes” and “putting on your socks”
Name the properties illustrated by each true statement. See Example 6. 71.
3#5
=
5#3
95. “putting on your left shoe” and “putting on your right shoe” 96. “reading the sports section” and “reading the comics section” 97. “mowing the lawn” and “trimming the hedges” 98. “baking a cake” and “eating the cake” 99. “dialing a number” and “turning on the cell phone”
72. 413 + 82 = 4 # 3 + 4 # 8
100. “feeding the dog” and “feeding the cat”
73. 2 + 1x + 52 = 12 + x2 + 5
Name the property illustrated by each step.
74. 1x + 92 + 3 = 19 + x2 + 3 75. 913 + 72 = 9 # 3 + 9 # 7 76. 1 # 9 = 9
101. a. 䉭 + (䊐 + ~ ) = (䊐 + ~ ) + 䉭 b.
= 1~ + 䊐2 + 䉭
c.
= ~ + 1䊐 + 䉭2
77. 14 # y2 # 9 = 4 # 1y # 92 78. 6 #
1 = 1 6
67
Review of Real Numbers 102. a. 1x + y2 + z = x + 1y + z2 b. = 1y + z2 + x c. = 1z + y2 + x
104. Explain why 1 is called the identity element for multiplication.
103. Explain why 0 is called the identity element for addition.
106. Write an example that shows that subtraction is not commutative.
Chapter
105. Write an example that shows that division is not commutative.
Vocabulary Check
Fill in each blank with one of the words or phrases listed below. set
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
inequality symbols
opposites
absolute value
numerator
denominator
grouping symbols
exponent
base
reciprocals
variable
equation
solution
The symbols ⬆, 6 , and 7 are called . A mathematical statement that two expressions are equal is called a(n) . The of a number is the distance between that number and 0 on the number line. A symbol used to represent a number is called a(n) . Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called . The number in a fraction above the fraction bar is called the . A(n) of an equation is a value for the variable that makes the equation a true statement. Two numbers whose product is 1 are called . In 23, the 2 is called the and the 3 is called the . The number in a fraction below the fraction bar is called the . Parentheses and brackets are examples of . A(n) is a collection of objects.
Chapter
Highlights
DEFINITIONS AND CONCEPTS
EXAMPLES Section 2
Symbols and Sets of Numbers
A set is a collection of objects, called elements, enclosed in braces. Natural Numbers: 5 1, 2, 3, 4, c6 Whole Numbers: 5 0, 1, 2, 3, 4, c6 Integers: 5c , 3, 2, 1, 0, 1, 2, 3, c6 Rational Numbers: {real numbers that can be expressed as a quotient of integers}
68
5 a, c, e 6 2 Given the set e 3.4, 23, 0, , 5,  4 f , list the numbers 3 that belong to the set of Natural numbers: 5 Whole numbers: 0, 5 Integers: 4, 0, 5
Irrational Numbers: {real numbers that cannot be expressed as a quotient of integers}
2 Rational numbers: 4, 3.4, 0, , 5 3 Irrational Numbers: 13
Real Numbers: {all numbers that correspond to a point on the number line}
2 Real numbers: 4, 3.4, 0, , 13, 5 3
Review of Real Numbers
DEFINITIONS AND CONCEPTS
EXAMPLES Section 2
Symbols and Sets of Numbers
A line used to picture numbers is called a number line.
3 2 1 0 1
The absolute value of a real number a, denoted by 0 a 0 , is the distance between a and 0 on the number line.
050 = 5
Symbols: = is equal to ⬆ is not equal to
7 = 7 3 ⬆ 3
7 is greater than
4 7 1
6 is less than
1 6 4
… is less than or equal to
6 … 6 18 Ú 
Ú is greater than or equal to Order Property for Real Numbers
000 = 0
1 3
3 2 1 0 1
For any two real numbers a and b, a is less than b if a is to the left of b on a number line. Section 3
3 0
0 3
2 3
0 2.5
2.5 0
Fractions and Mixed Numbers
A quotient of two integers is called a fraction. The numerator of a fraction is the top number. The denominator of a fraction is the bottom number. If a # b = c, then a and b are factors and c is the product. A fraction is in lowest terms or simplest form when the numerator and the denominator have no factors in common other than 1.
2 3
0 2 0 = 2
13 d numerator 17 d denominator
# 9 = 7 T T factor factor 13 is in simplest form. 17
To write a fraction in simplest form, factor the numerator and the denominator; then apply the fundamental principle.
Write in simplest form.
Two fractions are reciprocals if their product is 1. a b The reciprocal of is . b a To multiply fractions, numerator times numerator is the numerator of the product and denominator times denominator is the denominator of the product.
The reciprocal of
63 T product
2#3 3 6 = # = 14 2 7 7 6 25 is . 25 6
Perform the indicated operations. 2#3 6 = 5 7 35
To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
2 5 7 35 5 , = # = 9 7 9 2 18
To add fractions with the same denominator, add the numerators and place the sum over the common denominator.
5 3 8 + = 11 11 11
To subtract fractions with the same denominator, subtract the numerators and place the difference over the common denominator.
2 13 3 10 = = 15 15 15 3
Fractions that represent the same quantity are called equivalent fractions.
1#4 4 1 = # = 5 5 4 20 1 4 and are equivalent fractions. 5 20
69
Review of Real Numbers
DEFINITIONS AND CONCEPTS Section 4
EXAMPLES
Exponents, Order of Operations, Variable Expressions, and Equations
The expression a n is an exponential expression. The number a is called the base; it is the repeated factor. The number n is called the exponent; it is the number of times that the base is a factor.
43 = 4 # 4 # 4 = 64 72 = 7 # 7 = 49 82 + 517  32
Order of Operations
3#7
Simplify expressions in the following order. If grouping symbols are present, simplify expressions within those first, starting with the innermost set. Also, simplify the numerator and the denominator of a fraction separately.
= = =
1. Simplify exponential expressions. 2. Multiply or divide in order from left to right. 3. Add or subtract in order from left to right.
82 + 5142 21 64 + 5142 21 64 + 20 21
84 21 = 4 =
A symbol used to represent a number is called a variable.
Examples of variables are: q, x, z
An algebraic expression is a collection of numbers, variables, operation symbols, and grouping symbols.
Examples of algebraic expressions are: 5x, 21y  62,
To evaluate an algebraic expression containing a variable, substitute a given number for the variable and simplify.
q 2  3q + 1 6
Evaluate x 2  y 2 if x = 5 and y = 3. x 2  y 2 = 152 2  32 = 25  9 = 16
A mathematical statement that two expressions are equal is called an equation.
Examples of equations are: 3x  9 = 20 A = pr 2
A solution of an equation is a value for the variable that makes the equation a true statement.
Determine whether 4 is a solution of 5x + 7 = 27. 5x + 7 = 27 5142 + 7 ⱨ 27 20 + 7 ⱨ 27 27 = 27 True 4 is a solution.
Section 5 To Add Two Numbers with the Same Sign
1. Add their absolute values. 2. Use their common sign as the sign of the sum.
Adding Real Numbers Add. 10 + 7 = 17 3 + 1 82 =  11
To Add Two Numbers with Different Signs
1. Subtract their absolute values. 2. Use the sign of the number whose absolute value is larger as the sign of the sum.
70
25 + 5 =  20 14 + 1  92 = 5
Review of Real Numbers
DEFINITIONS AND CONCEPTS
EXAMPLES
Section 5
Adding Real Numbers (continued )
Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called opposites or additive inverses. The opposite of a number a is denoted by a.
The opposite of  7 is 7. The opposite of 123 is 123. 4 + 4 = 0
The sum of a number a and its opposite, a, is 0.
12 + 1 122 = 0
a + 1 a2 = 0
1 82 = 8
If a is a number, then 1 a2 = a.
1 142 = 14 Section 6
To subtract two numbers a and b, add the first number a to the opposite of the second number b.
Subtracting Real Numbers Subtract. 3  1 442 = 3 + 44 = 47
a  b = a + 1 b2
5  22 = 5 + 1 222 =  27  30  1 302 =  30 + 30 = 0 Section 7
Multiplying and Dividing Real Numbers Multiply or divide.
Quotient of two real numbers
42 1 = 42 # = 21 2 2
a 1 = a# b b
7 # 8 = 56
Multiplying and Dividing Real Numbers The product or quotient of two numbers with the same sign is a positive number. The product or quotient of two numbers with different signs is a negative number.
Products and Quotients Involving Zero
2 # 4 = 8
7 # 1 82 = 56 2 # 1 42 = 8
90 = 9 10
90 = 9 10
42 = 7 6
42 = 7 6
The product of 0 and any number is 0. 4 # 0 = 0
b # 0 = 0 and 0 # b = 0
3 0# a b = 0 4
The quotient of a nonzero number and 0 is undefined. b is undefined. 0 The quotient of 0 and any nonzero number is 0.
85 is undefined. 0
0 = 0 b
0 = 0 18 Section 8
0 = 0 47
Properties of Real Numbers
Commutative Properties Addition: a + b = b + a Multiplication: a # b = b # a
3 + 1 72 = 7 + 3 8 # 5 = 5 # 1 82
Associative Properties Addition: 1a + b2 + c = a + 1b + c2 Multiplication: 1a # b2 # c = a # 1b # c2
15 + 102 + 20 = 5 + 110 + 202 1 3 # 22 # 11 = 3 # 12 # 112 (continued)
71
Review of Real Numbers
DEFINITIONS AND CONCEPTS
EXAMPLES Section 8
Two numbers whose product is 1 are called multiplicative inverses or reciprocals. The reciprocal 1 1 of a nonzero number a is because a # = 1. a a Distributive Property
Properties of Real Numbers 1 The reciprocal of 3 is . 3 2 5 The reciprocal of  is  . 5 2
a1b + c2 = a # b + a # c
516 + 102 = 5 # 6 + 5 # 10 213 + x2 = 2 # 3 + 1 221x2
Identities
a + 0 = a
0 + a = a
a#1 = a
1#a = a
5 + 0 = 5
0 + 1 22 = 2
14 # 1 = 14
1 # 27 = 27
Inverses Additive or opposite: a + 1 a2 = 0
7 + 1 72 = 0
1 Multiplicative or reciprocal: b # = 1 b
3#
1 = 1 3
Chapter Review (2) Insert 6 , 7 , or = in the appropriate space to make the following statements true. 1. 8
10
3. 4
2. 7
5. 0 7 0
0 8 0
7.  0 1 0 9. 1.2
2
12 4. 2
5
6. 0 9 0 8. 0 14 0
1
3 10. 2
1.02
The following chart shows the gains and losses in dollars of Density Oil and Gas stock for a particular week. Day
8 9 1 142 3 4
TRANSLATING
Gain or Loss in Dollars
Monday
+1
Tuesday
2
Wednesday
+5
Thursday
+1
Friday
4
17. Which day showed the greatest loss?
Translate each statement into symbols. 11. Four is greater than or equal to negative three.
18. Which day showed the greatest gain?
12. Six is not equal to five.
(3) Write the number as a product of prime factors.
13. 0.03 is less than 0.3.
19. 36
14. New York City has 155 museums and 400 art galleries. Write an inequality comparing the numbers 155 and 400. (Source: Absolute Trivia.com) Given the following sets of numbers, list the numbers in each set that also belong to the set of:
a. Natural numbers c. Integers e. Irrational numbers
b. Whole numbers d. Rational numbers f. Real numbers
Perform the indicated operations. Write results in lowest terms. 21.
8 # 27 15 30
22.
21 7 , 8 32
23.
5 7 + 15 6
24.
3 3 4 20
25. 2
3 5 + 6 4 8
27. 5 , 1 15. e 6, 0, 1, 1 , 3, p, 9.62 f 2 16. e 3, 1.6, 2, 5,
72
11 , 15.1, 25, 2p f 2
20. 120
1 3
26. 7
1 2  2 6 3
28. 2 # 8
3 4
Review of Real Numbers Each circle represents a whole, or 1. Determine the unknown part of the circle. 29.
30. Z
~
q Q
45.
5 11
3 11
in.
in.
5 11
in.
in.
Octuplets were born in the U.S. in 2009. The following chart gives the octuplets’ birthweights. The babies are listed in order of birth. Baby
Gender
Baby A
boy
Baby B
girl
Baby C
boy
Baby D
girl
Baby E
boy
Baby F
boy
Baby G
boy
Baby H
boy
Birthweight (pounds) 1 2 1 2 8 1 3 16 3 2 16 3 1 4 9 2 16 13 1 16 7 2 16
Evaluate each expression if x = 6, y = 2, and z = 8. 49. 2x + 3y z x + 51. y 2y
50. x1y + 2z2 52. x 2  3y 2
53. The expression 180  a  b represents the measure of the unknown angle of the given triangle. Replace a with 37 and b with 80 to find the measure of the unknown angle.
2
33. What was the total weight of the boy octuplets? 34. What was the total weight of the girl octuplets? 35. Find the combined weight of all eight octuplets. 36. Which baby weighed the most? 37. Which baby weighed the least? 38. How much more did the heaviest baby weigh than the lightest baby?
(4) Choose the correct answer for each statement. 39. The expression 6 # 32 + 2 # 8 simplifies to a. 52
47. The difference of twenty and twelve is equal to the product of two and four. 48. The quotient of nine and two is greater than negative five.
3 11
1 a meter
46. 5[312 + 52  5]
4 + 6#4
Translate each word statement to symbols.
32.
√ meter
44. 8 + 312 # 6  12
4 + 0 6  2 0 + 82
TRANSLATING
Find the area and the perimeter of each figure. 31.
3 3 42. a b 4
2 2 41. a b 7 43. 311 + 2 # 52 + 4
?
?
Simplify each expression.
?
80
37
54. The expression 360  a  b  c represents the measure of the unknown angle of the given quadrilateral. Replace a with 93, b with 80, and c with 82 to find the measure of the unknown angle. 93
80
?
82
Decide whether the given number is a solution to the given equation. 55. Is x = 3 a solution of 7x  3 = 18? 56. Is x = 1 a solution of 3x 2 + 4 = x  1? (5) Find the additive inverse or the opposite. 2 58. 57. 9 3 59. 0 2 0
60.  0 7 0
Find the following sums. 61.  15 + 4
62. 6 + 1 112
1 1 63. + a b 16 4
64. 8 + 0 3 0
65. 4.6 + 1 9.32
66.  2.8 + 6.7
b. 448 c. 70 d. 64
40. The expression 68  5 # 23 simplifies to a. 232
b. 28
c. 38
d. 504
73
Review of Real Numbers 97. 318  52 = 3 # 8  3 # 152
(6) Perform the indicated operations. 67. 6  20
68. 3.1  8.4
69. 6  1  112
98. 4 + 1  42 = 0
70. 4  15
99. 2 + 13 + 92 = 12 + 32 + 9
71. 21  16 + 3(8  2)
72.
11  1 92 + 618  22 2 + 3#4
Evaluate each expression for x = 3, y = 6, and z = 9. Then choose the correct evaluation.
100. 2 # 8 = 8 # 2 101. 618 + 52 = 6 # 8 + 6 # 5 102. 13 # 82 # 4 = 3 # 18 # 42
2
73. 2x  y + z a. 15 74.
b. 3
c. 27
103. 4 #
d. 3
0 y  4x 0 2x a. 3 b. 1 c. 1
104. 8 + 0 = 8 Use the distributive property to write each expression without parentheses.
d. 3
75. At the beginning of the week, the price of Density Oil and Gas stock from Exercises 17 and 18 is $50 per share. Find the price of a share of stock at the end of the week. 76. Find the price of a share of stock by the end of the day on Wednesday. (7) Find the multiplicative inverse or reciprocal. 77. 6
78.
85.
106. 31z + y2 107. 17  x + 4z2 108.
1 16z  102 2
109. 413x + 52  7
MIXED REVIEW Insert 6 , 7 , or = in the space between each pair of numbers.
80. 1  221 142
79. 61 82
83.
105. 51y  22
110. 812y + 92  1
3 5
Simplify each expression.
81.
18 6
82.
42 3
84.
2 + 1 22
111.  0 11 0
31 22  5
86.
0 2
2
1 2
114. 14  20
115. 41 202
116.
4 5 a b 5 16
20 4
118. 0.51 0.32
1 12 # 2
88. 52  12  202 , 1  32 # 3
119. 8 , 2 # 4
If x =  5 and y = 2, evaluate each expression. 89. x  y
1 2
113. 7.2 + 18.12
117. 
2
112. 1
14
6 0
87.  4  1 3 + 52 ,
0 11.4 0
Perform the indicated operations. 2
41 32 + 1 82
2
1 = 1 4
4
2
90. x  y
3
TRANSLATING Translate each phrase to an expression. Use x to represent a number.
121.
120. 1  22 4
3  21 92 15  31 42
5 3 123.  , 8 4
122. 5 + 2[17  52 2 + 11  32]
124.
15 + 1 42 2 + 0 9 0 10  2 # 5
91. The product of −7 and a number 92. The quotient of a number and −13 93. Subtract a number from −20 94. The sum of −1 and a number (8) Name the property illustrated. 95. 6 + 5 = 5 + 1 62 96. 6 # 1 = 6
74
125. A trim carpenter needs a piece of quarter round molding 1 1 6 feet long for a bathroom. She finds a piece 7 feet long. 8 2 1 How long a piece does she need to cut from the 7 @foot@long 2 molding in order to use it in the bathroom?
Review of Real Numbers
Chapter Test Translate the statement into symbols. 1. The absolute value of negative seven is greater than five. 2. The sum of nine and five is greater than or equal to four. Simplify the expression.
30. 612 + 42 = 6 # 2 + 1 62 # 4 31.
1 162 = 1 6
3. 13 + 8
4. 13  1 22
32. Find the opposite of 9.
5. 12 , 4 # 3  6 # 2
6. 11321 32
1 33. Find the reciprocal of  . 3
7. 1 621 22
8.
0 16 0
The New Orleans Saints were 22 yards from the goal when the following series of gains and losses occurred.
8
0 6 0 + 2
8 0 1 5 11. 2 6
10.
13. 0.6 + 1.875
14. 31 42 2  80
15. 6[5 + 213  82  3]
12 + 3 # 8 16. 4
9.
17.
5  6 3 1 12. 5  1 4 8
1 221021 32
Gains and Losses in Yards First Down Second Down
5 10
Third Down
2
Fourth Down
29
34. During which down did the greatest loss of yardage occur?
6 Insert 6 , 7 , or = in the appropriate space to make each of the following statements true. 18. 3 20. 2
7
0 3 0
19. 4
8
21. 0 2 0
1  1 32
22. In the state of Massachusetts, there are 2221 licensed child care centers and 10,993 licensed homebased child care providers. Write an inequality statement comparing the numbers 2221 and 10,993. (Source: Children’s Foundation) 1 23. Given e 5, 1, 0, , 1, 7, 11.6, 17, 3p f , list the num4 bers in this set that also belong to the set of:
35. Was a touchdown scored? 36. The temperature at the Winter Olympics was a frigid 14 degrees below zero in the morning, but by noon it had risen 31 degrees. What was the temperature at noon?
? 0
31
14
a. Natural numbers b. Whole numbers c. Integers d. Rational numbers
37. A health insurance provider had net incomes of $356 million, $460 million, and  +166 million in 3 consecutive years. What was the health insurance provider’s total net income for these three years?
e. Irrational numbers f. Real numbers If x = 6, y =  2, and z = 3, evaluate each expression. 24. x 2 + y 2
25. x + yz
26. 2 + 3x  y
27.
y + z  1
38. A stockbroker decided to sell 280 shares of stock, which decreased in value by $1.50 per share yesterday. How much money did she lose?
x
Identify the property illustrated by each expression. 28. 8 + 19 + 32 = 18 + 92 + 3 29. 6 # 8 = 8 # 6
75
Review of Real Numbers
Answers to Selected Exercises Section 2 Practice Exercises 1. a. 6 b. 25
b. 7
c. 6
c. 25,  15,  99
7. a. 8
b. 9
c. 2.5
2. a. True b. False c. True d. True 3. a. 3 6 8 b. 15 Ú 9 c. 6 ⬆ 7 4.  10 5. a. 25 7 7 3 3 d. 25, ,  15,  ,  3.7, 8.8,  99 e. 15 f. 25, ,  15,  , 15,  3.7, 8.8,  99 6. a. 6 b. 7 c. = 3 4 3 4 5 d. e. 13 8. a. = b. 7 c. 6 d. 7 e. 6 11
Vocabulary & Readiness Check 2 1. whole
3. inequality
5. real
7. irrational
Exercise Set 1.2 1. 7 3. = 5. 6 7. 6 9. 32 6 212 11. 30 … 45 13. true 15. false 17. false 19. true 21. false 23. 8 6 12 25. 5 Ú 4 27. 15 ⬆  2 29. 14,494;  282 31.  28,000 33. 350;  126 35. whole, integers, rational, real 37. integers, rational, real 39. natural, whole, integers, rational, real 41. rational, real 43. irrational, real 45. false 47. true 49. true 51. true 53. false 55. 7 57. 7 59. 6 61. 6 63. 7 65. = 67. 6 69. 6 71. 2009 73. 2009, 2010 75. 280 million < 281 million 77. 49 million > 16 million 79. 38 million pounds less, or  38 million 81.  0.04 7  26.7 83. sun 85. sun 87. 20 … 25 89. 6 7 0 91.  12 6  10 93. answers may vary
Section 3 Practice Exercises 1. a. 2 # 2 # 3 # 3 6.
14 21
b. 2 # 2 # 2 # 5 # 5 46 77
7. a.
b.
1 14
7 8
2. a. 1 2
c.
8. 22
16 3
b. 11 15
c.
9. 40
7 25
3.
7 24
4. a.
27 16
b.
1 36
c.
5 2
5. a. 1
b.
6 5
c.
4 5
1 2
d.
5 6
Vocabulary & Readiness Check 3 1. fraction
3. product
5. factors, product
7. equivalent
Exercise Set 3 3 8 1 27. 2 16 53. 20 65 79. 21 1.
3.
5 7
5. 3 # 11 31. 15
33.
1 6
9. 2 # 2 # 5 35.
25 27
e.
8 125
11. 3 # 5 # 5
13. 3 # 3 # 5
15.
1 2
17.
2 3
19.
3 7
21.
3 5
30 61 21 49. 30 1 75. 7 12 23.
11 7 3 1 9 sq mi 39. sq ft 41. 43. 1 45. 47. 20 36 5 3 35 23 11 5 7 1 3 1 20 28 1 55. 57. 59. 61. 63. 65. 67. 69. 18 71. 2 73. 48 21 60 66 5 5 8 9 27 29 15 2 10 17 5 1 17 1 1 81. 83. 85. 87. 37 89. 91. 93. 5 95. 97. 55 ft 99. answers may vary 5 9 3 66 5 6 18 4 # # 2 2 3 1 9 11 21 12 2 3 101. 3 mi = = + = 103. 105. multiplexes 107. incorrect; 109. incorrect; 8 100 24 2#2#2#3 2 7 7 7 29.
6 7
7. 2 # 7 # 7
37.
25.
3 8
4 18 5 77. 7
51.
Section 4 Practice Exercises 1. a. 1
b. 25
19 8 c. 15 10 c. 2x + 9 ⬆ 25 b.
c.
1 100
d. 33
d. 9 7. No
8. a. 6x
2. a. 33 b. x  8
d. 51112 Ú x
Graphing Calculator Explorations 4 1. 625
3. 59,049
5. 30
7. 9857
9. 2376
Vocabulary & Readiness Check 4 1. base; exponent
76
3. variable
5. equation
7. solving
b. 11
c.
5 32 or 3 9 9
c. x # 9 or 9x
d. 36
d. 2x + 3
e.
3 16
e. 7 + x
3.
31 11
4. 4
9. a. x + 7 = 13
5.
9 22
6. a. 9
b. x  2 = 11
Review of Real Numbers Exercise Set 4 1. 243
3. 27
5. 1
27. 45
29. 0
31. 30
49. 1
51. 1
7. 5
9. 49
33. 2
53. 11
35.
55. 8
16 81
11. 7 18
27 10
37.
57. 45
1 125
13.
15. 1.44 7 5
39.
59. 27
17. 0.000064
41. 32
61. 132
63.
43.
37 18
23 27
19. 17
21. 20
45. a. 64
65. 16; 64; 144; 256
b. 43
23. 10 c. 19
67. yes
25. 21 d. 22
69. no
47. 9
71. no
73. yes
x 83. 3x + 22 85. 1 + 2 = 9 , 3 87. 3 ⬆ 4 , 2 89. 5 + x = 20 91. 7.6x = 17 4 93. 13  3x = 13 95. multiply 97. subtract 99. no; answers may vary 101. 14 in., 12 sq in. 103. 14 in., 9.01 sq in. 105. Rectangles with the same perimeter can have different areas. 107. 120  42 # 4 , 2 109. a. expression b. equation c. equation d. expression e. expression 111. answers may vary 113. answers may vary, for example,  2152  1 115. 12,000 sq ft 117. 51 mph 75. no
77. x + 15
79. x  5
81.
4
2.
Section 5 Practice Exercises 1.
2
ⴚ3
ⴚ2
3. a.  13
b.  32
ⴚ3
4.
8 ⴚ4 ⴚ3 ⴚ2 ⴚ1
0
1
2
3
4
5
ⴚ5 ⴚ4 ⴚ3 ⴚ2 ⴚ1
6
0
1
2
3
4
5 ⴚ5 ⴚ4 ⴚ3 ⴚ2 ⴚ1
5. a.  3 d. 3
b. 1
c.  0.2
10. a.  15
6. a.  1
3 5
b.
c. 5y
b.  6
1 d. 70
c. 0.5
7. a.  6
b.  6
5 9. a. 9
8. 4°F
0
1
b.  8
2
3
4
5
c.  6.2
d. 8
Vocabulary & Readiness Check 5 1. opposites
3. n
Exercise Set 5 1. 9
3.  14
27.  2.1 51.  18 75. 0
5. 1
29.  8
7.  12 31. 38
9.  5
11.  12 1 2 = 8 4 59. 146 ft
33.  13.1
35.
13.  4
15. 7 17.  2 19. 0 21.  19 23. 31 3 13 39. 41.  8 43.  59 45.  9 16 10 61.  +6.9 million 63.  16 65.  6 67. 2 69. 0
53. 19 55.  0.7 57.  6 2 79. yes 81. no 83. July 85. October 3 101. answers may vary 103. answers may vary
77. 
99. false
25.  47
37. 
87. 4.7°F
89.  3
91.  22
93. negative
47. 5
49. 11
71.  6
73.  2
95. positive
97. true
Section 6 Practice Exercises 1. a.  13 b. 13
b.  7
7. $357
c. 12
d.  2
8. a. 28
2. a. 10.9
b. 
1 2
19 20
3.  7
4. a.  6
b. 6.1
15. 5
17. 37
19.  6.4
21.  71
c. 
5. a.  20
b. 13
6. a. 2
b. 137
Vocabulary & Readiness Check 6 1. 7  x
3. x  7
5. 7  x
7.  10  1  142; d
Exercise Set 6 1.  10
3.  5
2 11
29. 
27.
51. 13
53. 6
5. 19 11 12
7.
1 6
31. 8.92 55. 9
57.  9
9. 2
11.  11
33. 13
35.  5
59.  7
75. 19,852 ft 77. 130° 79. 30° 81. no 95. answers may vary 97. 16 99.  20
13. 11 37.  1
39.  23
41.  26
43.  24
45. 3
23. 0
25. 4.1
47.  45
49.  4
7 1 63. 21 65. 67. 100°F 69. 265°F 71. 35,653 ft 73.  308 ft 5 4 83. no 85. yes 87.  5 + x 89.  20  x 91.  4.4; 2.6°; 12°; 23.5°; 15.3° 93. May 101. true; answers may vary 103. false; answers may vary 105. negative,  30,387 61.
Integrated Review 1. negative 12. 
9 9 ; 11 11
2. negative 13.  42
3. positive 14. 10
15. 2
4. 0
5. positive
16.  18
17.  7
6. 0
7. positive 18.  39
8. positive
19.  2
20.  9
1 1 9.  ; 7 7 21.  3.4
10.
12 12 ; 5 5
22.  9.8
11. 3; 3 23. 
25 28
77
Review of Real Numbers 5 24 13 37. 20 24. 
25.  4 38. 
26.  24
29 40
39. 4
27. 6 40. 9
28. 20 41.  1
29. 6
30. 61
42.  3
31.  6
43. 8
44. 10
32.  16 45. 47
33.  19
46.
34.  13
35.  4
36.  1
2 3
Section 7 Practice Exercises 1. a.  40 d.  64
b. 12 3 8
5. a.
b. undefined
c.  54 b.
2. a.  30
1 15
c. 
7 2
d. 
84 9. a. 5
c. undefined
b. 24 1 5
c. 0
d. 26
6. a.  8
b. 11
3. a.  0.046
b.  4
10. a.  9
c. 5
7. a. 3 5 c. 3
b. 33
b. 
4 15
c. 14
b.  16
4. a. 36
c. 
6 5
b.  36
d. 
1 18
c.  64
8. a. 0
11.  52
Graphing Calculator Explorations 7 1. 38
3.  441
5. 163.3
7. 54,499
9. 15,625
Vocabulary & Readiness Check 7 1. 0; 0
3. positive
5. negative
7. positive
Exercise Set 7 1.  24
3.  2
5. 50
7.  12
9. 0
11.  18
13.
3 10
15.
2 3
17.  7
19. 0.14
8 1 29.  121 31. 33.  30 35. 23 37.  7 39. true 41. false 43. 27. 27 4 3 1 1 1 11 53. 55. 57. 59. 61.  6.3 63.  9 65. 4 67.  4 51. 9 2 14 3 0.2 20 18 9 75. 3 77.  15 79. 81. 83.  1 85. 87.  4 89. 16 91.  3 7 27 2 3 # 101. 103.  21 105. 41 107.  134 109. 3 111. 0 113.  71 x or  71x 115. 2 or x , 1  332 121. 3 # 1  42 =  12; a loss of 12 yd 123. 51  202 =  100; a depth of 100 ft 125. 133. answers may vary
135. 1,  1
137. positive
139. not possible
141. negative
21.  800
16
45.  1
69. 0 16 7  16  x 93. 
yes
23.  28 47. 25
71.  5
49.  49
73. undefined
95. 2
97.
6 5
117.  29 + x
127. no  15 143.  2 + ; 7 3
25. 25
129. yes
99.  5 x 119.  33 131.  162F
145. 23  5 + 1  32 4 ;  16
Section 8 Practice Exercises 1. a. 8 # x b. 17 + x 2. a. 2 + 19 + 72 b. 1  4 # 22 # 7 3. a. x + 14 b.  30x 4. a. 5x  5y b.  24  12t c. 6x  8y  2z d.  3 + y e.  x + 7  2s f. x + 11 5. a. 51w + 32 b. 91w + z2 6. a. commutative property of multiplication b. associative property of addition c. identity element for addition d. multiplicative inverse property e. commutative property of addition f. additive inverse property g. commutative and associative properties of multiplication
Vocabulary & Readiness Check 8 1. commutative property of addition
3. distributive property
5. associative property of addition
7. opposites or additive inverses
Exercise Set 8 1. 16 + x 3. y # 1  42 5. yx 7. 13 + 2x 9. x # 1yz2 11. 12 + a2 + b 13. 14a2 # b 15. a + 1b + c2 17. 17 + b 19. 24y 21. y 23. 26 + a 25.  72x 27. s 29. 2 + x 31. 4x + 4y 33. 9x  54 35. 6x + 10 37. 28x  21 39. 18 + 3x 41.  2y + 2z 43.  21y  35 45. 5x + 20m + 10 47.  4 + 8m  4n 49.  5x  2 51.  r + 3 + 7p 53. 3x + 4 55.  x + 3y 57. 6r + 8 59.  36x  70 61.  16x  25 63. 411 + y2 65. 111x + y2 67.  115 + x2 69. 301a + b2 71. commutative property of multiplication 73. associative property of addition 75. distributive property 77. associative property of multiplication 79. identity element of addition 81. distributive property 83. commutative and associative 1 1 properties of multiplication 85.  8; 87.  x; 89. 2x;  2x 91. false 93. no 95. yes 97. yes 99. no 8 x 101. a. commutative property of addition b. commutative property of addition c. associative property of addition 103. answers may vary 105. answers may vary
Chapter Vocabulary Check 1. inequality symbols 2. equation 9. base; exponent 10. denominator
3. absolute value 4. variable 11. grouping symbols 12. set
5. opposites
6. numerator
7. solution
8. reciprocals
Chapter Review 1. 6
3. 7 5. 6 1 d.  6 , 0 , 1 , 1 , 3 , 9.62 2
78
7. = e. p
9. 7
11. 4 Ú  3 1 f.  6, 0, 1, 1 , 3, p, 9.62 2
13. 0.03 6 0.3 17. Friday
15. a. 1 , 3 19.
2#2#3#3
b. 0 , 1 , 3 c.  6 , 0 , 1 , 3 3 12 13 21. 23. 25. 9 25 10 8
27. 15
Review of Real Numbers
29.
7 12
31. A = 1
47. 20  12 = 2 # 4
1 5 sq m; P = 4 m 6 12 49. 18
51. 5
33. 14 53. 63°
1 lb 8 55. yes
35. 18
7 lb 16
57. 9
37. Baby E 59.  2
39. c
61.  11
41. 63. 
4 49
3 16
43. 37
45.
65.  13.9
18 7
67.  14
1 79.  48 81. 3 83. undefined 85. undefined 87.  12 89. 9 6 # 93.  20  x 95. commutative property of addition 97. distributive property 99. associative property of addition 91.  7 x or  7x 101. distributive property 103. multiplicative inverse property 105. 5y  10 107.  7 + x  4z 109.  12z  27 111. 6 1 5 3 113.  15.3 115.  80 117. 119. 16 121.  5 123. 125. 1 ft 4 6 8 69. 5
71.  19
73. a
75. $51
77. 
Chapter Test 1. 0  7 0 7 5
2. 9 + 5 Ú 4
3.  5
4.  11
5.  3
6.  39
7. 12
8.  2
9. undefined
10.  8
1 3 23. a. 1 , 7 11. 
5 8 b. 0 , 1 , 7 12. 4
16. 3 17. 0 18. 7 19. 7 20. 6 21. = 22. 2221 6 10,993 1 1 c.  5 ,  1 , 0 , 1 , 7 d.  5 ,  1 , 0 , , 1 , 7 , 11.6 e. 27 , 3p f.  5 ,  1 , 0 , , 1 , 7 , 11.6 , 27 , 3p 24. 40 25. 12 26. 22 27.  1 4 4 28. associative property of addition 29. commutative property of multiplication 30. distributive property 31. multiplicative inverse property 32. 9 33.  3 34. second down 35. yes 36. 17° 37. $650 million 38. $420 13. 1.275
14.  32
15.  48
79
80
Equations, Inequalities, and Problem Solving
From Chapter 2 of Beginning and Intermediate Algebra, Fifth Edition. Elayn MartinGay. Copyright © 2013 by Pearson Education, Inc. All rights reserved.
81
Equations, Inequalities, and Problem Solving Top 10 Countries by Number of InternetCrime Complaints
1 Simplifying Algebraic Expressions
2 The Addition and Multiplication Properties of Equality
2
3
3 Solving Linear
9
Equations Integrated Review– Solving Linear Equations
4 An Introduction to Problem Solving
5 Formulas and Problem Solving
7
1 8
5 10
Top Five States by Number of InternetCrime Complaints 1. California
4
6
2. Florida 3. Texas 4. New York 5. New Jersey
6 Percent and Mixture Problem Solving
7 Further Problem Solving
Internet Crime
8 Solving Linear
The Internet Crime Complaint Center (IC3) is a joint operation between the FBI and the National WhiteCollar Crime Center. The IC3 receives and refers criminal complaints occurring on the Internet. Of course, nondelivery of merchandise or payment are the highest reported offenses. In Section 6, Exercises 15 and 16, we analyze a bar graph on the yearly number of complaints received by the IC3.
Inequalities Much of mathematics relates to deciding which statements are true and which are false. For example, the statement x + 7 = 15 is an equation stating that the sum x + 7 has the same value as 15. Is this statement true or false? It is false for some values of x and true for just one value of x, namely 8. Our purpose in this chapter is to learn ways of deciding which values make an equation or an inequality true.
Ages of Persons Filing Complaints Over 60 13.6%
Under 20 3.2%
20–29 18.8%
50–59 22.1% 40–49 22.1%
2
82
30–39 20.2%
Equations, Inequalities, and Problem Solving
1
Simplifying Algebraic Expressions
OBJECTIVES 1 Identify Terms, Like Terms, and Unlike Terms.
2 Combine Like Terms.
As we explore in this section, an expression such as 3x + 2x is not as simple as possible because—even without replacing x by a value—we can perform the indicated addition. OBJECTIVE
1 Identifying Terms, Like Terms, and Unlike Terms Before we practice simplifying expressions, some new language of algebra is presented. A term is a number or the product of a number and variables raised to powers.
3 Use the Distributive Property to
Terms
Remove Parentheses.
4 Write Word Phrases as
y,
2x3,
5,
Algebraic Expressions.
3xz 2,
2 , y
0.8z
The numerical coefficient (sometimes also simply called the coefficient) of a term is the numerical factor. The numerical coefficient of 3x is 3. Recall that 3x means 3 # x. Term 3x y
3
5 0.7ab3c 5 z
Numerical Coefficient 3 y3 1 1 since means # y 3 5 5 5 0.7 1
y
1
5
5
Helpful Hint The term y means 1y and thus has a numerical coefficient of 1. The term z means 1z and thus has a numerical coefficient of 1.
EXAMPLE 1 a. 3y
Identify the numerical coefficient in each term. x c. y d. x e. b. 22z 4 7
Solution a. The numerical coefficient of 3y is 3. b. The numerical coefficient of 22z 4 is 22. c. The numerical coefficient of y is 1, since y is 1y. d. The numerical coefficient of x is 1, since x is 1x. x 1 x 1 e. The numerical coefficient of is , since means # x. 7 7 7 7 PRACTICE
1 a. t
Identify the numerical coefficient in each term. w d. 43x 4 b. 7x c. 5
e. b
Terms with the same variables raised to exactly the same powers are called like terms. Terms that aren’t like terms are called unlike terms.
83
Equations, Inequalities, and Problem Solving
Like Terms
Unlike Terms
3x, 2x
5x, 5x
6x 2y, 2x 2y, 4x 2y 2 3
2
Why? Same variable x but different powers x and x 2
7y, 3z, 8x 2
3 2
3
2ab c , ac b
Reason
6abc , 6ab
Why? Different variables 2
Why? Different variables and different powers
Helpful Hint In like terms, each variable and its exponent must match exactly, but these factors don’t need to be in the same order. 2x 2y and 3yx 2 are like terms.
EXAMPLE 2 a. 2x, 3x
2
Determine whether the terms are like or unlike. b. 4x2y, x2y, 2x2y
d. x4, x4
c. 2yz, 3zy
Solution a. Unlike terms, since the exponents on x are not the same. b. Like terms, since each variable and its exponent match. c. Like terms, since zy = yz by the commutative property. d. Like terms. PRACTICE
2
Determine whether the terms are like or unlike. b. 5q, 3q 2 y5 d. y 5, 2
a. 4xy, 5yx c. 3ab2, 2ab2, 43ab2
OBJECTIVE
2 Combining Like Terms An algebraic expression containing the sum or difference of like terms can be simplified by applying the distributive property. For example, by the distributive property, we rewrite the sum of the like terms 3x + 2x as 3x + 2x = 13 + 22x = 5x Also, y 2 + 5y 2 = 1y 2 + 5y 2 = 1 1 + 52y 2 = 4y 2 Simplifying the sum or difference of like terms is called combining like terms.
EXAMPLE 3 a. 7x  3x
Simplify each expression by combining like terms.
b. 10y 2 + y 2
c. 8x2 + 2x  3x
Solution a. 7x  3x = 17  32x = 4x b. 10y 2 + y 2 = 10y 2 + 1y 2 = 110 + 12y 2 = 11y 2 c. 8x2 + 2x  3x = 8x2 + 12  32x = 8x2  x d. 9n 2  5n 2 + n 2 = 19  5 + 12n 2 = 5n 2 PRACTICE
3
Simplify each expression by combining like terms.
a. 3y + 11y c. 5x  3x2 + 8x2
84
b. 4x2 + x2 d. 20y 2 + 2y 2  y 2
d. 9n 2  5n 2 + n 2
Equations, Inequalities, and Problem Solving The previous example suggests the following: Combining Like Terms To combine like terms, add the numerical coefficients and multiply the result by the common variable factors.
EXAMPLE 4
Simplify each expression by combining like terms.
a. 2x + 3x + 5 + 2
b. 5a  3 + a + 2 1 e.  b + b 2
d. 2.3x + 5x  6
c. 4y  3y 2
Solution Use the distributive property to combine like terms. a. 2x + 3x + 5 + 2 = 12 + 32x + 15 + 22 = 5x + 7 b. 5a  3 + a + 2 = 5a + 1a + 1 3 + 22 = 1 5 + 12a + 1 3 + 22 = 4a  1 2 c. 4y  3y These two terms cannot be combined because they are unlike terms. d. 2.3x + 5x  6 = 12.3 + 52x  6 = 7.3x  6 1 1 1 1 e.  b + b =  b + 1b = a  + 1b b = b 2 2 2 2 PRACTICE
4
Use the distributive property to combine like terms.
a. 3y + 8y  7 + 2
b. 6x  3  x  3
d. 9y + 3.2y + 10 + 3
e. 5z  3z 4
c.
3 t  t 4
OBJECTIVE
3 Using the Distributive Property Simplifying expressions makes frequent use of the distributive property to also remove parentheses. It may be helpful to study the examples below. ±(3a+2)=±1(3a+2)=±1(3a)+(±1)(2)=3a+2 means
–(3a+2)=–1(3a+2)=–1(3a)+(–1)(2)=–3a2 means
EXAMPLE 5
Find each product by using the distributive property to remove
parentheses. a. 513x + 22
b. 21y + 0.3z  12
c. 19x + y  2z + 62
Solution a. 5(3x+2)=5 # 3x+5 # 2 Apply the distributive property. = 15x + 10 Multiply.
85
Equations, Inequalities, and Problem Solving Apply the distributive
b. –2(y+0.3z1)=–2(y)+(–2)(0.3z)+(–2)(–1) property. = 2y  0.6z + 2 Multiply.
c. 19x + y  2z + 62 = 119x + y  2z + 62 Distribute 1 over each term. = 119x2  11y2  11 2z2  1162 = 9x  y + 2z  6 PRACTICE
5
Find each product by using the distributive property to remove parentheses.
a. 312x  72 c. 12x  y + z  22
b. 51x  0.5z  52
Helpful Hint If a ; < sign precedes parentheses, the sign of each term inside the parentheses is changed when the distributive property is applied to remove parentheses. Examples: 12x + 12 = 2x  1
1 5x + y  z2 = 5x  y + z
1x  2y2 = x + 2y
1 3x  4y  12 = 3x + 4y + 1
When simplifying an expression containing parentheses, we often use the distributive property in both directions—first to remove parentheses and then again to combine any like terms.
EXAMPLE 6 a. 312x  52 + 1
Simplify each expression. b. 214x + 72  13x  12
c. 9 + 314x  102
Solution a. 312x  52 + 1 = 6x  15 + 1 Apply the distributive property. = 6x  14 Combine like terms. b. 214x + 72  13x  12 = 8x  14  3x + 1 Apply the distributive property. Combine like terms. = 11x  13 Helpful Hint Don’t forget to use the distributive property and multiply before adding or subtracting like terms.
c. 9+3(4x10)=9+12x30 = 21 + 12x or 12x  21
Apply the distributive property. Combine like terms.
PRACTICE
6
Simplify each expression.
a. 419x + 12 + 6
EXAMPLE 7
b. 712x  12  16  3x2
c. 8  516x + 52
Write the phrase below as an algebraic expression. Then simplify
if possible. ;Subtract 4x  2 from 2x  3.