Chapter 1 - XYZ Custom Plus

Chapter PretestTranslate into symbols. [1.1]1. The difference of 15 and x is 12. 15 − x = 12 2. The product of 6 and a is 30. 6a = 30Simplify according to the rule for order of operations. [1.1]3. 10 + 2(7 − 3) − 4 2 2 4. 15 + 24 ÷ 6 − 3 2 10For each number, name the opposite, reciprocal, and absolute value. [1.2]5. −6 6, −​ 1_ 6 ​, 6 6. ​5 _3 ​ −​5_ 3 ​, ​3_ 5 ​, ​5_ 3 ​Add. [1.3] Subtract. [1.4]7. 5 + (−8) −3 8. ​| −2 + 7 |​ + ​| −3 + (−4) |​ 12 9. −7 − 4 −11 10. 5 − (6 + 3) − 3 −7Multiply. [1.6]11. 7(−4) −28 12. 3(−4)(−2) 24 13. 9 ​ − ​ 1 _3 ​ ​ −3 14. ​ −​ 3 _32 ​ ​​ −​ _ 278 ​Simplify using the rule for order of operations. [1.1, 1.6, 1.7]15. −2(3) − 7 −13 16. 2(3) 3 − 4(−2) 4 __−4(3) + 5(−2) ___4(3 − 5) − 2(−6 + 8)−10 17. ​ ​ 2 18. ​ ​ −6−5 − 64(−2) + 10Apply the associative property, and then simplify. [1.5, 1.6]19. 5 + (7 + 3x) 12 + 3x 20. 3(−5y) −15yMultiply by applying the distributive property. [1.5, 1.6]21. −5(2x − 3) −10x + 15 22. ​_1 ​( 6 x + 12) 2x + 43From the set of numbers {−3, − ​ 1_ 2 ​, 2, ​√— 5 ​, π} list all the elements that are in the following sets. [1.8]23. Integers−3, 224. Rational numbers 25. Irrational numbers−3, −​ 1_ 2 ​, 2 ​√ — 5 ​, π26. Real numbers−3, −​ 1_ 2 ​, 2, ​√— 5 ​, πGetting Ready for Chapter 1To get started in this book, we assume that you can do simple addition and multiplication problems with wholenumbers and decimals. To check to see that you are ready for this chapter, work each of the problems below.1. 5 ⋅ 5 ⋅ 51252. 12 ÷ 433. 1.7 − 1.20.54. 10 − 9.50.55. 7(0.2)1.46. 0.3(6)1.87. 14 − 958. 39 ÷ 1339. 5 ⋅ 94510. 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 23211. 15 + 142912. 28 ÷ 7413. 26 ÷ 21314. 125 − 814415. 630 ÷ 631016. 210 ÷ 102117. 2 ⋅ 3 ⋅ 5 ⋅ 721018. 3 ⋅ 7 ⋅ 1123119. 2 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 763020. 24 ÷ 832Chapter 1 The Basics

Notation and Symbols 1.1Suppose you have a checking account that costs you $15 a month, plus $0.05 foreach check you write. If you write 10 checks in a month, then the monthly chargefor your checking account will be15 + 10(0.05)Do you add 15 and 10 first and then multiply by 0.05? Or do you multiply 10 and0.05 first and then add 15? If you don’t know the answer to this question, you willafter you have read through this section.Because much of what we do in algebra involves comparison of quantities, wewill begin by listing some symbols used to compare mathematical quantities. Thecomparison symbols fall into two major groups: equality symbols and inequalitysymbols.We will let the letters a and b stand for (represent) any two mathematical quantities.When we use letters to represent numbers, as we are doing here, we callthe letters variables.ObjectivesA State Translate the between place value phrases for numbers written instandard in English notation. and expressions writtenin symbols.Write a whole number in expandedB Simplify form. expressions containingexponents.Write a number in words.DC Simplify expressions using the rulefor Write order a number of operations. from words.D Read tables and charts.E Recognize the pattern in a sequenceof numbers.Examples now playing atMathTV.com/booksA Variables: An Intuitive LookWhen you filled out the application for the school you are attending, there wasa space to fill in your first name. “First name” is a variable quantity because thevalue it takes depends on who is filling out the application. For example, if yourfirst name is Manuel, then the value of “First Name” is Manuel. However, if yourfirst name is Christa, then the value of “First Name” is Christa.If we denote “First Name” as FN, “Last Name” as LN, and “Whole Name” as WN,then we take the concept of a variable further and write the relationship betweenthe names this way:FN + LN = WN(We use the + symbol loosely here to represent writing the names together witha space between them.) This relationship we have written holds for all peoplewho have only a first name and a last name. For those people who have a middlename, the relationship between the names isFN + MN + LN = WNA similar situation exists in algebra when we let a letter stand for a number ora group of numbers. For instance, if we say “let a and b represent numbers,” thena and b are called variables because the values they take on vary. We use the variablesa and b in the following lists so that the relationships shown there are truefor all numbers that we will encounter in this book. By using variables, the followingstatements are general statements about all numbers, rather than specificstatements about only a few numbers.Comparison SymbolsEquality: a = b a is equal to b (a and b represent the same number)a ≠ b a is not equal to bInequality: a < b a is less than ba ≮ b a is not less than ba > b a is greater than ba ≯ b a is not greater than ba ≥ b a is greater than or equal to ba ≤ b a is less than or equal to b1.1 Notation and Symbols3

4Chapter 1 The BasicsNoteIn the past you mayhave used the notation3 × 5 to denotemultiplication. In algebra it is bestto avoid this notation if possible,because the multiplication symbol× can be confused with thevariable x when written by hand.The symbols for inequality, < and > , always point to the smaller of the twoquantities being compared. For example, 3 < x means 3 is smaller than x. In thiscase we can say “3 is less than x” or “x is greater than 3”; both statements are correct.Similarly, the expression 5 > y can be read as “5 is greater than y” or as “y isless than 5” because the inequality symbol is pointing to y, meaning y is thesmaller of the two quantities.Next, we consider the symbols used to represent the four basic operations:addition, subtraction, multiplication, and division.NoteFor each example inthe text there is acorresponding practiceproblem in the margin. After youread through an example in thetext, work the practice problemwith the same number in the margin.The answers to the practiceproblems are given on the samepage as the practice problems. Besure to check your answers as youwork these problems. The workedoutsolutions for the practiceproblems are given in the back ofthe book. So if you fi nd a practiceproblem that you cannot work correctly,you can look up the correctsolution to that problem in theback of the book.Operation SymbolsAddition: a + b The sum of a and bSubtraction: a − b The difference of a and bMultiplication: a ⋅ b, (a)(b), a(b), (a)b, ab The product of a and bDivision:When we encounter the word sum, the implied operation is addition. To find thesum of two numbers, we simply add them. Difference implies subtraction, productimplies multiplication, and quotient implies division. Notice also that there is morethan one way to write the product or quotient of two numbers.grouping Symbolsa ÷ b, a/b, a_b , b )__aThe quotient of a and bParentheses ( ) and brackets [ ] are the symbols used for grouping numberstogether. (Occasionally, braces { } are also used for grouping, although theyare usually reserved for set notation, as we shall see.)The following examples illustrate the relationship between the symbols forcomparing, operating, and grouping and the English language.pRACtiCE pROBlEmSWrite an equivalent expression inEnglish.1. 3 + 7 = 102. 9 − 6 < 43. 4(2 + 3) ≠ 64. 6(8 − 1) = 42ExAmplESMathematical Expression English Equivalent1. 4 + 1 = 5 The sum of 4 and 1 is 5.2. 8 − 1 < 10 The difference of 8 and 1 is less than 10.3. 2(3 + 4) = 14 Twice the sum of 3 and 4 is 14.4. 3x ≥ 15 The product of 3 and x is greater than or equal to 15.5.4_x = 8 − x_5. y = y − 2 The quotient of y and 2 is equal to the difference of y2and 2.BExponentsAnswers1. The sum of 3 and 7 is 10.2. The difference of 9 and 6 is lessthan 4.3. 4 times the sum of 2 and 3 is notequal to 6.4. Six times the difference of 8 and1 is 42.5. The quotient of 4 and x is equalto the difference of 8 and x.The last type of notation we need to discuss is the notation that allows us to writerepeated multiplications in a more compact form—exponents. In the expression2 3 , the 2 is called the base and the 3 is called the exponent. The exponent 3 tells usthe number of times the base appears in the product; that is,2 3 = 2 ⋅ 2 ⋅ 2 = 8The expression 2 3 is said to be in exponential form, whereas 2 ⋅ 2 ⋅ 2 is said to bein expanded form. Here are some additional examples of expressions involvingexponents.

1.1 Notation and Symbols5ExamplesExpand and multiply.6. 5 2 = 5 ⋅ 5 = 25 Base 5, exponent 27. 2 5 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 32 Base 2, exponent 58. 10 3 = 10 ⋅ 10 ⋅ 10 = 1,000 Base 10, exponent 3Expand and multiply.6. 7 27. 3 48. 10 5Notation and Vocabulary Here is how we read expressions containing exponents.MathematicalExpressionWritten Equivalent5 2 five to the second power5 3 five to the third power5 4 five to the fourth power5 5 five to the fifth power5 6 five to the sixth powerWe have a shorthand vocabulary for second and third powers because the area ofa square with a side of 5 is 5 2 , and the volume of a cube with a side of 5 is 5 3 .5 2 can be read “five squared.” 5 3 can be read “five cubed.”55Area = 5 25Volume = 5 3COrder of OperationsThe symbols for comparing, operating, and grouping are to mathematics whatpunctuation marks are to English. These symbols are the punctuation marks formathematics.Consider the following sentence:Paul said John is tall.It can have two different meanings, depending on how it is punctuated.1. “Paul,” said John, “is tall.”2. Paul said, “John is tall.”Let’s take a look at a similar situation in mathematics. Consider the followingmathematical statement:5 + 2 ⋅ 7If we add the 5 and 2 first and then multiply by 7, we get an answer of 49. However,if we multiply the 2 and the 7 first and then add 5, we are left with 19. Wehave a problem that seems to have two different answers, depending on whetherwe add first or multiply first. We would like to avoid this type of situation. Everyproblem like 5 + 2 ⋅ 7 should have only one answer. Therefore, we have developedthe following rule for the order of operations.55Answers6. 49 7. 81 8. 100,000

6Chapter 1 The BasicsRule (Order of Operations)When evaluating a mathematical expression, we will perform the operationsin the following order, beginning with the expression in the innermost parenthesesor brackets first and working our way out.1. Simplify all numbers with exponents, working from left to right if more thanone of these expressions is present.2. Then do all multiplications and divisions left to right.3. Perform all additions and subtractions left to right.Use the rule for order of operationsto simplify each expression.9. 4 + 6 ⋅ 7These next examples involve using the rule for order of operations.Example 9Simplify 5 + 8 ⋅ 2.Solution 5 + 8 ⋅ 2 = 5 + 16 Multiply 8 ⋅ 2 first= 2110. 18 ÷ 6 ⋅ 2Example 10Simplify 12 ÷ 4 ⋅ 2.Solution 12 ÷ 4 ⋅ 2 = 3 ⋅ 2 Work left to right= 611. 5[4 + 3(7 + 2 ⋅ 4)]Example 11Simplify 2[5 + 2(6 + 3 ⋅ 4)].Solution 2[5 + 2(6 + 3 ⋅ 4)] = 2[5 + 2(6 + 12)] Simplify within the innermost= 2[5 + 2(18)] grouping symbols first}= 2[5 + 36]= 2[41]} Next, simplify inside the brackets= 82 Multiply12. 12 + 8 ÷ 2 + 4 ⋅ 5Example 12Simplify 10 + 12 ÷ 4 + 2 ⋅ 3.Solution 10 + 12 ÷ 4 + 2 ⋅ 3 = 10 + 3 + 6 Multiply and divide left to right= 19 Add left to right13. 3 4 + 2 5 ÷ 8 − 5 2Example 13Simplify 2 4 + 3 3 ÷ 9 − 4 2 .Solution 2 4 + 3 3 ÷ 9 − 4 2 = 16 + 27 ÷ 9 − 16 Simplify numbers with exponents= 16 + 3 − 16 Then, divide}= 19 − 16Finally, add and subtract= 3 left to rightD Reading Tables and Bar ChartsThe table below shows the average amount of caffeine in a number of beverages.The diagram in Figure 1 is a bar chart. It is a visual presentation of the informationin Table 1. The table gives information in numerical form, whereas the chartgives the same information in a geometric way. In mathematics, it is important tobe able to move back and forth between the two forms.Answers9. 46 10. 6 11. 245 12. 3613. 60

1.1 Notation and Symbols7Table 1Caffeine Content of Hot DrinksDrinkcaffeine(6-ounce cup) (milligrams)Brewed coffee 100Instant coffee 70Tea 50Cocoa 5Decaffeinated coffee 4Caffeine (mg)1101009080706050403020100100Brewedcoffee70Instantcoffee50TeaFigure 15 4CocoaDecafcoffeeExample 14Referring to Table 1 and Figure 1, suppose you have 3cups of brewed coffee, 1 cup of tea, and 2 cups of decaf in one day. Write anexpression that will give the total amount of caffeine in these six drinks, thensimplify the expression.Solution From the table or the bar chart, we find the number of milligramsof caffeine in each drink; then we write an expression for the total amount ofcaffeine:14. Suppose you drink 2 cups ofinstant coffee, 1 cup of cocoa,and 3 cups of tea. Write anexpression that will give thetotal amount of caffeine inthese six drinks, and then simplifythe expression.3(100) + 50 + 2(4)Using the rule for order of operations, we get 358 total milligrams of caffeine.E Number Sequences and Inductive ReasoningSuppose someone asks you to give the next number in the sequence of numbersbelow. (The dots mean that the sequence continues in the same pattern forever.)2, 5, 8, 11, . . .If you notice that each number is 3 more than the number before it, you wouldsay the next number in the sequence is 14 because 11 + 3 = 14. When we reasonin this way, we are using what is called inductive reasoning. In mathematics weuse inductive reasoning when we notice a pattern to a sequence of numbers andthen use the pattern to extend the sequence.Example 15Find the next number in each sequence.Solutiona. 3, 8, 13, 18, . . .b. 2, 10, 50, 250, . . .c. 2, 4, 7, 11, . . .To find the next number in each sequence, we need to look for a patternor relationship.a. For the first sequence, each number is 5 more than the numberbefore it; therefore, the next number will be 18 + 5 = 23.b. For the sequence in part (b), each number is 5 times the numberbefore it; therefore, the next number in the sequence will be5 ⋅ 250 = 1,250.15. Find the next number in eachsequence.a. 3, 7, 11, 15, . . .b. 1, 3, 9, 27, . . .c. 2, 5, 9, 14, . . .Answer14. 2(70) + 5 + 3(50)= 295 milligrams

8Chapter 1 The Basicsc. For the sequence in part (c), there is no number to add each timeor multiply by each time. However, the pattern becomes apparentwhen we look at the differences between the numbers:24 7 11. . .23 4Proceeding in the same manner, we would add 5 to get the next term, giving us11 + 5 = 16.In the introduction to this chapter we mentioned the mathematician known asFibonacci. There is a special sequence in mathematics named for Fibonacci. Hereit is.Fibonacci sequence = 1, 1, 2, 3, 5, 8, . . .Can you see the relationship among the numbers in this sequence? Start with two1’s, then add two consecutive members of the sequence to get the next number.Here is a diagram.1 1 2 3 5 8Sometimes we refer to the numbers in a sequence as terms of the sequence.16. Write the first 10 terms of thefollowing sequence.2, 2, 4, 6, 10, 16, . . .Example 16Write the first 10 terms of the Fibonacci sequence.Solution The first six terms are given above. We extend the sequence by adding5 and 8 to obtain the seventh term, 13. Then we add 8 and 13 to obtain 21.Continuing in this manner, the first 10 terms in the Fibonacci sequence are1, 1, 2, 3, 5, 8, 13, 21, 34, 55Each section of the book will end with some problems and questions like the onesthat follow. They are for you to answer after you have read through the sectionbut before you go to class. All of them require that you give written responses incomplete sentences. Writing about mathematics is a valuable exercise. If youwrite with the intention of explaining and communicating what you know tosomeone else, you will find that you understand the topic you are writing abouteven better than you did before you started writing. As with all problems in thiscourse, you want to approach these writing exercises with a positive point ofview. You will get better at giving written responses to questions as you progressthrough the course. Even if you never feel comfortable writing about mathematics,just the process of attempting to do so will increase your understanding andability in mathematics.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answers15. a. 19 b. 81 c. 2016. 2, 2, 4, 6, 10, 16, 26, 42, 68, 1101. What is a variable?2. Write the first step in the rule for order of operations.3. What is inductive reasoning?4. Explain the relationship between an exponent and its base.

1.1 Problem Set9Problem Set 1.1Answers appear in the Instructor’s Edition only.A For each sentence below, write an equivalent expression in symbols. [Examples 1–5]1. The sum of x and 5 is 14.x + 5 = 142. The difference of x and 4 is 8.x − 4 = 83. The product of 5 and y is less than 30.5y < 304. The product of 8 and y is greater than 16.8y > 165. The product of 3 and y is less than or equal to the sumof y and 6.3y ≤ y + 66. The product of 5 and y is greater than or equal to the differenceof y and 16.5y ≥ y − 167. The quotient of x and 3 is equal to the sum of x and 2. 8. The quotient of x and 2 is equal to the difference of x and_​ x 3 ​= x + 2 4._​ x 2 ​ = x − 4B Expand and multiply. [Examples 6–8]9. 3 2910. 4 21611. 7 24912. 9 28113. 2 3814. 3 32715. 4 36416. 5 312517. 2 41618. 3 48119. 10 210020. 10 410,00021. 11 212122. 111 212,321C Use the rule for order of operations to simplify each expression as much as possible. [Examples 9–13]23. 2 ⋅ 3 + 51124. 8 ⋅ 7 + 15725. 2(3 + 5)1626. 8(7 + 1)6427. 5 + 2 ⋅ 61728. 8 + 9 ⋅ 44429. (5 + 2) ⋅ 64230. (8 + 9) ⋅ 46831. 5 ⋅ 4 + 5 ⋅ 23032. 6 ⋅ 8 + 6 ⋅ 36633. 5(4 + 2)3034. 6(8 + 3)6635. 8 + 2(5 + 3)2436. 7 + 3(8 − 2)2537. (8 + 2)(5 + 3)8038. (7 + 3)(8 − 2)60Selected exercises available online at www.webassign.net/brookscole

10Chapter 1 The Basics39. 20 + 2(8 − 5) + 12740. 10 + 3(7 + 1) + 23641. 5 + 2(3 ⋅ 4 − 1) + 83542. 11 − 2(5 ⋅ 3 − 10) + 2343. 8 + 10 ÷ 21344. 16 − 8 ÷ 41445. 4 + 8 ÷ 4 − 2446. 6 + 9 ÷ 3 + 21147. 3 + 12 ÷ 3 + 6 ⋅ 53748. 18 + 6 ÷ 2 + 3 ⋅ 43349. 3 ⋅ 8 + 10 ÷ 2 + 4 ⋅ 23750. 5 ⋅ 9 + 10 ÷ 2 + 3 ⋅ 35951. (5 + 3)(5 − 3)1652. (7 + 2)(7 − 2)4553. 5 2 − 3 21654. 7 2 − 2 24555. (4 + 5) 28156. (6 + 3) 28157. 4 2 + 5 24158. 6 2 + 3 24559. 3 ⋅ 10 2 + 4 ⋅ 10 + 534560. 6 ⋅ 10 2 + 5 ⋅ 10 + 465461. 2 ⋅ 10 3 + 3 ⋅ 10 2 + 4 ⋅ 10 + 52,34562. 5 ⋅ 10 3 + 6 ⋅ 10 2 + 7 ⋅ 10 + 85,67863. 10 − 2(4 ⋅ 5 − 16)264. 15 − 5(3 ⋅ 2 − 4)565. 4[7 + 3(2 ⋅ 9 − 8)]14866. 5[10 + 2(3 ⋅ 6 − 10)]13067. 5(7 − 3) + 8(6 − 4)3668. 3(10 − 4) + 6(12 − 10)3069. 3(4 ⋅ 5 − 12) + 6(7 ⋅ 6 − 40)3670. 6(8 ⋅ 3 − 4) + 5(7 ⋅ 3 − 1)22071. 3 4 + 4 2 ÷ 2 3 − 5 25872. 2 5 + 6 2 ÷ 2 2 − 3 23273. 5 2 + 3 4 ÷ 9 2 + 6 26274. 6 2 + 2 5 ÷ 4 2 + 7 287

1.1 Problem Set11We are assuming that you know how to do arithmetic with decimals. Here are some problems to practice. Simplify eachexpression.75. 0.08 + 0.09 0.17 76. 0.06 + 0.04 0.10 77. 0.10 + 0.12 0.22 78. 0.08 + 0.06 0.1479. 4.8 − 2.5 2.3 80. 6.3 − 4.8 1.5 81. 2.07 + 3.48 5.55 82. 4.89 + 2.31 7.2083. 0.12(2,000) 240 84. 0.09(3,000) 270 85. 0.25(40) 10 86. 0.75(40) 3087. 510 ÷ 0.17 3,000 88. 400 ÷ 0.1 4,000 89. 240 ÷ 0.12 2,000 90. 360 ÷ 0.12 3,000Use a calculator to find the following quotients. Round your answers to the nearest hundredth, if rounding is necessary.91. 37.80 ÷ 1.07 35.33 92. 85.46 ÷ 4.88 17.51 93. 555 ÷ 740 0.75 94. 740 ÷ 108 6.8595. 70 ÷ 210 0.33 96. 15 ÷ 80 0.19 97. 6,000 ÷ 22 272.73 98. 51,000 ÷ 17 3,000Simplify each expression.99. 20 ÷ 2 ⋅ 10100100. 40 ÷ 4 ⋅ 550101. 24 ÷ 8 ⋅ 39102. 24 ÷ 4 ⋅ 636103. 36 ÷ 6 ⋅ 318104. 36 ÷ 9 ⋅ 28105. 48 ÷ 12 ⋅ 28106. 48 ÷ 8 ⋅ 318107. 16 − 8 + 412108. 16 − 4 + 820109. 24 − 14 + 818110. 24 − 16 + 614111. 36 − 6 + 1242112. 36 − 9 + 2047113. 48 − 12 + 1753114. 48 − 13 + 1550

12Chapter 1 The BasicsD Applying the Concepts [Example 14]115. Babies According to the chart, in 2002 the number ofbabies born to mothers between the ages of 20 and29 was 2,082,497. Write this number in words.Two million, eighty-two thousand, four hundred ninety-seven.116. Cars The map shows the highest producers of carsin 2002. According to the map, Germany produced6,213,000 cars. Write this number in words.Six million, two hundred thirteen thousand.20-somethings have most babiesUnder 15 7,31515–19 425,49320–29 2,082,49730–39 1,405,14640–44 95,78845–49 5,22450–54 263Source: National Center for Health StatisticsWhere Are Our Cars Coming From?United StatesFrance 3,019,000GermanySouth KoreaJapanSource: www.oica.net6,213,0004,086,00010,781,00011,596,000Food Labels In 1993 the government standardized the way in which nutrition information was presented on the labels ofmost packaged food products. Figure 2 shows a standardized food label from a package of cookies that I ate at lunch theday I was writing the problems for this problem set. Use the information in Figure 2 to answer the following questions.117. How many cookies are in the package?10118. If I paid $0.50 for the package of cookies, how much dideach cookie cost?$0.05119. If the “calories” category stands for calories per serving,how many calories did I consume by eating thewhole package of cookies?420120. Suppose that, while swimming, I burn 11 calories eachminute. If I swim for 20 minutes, will I burn enoughcalories to cancel out the calories I added by eating 5cookies?Yes, by 10 caloriesNutrition FactsServing Size 5 Cookies(about 43 g)Servings Per Container 2Calories 210Fat Calories 90* Percent Daily Values (DV) arebased on a 2,000 calorie diet.Amount/servingTotal Fat 9 gSat. Fat 2.5 gCholest. less than 5 mgSodium 110 mg%DV*15%12%2%5%Amount/servingTotal Carb. 30 gFiber 1 gSugars 14 gProtein 3 gVitamin A 0% • Vitamin C 0% • Calcium 2% • Iron 8%%DV*10%2%Sandwich cremesFigure 2

1.1 Problem Set13Education The chart shows the average income for peoplewith different levels of education. This data is publishedrecently by the U.S. Census. Use the information in thechart to answer the following questions.Who’s in the Money?100,00080,00060,000$51,568$67,073$93,33340,00020,000$19,041$28,6310No H.S.DiplomaHigh SchoolGrad/ GEDBachelor’sDegreeMaster’sDegreePhDSource: U.S. Census Bureau121. At a high-school reunion seven friends get together.One of them has a Ph.D., two have master’s degrees,three have bachelor’s degrees, and one did not goto college. Use order of operations to find their combinedannual income.$410,814122. At a dinner party for six friends, one has a master’sdegree, two have bachelor’s degrees, two have highschool diplomas, and one did not finish high school.Use order of operations to find their combined annualincome.$246,512123. Reading Tables and Charts The following table andbar chart give the amount of caffeine in five differentsoft drinks. How much caffeine is in each of thefollowing?a. A 6-pack of Jolt 600 mgb. 2 Coca-Colas plus 3 Tabs 231 mgCaffeine Content in Soft Drinks124. Reading Tables and Charts The following table and barchart give the amount of caffeine in five different nonprescriptiondrugs. How much caffeine is in each of thefollowing?a. A box of 12 Excedrin 780 mgb. 1 Dexatrim plus 4 Excedrin 460 mgCaffeine Content in Nonprescription DrugsDrinkCaffeine (milligrams)Nonprescription DrugCaffeine (milligrams)Jolt 100Tab 47Coca-Cola 45Diet Pepsi 367 UP 0Dexatrim 200NoDoz 100Excedrin 65Triaminicin tablets 30Dristan tablets 16Caffeine (mg)Caffeine (mg)100806040200100Jolt47 45TabCoca-Cola36DietPepsi07UP20018016014012010080604020020010065Dexatrim NoDoz Excedrin30 16Triaminicin Dristan

14Chapter 1 The Basics125. Reading Tables and Charts The following bar chartgives the number of calories burned by a 150-poundperson during 1 hour of various exercises. Theaccompanying table should display the same information.Use the bar chart to complete the table.ActivityCalories Burned by 150-Pound PersonCalories Burned in 1 HourBicycling 374Bowling 265Handball 680Jogging 680Skiing 544126. Reading Tables and Charts The following bar chart givesthe number of calories consumed by eating some popularfast foods. The accompanying table should displaythe same information. Use the bar chart to completethe table.Calories in Fast FoodFoodCaloriesMcDonald’s Hamburger 270Burger King Hamburger 260Jack in the Box Hamburger 280McDonald’s Big Mac 510Burger King Whopper 630Calories7006005004003002001000Calories Burned in 1 Hour by a 150-Pound Person680 680544374265Bicycling Bowling Handball Jogging SkiingActivityCalories7006005004003002001000McDonald's Burger KingHamburger HamburgerCalories in Fast Food270 260 280Jack in theBoxHamburger510McDonald'sBig Mac630Burger KingWhopperE Find the next number in each sequence. [Examples 15–16]127. 1, 2, 3, 4, . . . (The sequence of counting numbers)5128. 0, 1, 2, 3, . . . (The sequence of whole numbers)4129. 2, 4, 6, 8, . . . (The sequence of even numbers)10130. 1, 3, 5, 7, . . . (The sequence of odd numbers)9131. 1, 4, 9, 16, . . . (The sequence of squares)25132. 1, 8, 27, 64, . . . (The sequence of cubes)125133. 2, 2, 4, 6, . . . (A Fibonacci-like sequence)10134. 5, 5, 10, 15, . . . (A Fibonacci-like sequence)25

Real Numbers 1.2Table 1 and Figure 1 give the record low temperature, in degrees Fahrenheit, foreach month of the year in the city of Jackson, Wyoming. Notice that some of thesetemperatures are represented by negative numbers.Temperature (°F)3020100-10-20-30-40-50Jan Feb Mar Apr May June July Aug Sept Oct Nov DecFIGuRE 1TABLE 1RECORD lOW tEmpERAtuRES fOR JACkSON, WyOmiNgObjectivesA State Locate the and place label value points for on numbers the instandard number line. notation.B Write Change a whole a fraction number to an in equivalent expandedform. fraction with a new denominator.C Write Simplify a number expressions in words. containingabsolute value.Write a number from words.D Identify the opposite of a number.EfMultiply fractions.Identify the reciprocal of a number.g Find the perimeter and area ofsquares, rectangles, and triangles.Examples now playing atMathTV.com/bookstemperaturetemperaturemonth (Degrees fahrenheit) month (Degrees fahrenheit)January −50 July 24February −44 August 18March −32 September 14April −5 October 2May 12 November −27June 19 December −49A The Number LineIn this section we start our work with negative numbers. To represent negativenumbers in algebra, we use what is called the real number line. Here is how weconstruct a real number line: We first draw a straight line and label a convenientpoint on the line with 0. Then we mark off equally spaced distances in both directionsfrom 0. Label the points to the right of 0 with the numbers 1, 2, 3, . . . (thedots mean “and so on”). The points to the left of 0 we label in order, −1, −2, −3,. . . . Here is what it looks like.NoteIf there is no sign(+ or −) in front of anumber, the number isassumed to be positive (+).Negative directionPositive direction−5 −4 −3 −2 −1 0 1 2 3 4 5Negative numbersPositive numbersOriginThe numbers increase in value going from left to right. If we “move” to the right,we are moving in the positive direction. If we move to the left, we are moving inthe negative direction. When we compare two numbers on the number line, thenumber on the left is always smaller than the number on the right. For instance,−3 is smaller than −1 because it is to the left of −1 on the number line.NoteThere are other numberson the numberline that you maynot be as familiar with. They areirrational numbers such as π, √ — 2 ,√ — 3 . We will introduce these numberslater in the chapter.1.2 Real Numbers15

16Chapter 1 The BasicsPractice Problems1. Locate and label the points_associated with −2, − ​ 1 ​, 0, 1.5,22.75.−3 −2 −1 0 1 2 3Example 1Locate and label the points on the real number line associatedwith the numbers −3.5, −1 ​_ 1 4 ​, ​1 _2 ​, ​3 _​, 2.5.4Solution We draw a real number line from −4 to 4 and label the points inquestion.−3.5 −1 1 412342.5−4 −3 −2 −1 0 1 2 3 4DefinitionThe number associated with a point on the real number line is called thecoordinate of that point.In the preceding example, the numbers ​_ 1 2 ​, ​3 _4 ​, 2.5, −3.5, and −1 ​1 _​are the coordinatesof the points they4represent.DefinitionThe numbers that can be represented with points on the real number lineare called real numbers.Real numbers include whole numbers, fractions, decimals, and other numbersthat are not as familiar to us as these.Fractions on the Number LineAs we proceed through Chapter 1, from time to time we will review some of themajor concepts associated with fractions. To begin, here is the formal definitionof a fraction.DefinitionIf a and b are real numbers, then the expression​_a b ​ b ≠ 0is called a fraction. The top number a is called the numerator, and thebottom number b is called the denominator. The restriction b ≠ 0 keeps usfrom writing an expression that is undefined. (As you will see, division byzero is not allowed.)Answer1. See graph in the appendix titled“Solutions to Selected PracticeProblems.”The number line can be used to visualize fractions. Recall that for the fraction ​ a _b ​,a is called the numerator and b is called the denominator. The denominator indicatesthe number of equal parts in the interval from 0 to 1 on the number line.The numerator indicates how many of those parts we have. If we take that part ofthe number line from 0 to 1 and divide it into three equal parts, we say that wehave divided it into thirds (Figure 2). Each of the three segments is ​_ 1 ​(one third) of3the whole segment from 0 to 1.13130 12133Figure 213

1.2 Real Numbers17Two of these smaller segments together are ​_ 2 ​(two thirds) of the whole segment.3And three of them would be ​_ 3 ​(three thirds), or the whole segment.3Let’s do the same thing again with six equal divisions of the segment from 0 to1 (Figure 3). In this case we say each of the smaller segments has a length of ​ 1 _6 ​(one sixth).1616161616160 12345 166666Figure 3B Equivalent FractionsThe same point we labeled with ​_ 1 3 ​in Figure 2 is labeled with ​2 _​in Figure 3. Likewise,the point we labeled earlier with ​6_ 2 3 ​is now labeled ​4 _​. It must be true then6that_​ 2 6 ​= ​1 _3 ​ and ​4 _6 ​= ​2 _3 ​Actually, there are many fractions that name the same point as ​_ 1 ​. If we were to3divide the segment between 0 and 1 into 12 equal parts, 4 of these 12 equal parts​ ​ 4 _12 ​ ​would be the same as ​ 2 __​ 4 12 ​= ​2 _6 ​= ​1 _3 ​6 ​ o r ​1 _​; that is,3Even though these three fractions look different, each names the same point onthe number line, as shown in Figure 4. All three fractions have the same valuebecause they all represent the same number.0 123333= 10 12345 6= 166666 60 1 2 3 4 5 6 7 8 9 10 11 12= 112 12 12 12 12 12 12 12 12 12 12 12Figure 4DefinitionFractions that represent the same number are said to be equivalent.Equivalent fractions may look different, but they must have the same value.It is apparent that every fraction has many different representations, each ofwhich is equivalent to the original fraction. The next two properties give us a wayof changing the terms of a fraction without changing its value.

18Chapter 1 The BasicsProperty 1Multiplying the numerator and denominator of a fraction by the samenonzero number never changes the value of the fraction.Property 2Dividing the numerator and denominator of a fraction by the same nonzeronumber never changes the value of the fraction.2. Write ​_ 5 ​as an equivalent fractionwith denominator848.Example 2Write ​_3 ​as an equivalent fraction with denominator 20.4Solution The denominator of the original fraction is 4. The fraction we are tryingto find must have a denominator of 20. We know that if we multiply 4 by 5, weget 20. Property 1 indicates that we are free to multiply the denominator by 5 aslong as we do the same to the numerator._​ 3 4 ​= ​3 _ ⋅ 5 ​= ​15_4 ⋅ 5 20 ​The fraction ​_1520 ​is equivalent to the fraction ​3 _4 ​.C Absolute ValuesRepresenting numbers on the number line lets us give each number two importantproperties: a direction from zero and a distance from zero. The directionfrom zero is represented by the sign in front of the number. (A number without asign is understood to be positive.) The distance from zero is called the absolutevalue of the number, as the following definition indicates.DefinitionThe absolute value of a real number is its distance from zero on thenumber line. If x represents a real number, then the absolute value of x iswritten ​| x |​.Write each expression withoutabsolute value bars.3. ​| 7 |​Example 3Write the expression ​| 5 |​without absolute value bars.Solution ​| 5 |​ = 5 The number 5 is 5 units from zero4. ​| −7 |​5. ​| − ​3 _4 ​ | ​Answers2. ​ 30 _48 ​ 3. 7 4. 7 5. ​3 _4 ​Example 4Write the expression ​| −5 |​without absolute value bars.Solution ​| −5 |​ = 5 The number −5 is 5 units from zeroExample 5Write the expression ​| − ​1 _2 | ​ ​without absolute value bars.Solution​| − ​1 _2 ​ | ​ = ​1 _2 ​ The number − ​1 _2 ​ is ​1 _​ unit from zero2The absolute value of a number is never negative. It is the distance the numberis from zero without regard to which direction it is from zero. When workingwith the absolute value of sums and differences, we must simplify the expression

1.2 Real Numbers19inside the absolute value symbols first and then find the absolute value of thesimplified expression.ExamplesSimplify each expression.6. ​| 8 − 3 |​ = ​| 5 |​ = 57. ​| 3 ⋅ 2 3 + 2 ⋅ 3 2 |​ = ​| 3 ⋅ 8 + 2 ⋅ 9 |​ = ​| 24 + 18 |​ = ​| 42 |​ = 428. ​| 9 − 2 |​ − ​| 6 − 8 |​ = ​| 7 |​ − ​| −2 |​ = 7 − 2 = 5Simplify each expression.6. ​| 7 − 2 |​7. ​| 2 ⋅ 3 2 + 5 ⋅ 2 2 |​8. ​| 10 − 4 |​ − ​| 9 − 11 |​D OppositesAnother important concept associated with numbers on the number line is that ofopposites. Here is the definition.DefinitionNumbers the same distance from zero but in opposite directions from zeroare called opposites.Example 9Give the opposite of 5.NumberOppositeGive the opposite of each number.9. 8Solution 5 −5 5 and −5 are oppositesExample 10Give the opposite of −3.NumberOpposite10. −5Solution −3 3 −3 and 3 are oppositesExample 11SolutionGive the opposite of ​_1 4 ​.Number Opposite​_1 4 ​ − ​1 _4 ​ ​1 _4 ​ and − ​1 _​ are opposites411. − ​ 2 _3 ​Example 12Give the opposite of −2.3.NumberOpposite12. −4.2Solution −2.3 2.3 −2.3 and 2.3 are oppositesEach negative number is the opposite of some positive number, and each positivenumber is the opposite of some negative number. The opposite of a negativenumber is a positive number. In symbols, if a represents a positive number, then−(−a) = aOpposites always have the same absolute value. And, when you add any twoopposites, the result is always zero:a + (−a) = 0Answers6. 5 7. 38 8. 4 9. −8 10. 511. ​_ 2 ​ 12. 4.23

20Chapter 1 The BasicsE Multiplication with FractionsThe last concept we want to cover in this section is the concept of reciprocals.Understanding reciprocals requires some knowledge of multiplication with fractions.To multiply two fractions, we simply multiply numerators and multiplydenominators._13. Multiply 2 3 ⋅ _ 7 9NoteIn past math classesyou may have writtenfractions like 7_ 3(improper fractions) as mixednumbers, such as 2 1_ 3. In algebrait is usually better to write themas improper fractions rather thanmixed numbers.14. Multiply 8 1 _5 .ExAmplE 13SOlutiON_Multiply 3 4 ⋅ _ 5 7 .The product of the numerators is 15, and the product of the denominatorsis 28:3_4 ⋅ _ 5 7 = _ 3 ⋅ 54 ⋅ 7 = _ 1528ExAmplE 14Multiply 7 _ 1 3 ._SOlutiON The number 7 can be thought of as the fraction 7 1 :7 1 _3 = 7 _1 1 __3 = 7 ⋅ 11 ⋅ 3 = _ 7 315. Expand and multiply 11 _12 2 .ExAmplE 15SOlutiONhaveExpand and multiply _ 2 3 3 .Using the definition of exponents from the previous section, we 2 _3 3 = 2 _3 ⋅ 2 _3 ⋅ 2 _3 = 8 _27fReciprocalsWe are now ready for the definition of reciprocals.DefinitionTwo numbers whose product is 1 are called reciprocals.ExAmplESGive the reciprocal of each number.Give the reciprocal of each number.16. 6Number16. 5Reciprocal1_5_Because 5 1 _=5 5 _1 1 _=5 5 5 = 117. 317. 21_2_Because 2 1 _=2 2 _1 1 _=2 2 2 = 118. 1 _218.1_3_3 Because 1 3 (3) = _ 1 _3 3 _=1 3 3 = 119. 2 _319.3_44_3Because 3 _4 4 __=3 1212 = 1Answers13. 14 _2714. 8 _515. 121 _14416. 1 _617. 1 _318. 2 19. 3 _2

1.2 Real Numbers21Although we will not develop multiplication with negative numbers until laterin the chapter, you should know that the reciprocal of a negative number is also a_negative number. For example, the reciprocal of −4 is − ​ 1 4 ​.Previously we mentioned that a variable is a letter used to represent a numberor a group of numbers. An expression that contains any combination of numbers,variables, operation symbols, and grouping symbols is called an algebraic expression(sometimes referred to as just an expression). This definition includes the useof exponents and fractions. Each of the following is an algebraic expression.3x + 5 4t 2 − 9 x 2 − 6xy + y 2 −15x 2 y 4 z 5 ​_a 2 − 9​ ​(x__− 3)(x + 2)​a − 3 4xIn the last two expressions, the fraction bar separates the numerator from thedenominator and is treated the same as a pair of grouping symbols; it groups thenumerator and denominator separately.The Value of an Algebraic ExpressionAn expression such as 3x + 5 will take on different values depending on whatx is. If we were to let x equal 2, the expression 3x + 5 would become 11. On theother hand, if x is 10, the same expression has a value of 35:When x = 2 When x = 10the expression 3x + 5 the expression 3x + 5becomes 3(2) + 5 becomes 3(10) + 5= 6 + 5 = 30 + 5= 11 = 35Table 2 lists some other algebraic expressions, along with specific values forthe variables and the corresponding value of the expression after the variable hasbeen replaced with the given number.Table 2Original Value of Value ofExpression the Variable the Expression5x + 2 x = 4 5(4) + 2 = 20 + 2= 223x − 9 x = 2 3(2) − 9 = 6 − 9= −34t 2 − 9 t = 5 4(5 2 ) − 9 = 4(25) − 9= 100 − 9= 91​_a 2 − 9a − 3 ​ a = 8 _ 8 2 − 9​= ​64_ − 98 − 3 8 − 3 ​= ​ 55 _5 ​= 11

22NoteThe vertical linelabeled h in the triangleis its height, oraltitude. It extends from the topof the triangle down to the base,meeting the base at an angle of90°. The altitude of a triangle isalways perpendicular to the base.The small square shown where thealtitude meets the base is usedto indicate that the angle formedis 90°.Chapter 1 The Basicsg Formulas for Area and PerimeterfACtS fROm gEOmEtRy: formulas for Area and perimeterA square, rectangle, and triangle are shown in the following figures. Notethat we have labeled the dimensions of each with variables. The formulasfor the perimeter and area of each object are given in terms of itsdimensions.A SquareA RectangleA TriangleslawwhcssblsPerimeter = 4sArea = s 2Perimeter = 2l + 2wArea = lwPerimeter = a + b + c1Area = bh2The formula for perimeter gives us the distance around the outside of theobject along its sides, whereas the formula for area gives us a measure of theamount of surface the object has.20. Find the perimeter and area ofthe figures in Example 20, withthe following changes:a. s = 4 feetb. l = 7 inchesw = 5 inchesc. a = 40 metersb = 50 meters (base)c = 30 metersh = 24 metersAnswer20. a. Perimeter = 16 feetArea = 16 square feetb. Perimeter = 24 inchesArea = 35 square inchesc. Perimeter = 120 metersArea = 600 square metersExAmplE 20Find the perimeter and area of each figure.a. b. c.SOlutiON5 ft 8 in.6 in.20 m12 m25 m15 mWe use the preceding formulas to find the perimeter and the area. Ineach case, the units for perimeter are linear units, whereas the units for area aresquare units.a. Perimeter = 4s = 4 ⋅ 5 feet = 20 feetArea = s 2 = (5 feet) 2 = 25 square feetb. Perimeter = 2l + 2w = 2(8 inches) + 2(6 inches) = 28 inchesArea = lw = (8 inches)(6 inches) = 48 square inchesc. Perimeter = a + b + c= (20 meters) + (25 meters) + (15 meters)= 60 meters_Area = 1 2 bh = _ 1 (25 meters)(12 meters) = 150 square meters2Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. What is a real number?2. Explain multiplication with fractions.3 . How do you find the opposite of a number?

1.2 Problem Set23Problem Set 1.2A Draw a number line that extends from −5 to +5. Label the points with the following coordinates. [Example 1]1. 5 2. −2 3. −4 4. −35. 1.5 6. −1.5 7. ​ 9 _4 ​ 8. ​8 _3 ​−4 −3 −2 −1.51.594835−5−4 −3 −2 −1 0 1 2 3 45B Write each fraction of the following as an equivalent fraction with denominator 24. [Example 2]9. ​ 3 _4 ​10. ​_5 6 ​11. ​_1 2 ​12. ​_1 8 ​13. ​_5 8 ​14. ​_712 ​24 ​ ​_ 2024 ​ ​_ 1224 ​ ​_ 324 ​ ​_ 1524 ​ ​_ 1424 ​​_18B Write each fraction as an equivalent fraction with denominator 60.15. ​_3 5 ​16. ​_512 ​17. ​_1130 ​18. ​_910 ​​_3660 ​ ​_ 2560 ​ ​_ 2260 ​ ​_ 5460 ​Fill in the missing numerator so the fractions are equal. [Example 2]19. ​ 1 _2 ​= ​2 _4 ​ 20. ​1 _5 ​= ​ 4 _20 ​ 21. ​5 _9​= ​25_45 ​ 22. ​2 _5​= ​18_45 ​ 23. ​3 _4 ​= ​6 _8 ​ 24. ​1 _2 ​= ​4 _8 ​C D F For each of the following numbers, give the opposite, the reciprocal, and the absolute value. (Assume all variablesare nonzero.) [Example 9–12, 16–19]25. 10−10, ​ 1 _10 ​, 10 26. 8−8, ​ 1 _8 ​, 8 27. ​ 3 _4 ​−​ 3 _4 ​, ​ 4 _3 ​, ​ 3 _4 ​ 28. ​ 5 _7 ​−​ 5 _7 ​, ​ 7 _5 ​, ​ 5 _29. ​_112 ​7 ​ _−​ 112 ​, ​ _ 21130. ​_163 ​11​, ​_2 ​ _−​ 163 ​, ​ _ 31616​, ​_3 ​31. −33, −​ 1 _3 ​, 3 32. −55, −​ 1 _5 ​, 5 33. −​ 2 _5 ​​_ 2 5 ​, −​5 _2 ​, ​ _ 25 ​ 34. −​ 3 _8 ​​_ 3 8 ​, −​8 _3 ​, ​ _ 38 ​ 35. x−x, ​ 1 _x ​, ​ | x |​36. a−a, ​ 1 _a ​, ​ | a |​Place one of the symbols < or > between each of the following to make the resulting statement true.37. −5 < −3 38. −8 < −1 39. −3 > −7 40. −6 < 5Selected exercises available online at www.webassign.net/brookscole

24Chapter 1 The Basics41. ​| −4 |​ > −​| −4 |​ 42. 3 > −​| −3 |​ 43. 7 > −​| −7 |​ 44. −7 < ​| −7 |​45. − ​ 3 _4 ​ < − ​ 1 _4 ​ 46. −​2 _3 ​< −​1 _3 ​ 47. −​3 _2 ​< −​3 _4 ​ 48. −​8 _3_​> −​173 ​C Simplify each expression. [Example 3–8]49. ​| 8 − 2 |​650. ​| 6 − 1 |​551. ​| 5 ⋅ 2 3 − 2 ⋅ 3 2 |​2252. ​| 2 ⋅ 10 2 + 3 ⋅ 10 |​23053. ​| 7 − 2 |​ − ​| 4 − 2 |​354. ​| 10 − 3 |​ − ​| 4 − 1 |​455. 10 − ​| 7 − 2(5 − 3) |​756. 12 − ​| 9 − 3(7 − 5) |​957. 15 − ​| 8 − 2(3 ⋅ 4 − 9) |​ − 10358. 25 − ​| 9 − 3(4 ⋅ 5 − 18) |​ − 202E Multiply the following. [Example 13–15]59. ​_2 3 ​⋅ ​4 _5 ​ 60. ​_1 4 ​⋅ ​3 _5 ​ 61. ​_1 2 ​(3)62. ​_1 3 ​(2)63. ​_1 4 ​(5)64. ​_1 5 ​(4)​_815 ​ ​_ 320 ​ ​_ 3 2 ​ ​_ 2 3 ​ ​_ 5 4 ​ ​_ 4 5 ​65. ​ 4 _3 ​⋅ ​3 _4 ​66. ​ 5 _7 ​⋅ ​7 _5 ​67. 6 ​ ​ 1 _6 ​ ​68. 8 ​ ​ 1 _8 ​ ​69. 3 ⋅ ​ 1 _3 ​70. 4 ⋅ ​ 1 _4 ​111111Multiply.71. a. ​_1 2 ​(4) 272. a. ​_1 4 ​(8) 273. a. ​_3 2 ​(4) 674. a. ​_3 4 ​(8) 6b. ​_1 2 ​(8) 4b. ​_1 4 ​(24) 6b. ​_3 ​(8) 12b. ​_3 ​(24) 182 4c. ​_1 2 ​(16) 8c. ​_1 4 ​(16) 4c. ​_3 ​(16) 24c. ​_3 ​(16) 122 4d. ​_1 2 ​(0.06) 0.03 d. ​_1 4 ​(0.20) 0.05 d. ​_3 2 (0.06) 0.09 d. ​_3 ​(0.20) 0.154Expand and multiply.75. ​ _​ 3 24 ​ ​ ​76. ​ ​ 5 2_6 ​ ​ ​77. ​ ​ 2 3_3 ​ ​ ​78. ​ ​ 1 3_2 ​ ​ ​79. ​ ​ 1 4_10 ​ ​ ​ 80. ​ ​ 1 5_10 ​ ​ ​​_916 ​ ​_ 2536 ​ ​_ 827 ​ ​_ 1 8 ​ ​1_10,000 ​ ​1_100,000 ​

1.2 Problem Set25Problems here involve finding the value of an algrebraic expression.81. Find the value of 2x − 6 whena. x = 5 4b. x = 10 14c. x = 15 24d. x = 20 3484. Find the value of each expressionwhen x is 3.a. x + 3 6b. 3x 9c. x 2 982. Find the value of 2(x − 3) whena. x = 5 4b. x = 10 14c. x = 15 24d. x = 20 3485. Find the value of each expressionwhen x is 4.a. x 2 + 1 17b. (x + 1) 2 25c. x 2 + 2x + 1 2583. Find the value of each expressionwhen x is 10.a. x + 2 12b. 2x 20c. x 2 100d. 2 x 1,02486. Find the value of b 2 − 4ac whena. a = 2, b = 6, c = 3 12b. a = 1, b = 5, c = 6 1c. a = 1, b = 2, c = 1 0d. 3 x 27Find the next number in each sequence.87. 1, ​_1 3 ​, ​1 _5 ​, ​1 _​, . . . (Reciprocals of odd numbers)88. ​_17​_ 1 9 ​ ​_ 110 ​2 ​, ​1 _4 ​, ​1 _6 ​, ​1 _​, . . . (Reciprocals of even numbers)889. 1, ​_1 4 ​, ​1 _9 ​, ​ _ 1 ​, . . . (Reciprocals of squares)1690. 1, ​_1 8 ​, ​ _ 127 ​, ​ _ 1 ​, . . . (Reciprocals of cubes)64​_125 ​ ​_ 1125 ​G Find the perimeter and area of each figure. [Example 20]91.92.1 inch15 millimeters15 millimeters1 inch4 inches; 1 square inch60 millimeters; 225 square millimeters93.94.0.75 inches1.5 centimeters1.5 inches4.5 centimeters4.5 inches; 1.125 square inches12 centimeters; 6.75 square centimeters

26Chapter 1 The Basics95.96.2.75 cm2.5 cm3.5 cm1.8 in.1 in.1.2 in.4 cm10.25 centimeters; 5 square centimeters5 inches; 1 square inch2 in.Applying the Concepts97. football yardage A football team gains 6 yards on oneplay and then loses 8 yards on the next play. To whatnumber on the number line does a loss of 8 yards correspond?The total yards gained or lost on the twoplays corresponds to what negative number?−8, −298. Checking Account Balance A woman has a balance of $20in her checking account. If she writes a check for $30,what negative number can be used to represent the newbalance in her checking account?−$10temperature In the United States, temperature is measured on the Fahrenheit temperature scale. On this scale, water boilsat 212 degrees and freezes at 32 degrees. To denote a temperature of 32 degrees on the Fahrenheit scale, we write32°F, which is read “32 degrees Fahrenheit”Use this information for Problems 99 and 100.99. temperature and Altitude Marilyn is flying from Seattleto San Francisco on a Boeing 737 jet. When the planereaches an altitude of 35,000 feet, the temperatureoutside the plane is 64 degrees below zero Fahrenheit.Represent the temperature with a negative number. Ifthe temperature outside the plane gets warmer by 10degrees, what will the new temperature be?−64°F; −54°F101. google Earth The Google image shows theStratosphere Tower in Las Vegas, Nevada. The_Stratosphere Tower is 23 the height of the Space12Needle in Seattle, Washington. If the Space Needleis about 600 feet tall, how tall is the StratosphereTower?1150 feet tallBuildings ©2008 Sanborn100. temperature Change At 10:00 in the morning in WhiteBear Lake, Minnesota, John notices the temperatureoutside is 10 degrees below zero Fahrenheit. Write thetemperature as a negative number. An hour later it haswarmed up by 6 degrees. What is the temperature at11:00 that morning?−10°F; −4°F102. Skyscrapers The chart shows the heights of the threetallest buildings in the world. The TransamericaPyramid in San Francisco, California, is half the heightof the Taipei 101 tower. How tall is the TransamericPyramid?835 FeetSuch Great HeightsTaipei 101Taipei, TaiwanPetronas Tower 1 & 2Kuala Lumpur, Malaysia1,483 ft1,670 ft Sears TowerChicago, USA1,450 ftBuildings ©2008 SanbornImage © 2008 DigitalGlobeSource: www.tenmojo.com

1.2 Problem Set27Wind Chill Table 2 is a table of wind chill temperatures. The top row gives the air temperature, and the first column iswind speed in miles per hour. The numbers within the table indicate how cold the weather will feel. For example, if thethermometer reads 30°F and the wind is blowing at 15 miles per hour, the wind chill temperature is 9°F. Use Table 2 toanswer Questions 103 through 106.TABLE 2WiND CHill tEmpERAtuRESAir temperature (°f)Wind Speed(mph) 30° 25° 20° 15° 10° 5° 0° −5°10 16° 10° 3° −3° −9° −15° −22° −27°15 9° 2° −5° −11° −18° −25° −31° −38°20 4° −3° −10° −17° −24° −31° −39° −46°25 1° −7° −15° −22° −29° −36° −44° −51°30 −2° −10° −18° −25° −33° −41° −49° −56°103. Reading tables Find the wind chill temperature if thethermometer reads 20°F and the wind is blowing at25 miles per hour.−15°F104. Reading tables Which will feel colder: a day with an airtemperature of 10°F with a 25-mile-per-hour wind, or aday with an air temperature of −5° F and a 10-mile-perhourwind?A day with an air temperature of 10°F with a 25-mile-per-hour wind105. Reading tables Find the wind chill temperature if thethermometer reads 5°F and the wind is blowing at15 miles per hour.−25°F106. Reading tables Which will feel colder: a day with an airtemperature of 5°F with a 20-mile-per-hour wind, or aday with an air temperature of −5° F and a 10-mile-perhourwind?A day with an air temperature of 5°F with a 20-mile-per-hour wind107. Scuba Diving Steve is scuba diving near his home inMaui. At one point he is 100 feet below the surface.Represent this number with a negative number. If hedescends another 5 feet, what negative number willrepresent his new position?−100 feet; −105 feet108. google Earth The Google Earth map shows YellowstoneNational Park. If the boundary of the park is roughly arectangle with a length of 63 miles and a width of 54miles, find the perimeter and the area of the park.P = 234 miles, A = 3,402 square miles100 ft5 ftImage © 2008 DigitalGlobe

28Chapter 1 The BasicsCalories and Exercise Table 3 gives the amount of energy expended per hour for various activities for a person weighing120, 150, or 180 pounds. Use Table 3 to answer questions 109 – 112.109. Suppose you weigh 120 pounds. How many calorieswill you burn if you play handball for 2 hours andthen ride your bicycle for an hour?1,387 caloriesTable 3Energy Expended from ExercisingCalories per HourActivity 120 lb 150 lb 180 lbBicycling 299 374 449Bowling 212 265 318Handball 544 680 816Horseback trotting 278 347 416Jazzercise 272 340 408110. How many calories are burned by a person weighing150 pounds who jogs for ​_ 1 ​hour and then goes bicyclingfor 2 hours?21,088 caloriesJogging 544 680 816Skiing (downhill) 435 544 653111. Two people go skiing. One weighs 180 pounds andthe other weighs 120 pounds. If they ski for 3 hours,how many more calories are burned by the personweighing 180 pounds?654 more calories112. Two people spend 3 hours bowling. If one weighs 120pounds and the other weighs 150 pounds, how manymore calories are burned during the evening by theperson weighing 150 pounds?159 more calories

Addition of Real NumbersSuppose that you are playing a friendly game of poker with some friends, andyou lose $3 on the first hand and $4 on the second hand. If you represent winningwith positive numbers and losing with negative numbers, how can you translatethis situation into symbols? Because you lost $3 and $4 for a total of $7, one wayto represent this situation is with addition of negative numbers:(−$3) + (−$4) = −$7From this equation, we see that the sum of two negative numbers is a negativenumber. To generalize addition with positive and negative numbers, we use thenumber line.Because real numbers have both a distance from zero (absolute value) and adirection from zero (sign), we can think of addition of two numbers in terms ofdistance and direction from zero.1.3ObjectivesA Add any combination of positiveand negative numbers.B Simplify expressions using the rulefor order of operations.C Extend an arithmetic sequence.Examples now playing atMathTV.com/booksAAdding Positive and Negative NumbersLet’s look at a problem for which we know the answer. Suppose we want to addthe numbers 3 and 4. The problem is written 3 + 4. To put it on the number line,we read the problem as follows:1. The 3 tells us to “start at the origin and move 3 units in the positive direction.”2. The + sign is read “and then move.”3. The 4 means “4 units in the positive direction.”To summarize, 3 + 4 means to start at the origin, move 3 units in the positivedirection, and then move 4 units in the positive direction.Start3 units 4 unitsEnd−7−6−5−4 −3 −2 −1 0 1 2 3 4 5 6 7We end up at 7, which is the answer to our problem: 3 + 4 = 7.Let’s try adding other combinations of positive and negative 3 and 4 on thenumber line.Example 1Add 3 + (−4).Solution Starting at the origin, move 3 units in the positive direction and then4 units in the negative direction.Practice Problems1. Add 2 + (−5). You can use thenumber line in Example 1 if youlike.EndStart3 units4 units−7−6−5−4 −3 −2 −1 0 1 2 3 4 5 6 7We end up at −1; therefore, 3 + (−4) = −1.1.3 Addition of Real NumbersAnswer1. −329

30Chapter 1 The Basics2. Add −2 + 5.Example 2Add −3 + 4.Solution Starting at the origin, move 3 units in the negative direction and then4 units in the positive direction.4 units3 unitsStartEnd−7−6−5−4 −3 −2 −1 0 1 2 3 4 5 6 7We end up at +1; therefore, −3 + 4 = 1.3. Add −2 + (−5).Example 3Add −3 + (−4).Solution Starting at the origin, move 3 units in the negative direction and then4 units in the negative direction.End4 units3 unitsStart−7−6−5−4 −3 −2 −1 0 1 2 3 4 5 6 7We end up at −7; therefore, −3 + (−4) = −7.Here is a summary of what we have just completed:3 + 4 = 73 + (−4) = −1−3 + 4 = 1−3 + (−4) = −7Let’s do four more problems on the number line and then summarize ourresults into a rule we can use to add any two real numbers.4. Show 9 + (−4).Example 4Show that 5 + 7 = 12.SolutionStart5 units 7 unitsEnd−3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 125. Show 7 + (−3).Example 5Show that 5 + (−7) = −2.SolutionEnd7 unitsStart5 units−3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 126. Show −10 + 12.Example 6Show that −5 + 7 = 2.Solution7 unitsEnd5 unitsStartAnswers2. 3 3. −7 4. 5 5. 4 6. 2−12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3

1.3 Addition of Real Numbers31ExAmplE 7Show that −5 + (−7) = −12.7. Show −4 + (−6).SOlutiONEnd7 units5 unitsStart−12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3If we look closely at the results of the preceding addition problems, we can seethat they support (or justify) the following rule.RuleTo add two real numbers with1. The same sign: Simply add their absolute values and use the common sign.(Both numbers are positive, the answer is positive. Both numbers are negative,the answer is negative.)2. Different signs: Subtract the smaller absolute value from the larger. Theanswer will have the sign of the number with the larger absolute value.NoteThis rule is what wehave been workingtoward. The rule isvery important. Be sure that youunderstand it and can use it. Theproblems we have done up to thispoint have been done simply tojustify this rule. Now that we havethe rule, we no longer need todo our addition problems on thenumber line.This rule covers all possible combinations of addition with real numbers. Youmust memorize it. After you have worked a number of problems, it will seemalmost automatic.ExAmplE 8Add all combinations of positive and negative 10 and 13.SOlutiON Rather than work these problems on the number line, we use therule for adding positive and negative numbers to obtain our answers:10 + 13 = 2310 + (−13) = −3−10 + 13 = 38. Add all combinations of positiveand negative 9 and 12.9 + 12 =9 + (−12) =−9 + 12 =−9 + (−12) =−10 + (−13) = −23ExAmplE 9Add all combinations of positive and negative 12 and 17.SOlutiON Applying the rule for adding positive and negative numbers, wehave12 + 17 = 2912 + (−17) = −5−12 + 17 = 5−12 + (−17) = −299. Add all combinations of positiveand negative 6 and 10.6 + 10 =6 + (−10) =−6 + 10 =−6 + (−10) =BOrder of OperationsExAmplE 10Add −3 + 2 + (−4).SOlutiONApplying the rule for order of operations, we add left to right:10. Add −5 + 2 + (−7).−3 + 2 + (−4) = −1 + (−4)= −5Answers7. −10 8. 21, −3, 3, −219. 16, −4, 4, −16 10. −10

32Chapter 1 The Basics11. Add −4 + [2 + (−3)] + (−1).Example 11Add −8 + [2 + (−5)] + (−1).SolutionAdding inside the brackets first and then left to right, we have−8 + [2 + (−5)] + (−1) = −8 + (−3) + (−1)= −11 + (−1)= −1212. Simplify−5 + 3(−4 + 7) + (−10).Example 12Simplify −10 + 2(−8 + 11) + (−4).Solution First, we simplify inside the parentheses. Then, we multiply. Finally,we add left to right:−10 + 2(−8 + 11) + (−4) = −10 + 2(3) + (−4)= −10 + 6 + (−4)= −4 + (−4)= −8CArithmetic SequencesThe pattern in a sequence of numbers is easy to identify when each number inthe sequence comes from the preceding number by adding the same amount.DefinitionAn arithmetic sequence is a sequence of numbers in which each number(after the first number) comes from adding the same amount to the numberbefore it.13. Find the next two numbers ineach of the following arithmeticsequences.a. 5, 9, 13, . . .b. 6, 6.5, 7, . . .c. 4, 0, −4, . . .Example 13Each sequence below is an arithmetic sequence. Find thenext two numbers in each sequence.a. 7, 10, 13, . . .Solutionb. 9.5, 10, 10.5, . . .c. 5, 0, −5, . . .Because we know that each sequence is arithmetic, we know to lookfor the number that is added to each term to produce the next consecutive term.a. 7, 10, 13, . . . : Each term is found by adding 3 to the term before it. Therefore,the next two terms will be 16 and 19.b. 9.5, 10, 10.5, . . . : Each term comes from adding 0.5 to the term before it.Therefore, the next two terms will be 11 and 11.5.c. 5, 0, −5, . . . : Each term comes from adding −5 to the term before it.Therefore, the next two terms will be −5 + (−5) = −10 and−10 + (−5) = −15.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answers11. −6 12. −613. a. 17, 21 b. 7.5, 8c. −8, −121. Explain how you would add 3 and −5 on the number line.2. How do you add two negative numbers?3. What is an arithmetic sequence?4. Why is the sum of a number and its opposite always 0?

1.3 Problem Set33Problem Set 1.3A [Example 8–9]1. Add all combinations of positive and negative 3 and 5.3 + 5 = 8, 3 + (−5) = −2−3 + 5 = 2, −3 + (−5) = −82. Add all combinations of positive and negative 6 and 4.6 + 4 = 10, 6 + (−4) = 2,−6 + 4 = −2, −6 + (−4) = −103. Add all combinations of positive and negative 15 and20.15 + 20 = 35, 15 + (−20) = −5,−15 + 20 = 5, −15 + (−20) = −354. Add all combinations of positive and negative 18 and 12.18 + 12 = 30, 18 + (−12) = 6,−18 + 12 = −6, −18 + (−12) = −30A Work the following problems. You may want to begin by doing a few on the number line. [Example 1–7]5. 6 + (−3)36. 7 + (−8)−17. 13 + (−20)−78. 15 + (−25)−109. 18 + (−32)10. 6 + (−9)11. −6 + 312. −8 + 7−14−3−3−113. −30 + 514. −18 + 615. −6 + (−6)16. −5 + (−5)−25−12−12−1017. −9 + (−10)18. −8 + (−6)19. −10 + (−15)20. −18 + (−30)−19−14−25−48Selected exercises available online at www.webassign.net/brookscole

34Chapter 1 The BasicsB Work the following problems using the rule for addition of real numbers. You may want to refer back to the rule fororder of operations. [Example 10–11]21. 5 + (−6) + (−7)−822. 6 + (−8) + (−10)−1223. −7 + 8 + (−5)−424. −6 + 9 + (−3)025. 5 + [6 + (−2)] + (−3)626. 10 + [8 + (−5)] + (−20)−727. [6 + (−2)] + [3 + (−1)]628. [18 + (−5)] + [9 + (−10)]1229. 20 + (−6) + [3 + (−9)]830. 18 + (−2) + [9 + (−13)]1231. −3 + (−2) + [5 + (−4)]−432. −6 + (−5) + [−4 + (−1)]−1633. (−9 + 2) + [5 + (−8)] + (−4)−1434. (−7 + 3) + [9 + (−6)] + (−5)−635. [−6 + (−4)] + [7 + (−5)] + (−9)−1736. [−8 + (−1)] + [8 + (−6)] + (−6)−1337. (−6 + 9) + (−5) + (−4 + 3) + 7438. (−10 + 4) + (−3) + (−3 + 8) + 62B The problems that follow involve some multiplication. Be sure that you work inside the parentheses first, then multiply,and finally, add left to right. [Example 12]39. −5 + 2(−3 + 7)340. −3 + 4(−2 + 7)1741. 9 + 3(−8 + 10)1542. 4 + 5(−2 + 6)2443. −10 + 2(−6 + 8) + (−2)−844. −20 + 3(−7 + 10) + (−4)−1545. 2(−4 + 7) + 3(−6 + 8)1246. 5(−2 + 5) + 7(−1 + 6)50

1.3 Problem Set35Add.47. 3.9 + 7.1 11 48. 4.7 + 4.3 9 49. 8.1 + 2.7 10.8 50. 2.4 + 7.3 9.7Solve.51. Find the value of 0.06x + 0.07y when x = 7,000 andy = 8,000. 98052. Find the value of 0.06x + 0.07y when x = 7,000 andy = 3,000. 63053. Find the value of 0.05x + 0.10y when x = 10 andy = 12. 1.7054. Find the value of 0.25x + 0.10y when x = 3 andy = 11. 1.85C Each sequence below is an arithmetic sequence. In each case, find the next two numbers in the sequence. [Example 13]55. 3, 8, 13, 18, . . .23, 2856. 1, 5, 9, 13, . . .17, 2157. 10, 15, 20, 25, . . .30, 3558. 10, 16, 22, 28, . . .34, 4059. 20, 15, 10, 5, . . .0, −560. 24, 20, 16, 12, . . .8, 461. 6, 0, −6, . . .−12, −1862 1, 0, −1, . . .−2, −363. 8, 4, 0, . . .64. 5, 2, −1, . . .−4, −8−4, −765. Is the sequence of odd numbers an arithmeticsequence?Yes66. Is the sequence of squares an arithmetic sequence?NoRecall that the word sum indicates addition. Write the numerical expression that is equivalent to each of the followingphrases and then simplify.67. The sum of 5 and 95 + 9 = 1468. The sum of 6 and −36 + (−3) = 369. Four added to the sum of −7 and −5[−7 + (−5)] + 4 = −870. Six added to the sum of −9 and 1(−9 + 1) + 6 = −271. The sum of −2 and −3 increased by 10[−2 + (−3)] + 10 = 572. The sum of −4 and −12 increased by 2[−4 + (−12)] + 2 = −14Answer the following questions.73. What number do you add to −8 to get −5?374. What number do you add to 10 to get 4?−675. The sum of what number and −6 is −9?−376. The sum of what number and −12 is 8?20

36Chapter 1 The BasicsApplying the Concepts77. Music The chart shows the country music artists thatearned the most in 2007. How much did Toby Keithand Tim McGraw make combined?$71.3 million78. Computers The chart shows the three countries with themost computers. What is the total number of computersthat can be found in the three countries?372.9 millionTop-Earning Country Music StarsWho’s Connected?Toby KeithRascal Flatts$48.2 million$40.4 millionUnited States240.5Tim McGraw$23.1 millionJapan77.9Kenny ChesneyBrooks & Dunn$22.3 million$20.4 millionGermany54.5Millions of computersSource: www.forbes.comSource: Computer Industry Almanac Inc.79. Temperature Change The temperature atnoon is 12 degrees below 0 Fahrenheit. By1:00 it has risen 4 degrees. Write anexpression using the numbers −12 and 4to describe this situation.−12 + 420100–1080. Stock Value On Monday a certain stock gains 2 points.On Tuesday it loses 3 points. Write an expression usingpositive and negative numbers with addition to describethis situation and then simplify.2 + (−3) = −1˚Fahrenheit81. Gambling On three consecutive hands of draw poker, agambler wins $10, loses $6, and then loses another $8.Write an expression using positive and negative numbersand addition to describe this situation and thensimplify.10 + (−6) + (−8) = −$482. Number Problem You know from your past experiencewith numbers that subtracting 5 from 8 results in3 (8 − 5 = 3). What addition problem that starts with thenumber 8 gives the same result?8 + (−5) = 383. Checkbook Balance Suppose that you balance yourcheckbook and find that you are overdrawn by $30;that is, your balance is −$30. Then you go to the bankand deposit $40. Translate this situation into an additionproblem, the answer to which gives the newbalance in your checkbook.−30 + 40 = 1084. Checkbook Balance The balance in your checkbook is−$25. If you make a deposit of $75, and then write acheck for $18, what is the new balance?$32

Subtraction of Real NumbersSuppose that the temperature at noon is 20° Fahrenheit and 12 hours later, atmidnight, it has dropped to −15° Fahrenheit. What is the difference between thetemperature at noon and the temperature at midnight? Intuitively, we know thedifference in the two temperatures is 35°. We also know that the word differenceindicates subtraction. The difference between 20 and −15 is written20 − (−15)It must be true that 20 − (−15) = 35. In this section we will see how our definitionfor subtraction confirms that this last statement is in fact correct.A Subtracting Positive and Negative NumbersIn the previous section we spent some time developing the rule for addition ofreal numbers. Because we want to make as few rules as possible, we can definesubtraction in terms of addition. By doing so, we can then use the rule for additionto solve our subtraction problems.1.4ObjectivesA Subtract any combination ofpositive and negative numbers.B Simplify expressions using the rulefor order of operations.C Translate sentences from Englishinto symbols and then simplify.D Find the complement and thesupplement of an angle.Examples now playing atMathTV.com/booksRuleTo subtract one real number from another, simply add its opposite.Algebraically, the rule is written like this: If a and b represent two real numbers,then it is always true thata − b = a + (−b)To subtract b add the opposite of bThis is how subtraction is defined in algebra. This definition of subtraction willnot conflict with what you already know about subtraction, but it will allow youto do subtraction using negative numbers.Example 1Subtract all possible combinations of positive and negative7 and 2.Solution7 − 2 = 7 + (−2) = 5−7 − 2 = −7 + (−2) = −9}}Subtracting 2 is the sameas adding −27 − (−2) = 7 + 2 = 9Subtracting −2 is the same−7 − (−2) = −7 + 2 = −5 as adding 2Notice that each subtraction problem is first changed to an addition problem.The rule for addition is then used to arrive at the answer.We have defined subtraction in terms of addition, and we still obtain answersconsistent with the answers we are used to getting with subtraction. Moreover, wenow can do subtraction problems involving both positive and negative numbers.As you proceed through the following examples and the problem set, you willbegin to notice shortcuts you can use in working the problems. You will notalways have to change subtraction to addition of the opposite to be able to getanswers quickly. Use all the shortcuts you wish as long as you consistently getthe correct answers.1.4 Subtraction of Real NumbersPractice Problems1. Subtract7 − 4 =−7 − 4 =7 − (−4) =−7 − (−4) =Answer1. 3, −11, 11, −337

38Chapter 1 The Basics2. Subtract6 − 4 =−6 − 4 =6 − (−4) =−6 − (−4) =Example 2Subtract all combinations of positive and negative 8 and 13.Solution8 − 13 = 8 + (−13) = −5−8 − 13 = −8 + (−13) = −21}}Subtracting +13 is thesame as adding −138 − (−13) = 8 + 13 = 21Subtracting −13 is the−8 − (−13) = −8 + 13 = 5 same as adding +13BUsing Order of OperationsSimplify each expression as muchas possible.3. 4 + (−2) − 3Example 3Simplify the expression 7 + (−3) − 5 as much as possible.Solution 7 + (−3) − 5 = 7 + (−3) + (−5) Begin by changing all= 4 + (−5) subtractions to additions= −1 Then add left to right4. 9 − 4 − 5Example 4Simplify the expression 8 − (−2) − 6 as much as possible.Solution 8 − (−2) − 6 = 8 + 2 + (−6) Begin by changing all= 10 + (−6) subtractions to additions= 4 Then add left to right5. −3 − (−5 + 1) − 6Example 5Simplify the each expression −2 − (−3 + 1) − 5 as muchas possible.Solution −2 − (−3 + 1) − 5 = −2 − (−2) − 5 Do what is in the parentheses first= −2 + 2 + (−5)= −5The next two examples involve multiplication and exponents as well as subtraction.Remember, according to the rule for order of operations, we evaluate thenumbers containing exponents and multiply before we subtract.6. Simplify 3 ⋅ 9 − 4 ⋅ 10 − 5 ⋅ 11.Example 6Simplify 2 ⋅ 5 − 3 ⋅ 8 − 4 ⋅ 9.SolutionFirst, we multiply left to right, and then we subtract.2 ⋅ 5 − 3 ⋅ 8 − 4 ⋅ 9 = 10 − 24 − 36= −14 − 36= −507. Simplify 4 ⋅ 2 5 − 3 ⋅ 5 2 .Example 7Simplify 3 ⋅ 2 3 − 2 ⋅ 4 2 .Solution We begin by evaluating each number that contains an exponent.Then we multiply before we subtract:3 ⋅ 2 3 − 2 ⋅ 4 2 = 3 ⋅ 8 − 2 ⋅ 16= 24 − 32= −8Answers2. 2, −10, 10, −2 3. −1 4. 05. −5 6. −68 7. 53

1.4 Subtraction of Real Numbers39CWriting Words in SymbolsExample 8Subtract 7 from −3.Solution First, we write the problem in terms of subtraction. We then changeto addition of the opposite:8. Subtract 8 from −5.−3 − 7 = −3 + (−7)= −10Example 9Subtract −5 from 2.9. Subtract −3 from 8.Solution Subtracting −5 is the same as adding +5:2 − (−5) = 2 + 5= 7Example 10Find the difference of 9 and 2.10. Find the difference of 8 and 6.SolutionWritten in symbols, the problem looks like this:9 − 2 = 7The difference of 9 and 2 is 7.Example 11Find the difference of 3 and −5.SolutionSubtracting −5 from 3 we have11. Find the difference of 4 and−7.3 − (−5) = 3 + 5= 8DComplementary and Supplementary AnglesWe can apply our knowledge of algebra to help solve some simple geometryproblems. Before we do, however, we need to review some of the vocabularyassociated with angles.DefinitionIn geometry, two angles that add to 90° are called complementary angles.In a similar manner, two angles that add to 180° are called supplementaryangles. The diagrams at the left illustrate the relationships between anglesthat are complementary and between angles that are supplementary.yxyxComplementary angles: x + y = 90°Supplementary angles: x + y = 180°Answers8. −13 9. 11 10. 2 11. 11

40Chapter 1 The Basics12. Find x in each of the followingdiagrams.a.Example 12Find x in each of the following diagrams.x45°x30°x45°b.SolutionWe use subtraction to find each angle.x60°a. Because the two angles are complementary, we can find x by subtracting30° from 90°:x = 90° − 30° = 60°We say 30° and 60° are complementary angles. The complement of 30° is60°.b. The two angles in the diagram are supplementary. To find x, we subtract45° from 180°:x = 180° − 45° = 135°We say 45° and 135° are supplementary angles. The supplement of 45° is135°.Subtraction and Taking AwayFor some people taking algebra for the first time, subtraction of positive and negativenumbers can be a problem. These people may believe that −5 − 9 should be −4 or 4,not −14. If this is happening to you, you probably are thinking of subtraction in termsof taking one number away from another. Thinking of subtraction in this way workswell with positive numbers if you always subtract the smaller number from the larger.In algebra, however, we encounter many situations other than this. The definition ofsubtraction, that a − b = a + (−b), clearly indicates the correct way to use subtraction;that is, when working subtraction problems, you should think “addition of the opposite,”not “take one number away from another.” To be successful in algebra, you needto apply properties and definitions exactly as they are presented here.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Why do we define subtraction in terms of addition?2. Write the definition for a − b.3. Explain in words how you would subtract 3 from −7.4. What are complementary angles?Answers12. a. 45° b. 120°

1.4 Problem Set41Problem Set 1.4The following problems are intended to give you practice with subtraction of positive and negative numbers. Rememberin algebra subtraction is not taking one number away from another. Instead, subtracting a number is equivalent to addingits opposite.A Subtract. [Examples 1–2}1. 5 − 8−32. 6 − 7−13. 3 − 9−64. 2 − 7−55. 5 − 506. 8 − 807. −8 − 28. −6 − 39. −4 − 1210. −3 − 15−10−9−16−1811. −6 − 612. −3 − 313. −8 − (−1)14. −6 − (−2)−12−6−7−415. 15 − (−20)3516. 20 − (−5)2517. −4 − (−4)018. −5 − (−5)0B Simplify each expression by applying the rule for order of operations. [Examples 3–5]19. 3 − 2 − 5−420. 4 − 8 − 6−1021. 9 − 2 − 3422. 8 − 7 − 12−1123. −6 − 8 − 1024. −5 − 7 − 925. −22 + 4 − 1026. −13 + 6 − 5−24−21−28−1227. 10 − (−20) − 52528. 15 − (−3) − 20−229. 8 − (2 − 3) − 5430. 10 − (4 − 6) − 8431. 7 − (3 − 9) − 6732. 4 − (3 − 7) − 8033. 5 − (−8 − 6) − 21734. 4 − (−3 − 2) − 18Selected exercises available online at www.webassign.net/brookscole

42Chapter 1 The Basics35. −(5 − 7) − (2 − 8)836. −(4 − 8) − (2 − 5)737. −(3 − 10) − (6 − 3)438. −(3 − 7) − (1 − 2)539. 16 − [(4 − 5) − 1]1840. 15 − [(4 − 2) − 3]1641. 5 − [(2 − 3) − 4]1042. 6 − [(4 − 1) − 9]1243. 21 − [−(3 − 4) − 2] − 51744. 30 − [−(10 − 5) − 15] − 2525B The following problems involve multiplication and exponents. Use the rule for order of operations to simplify eachexpression as much as possible. [Examples 6–7]45. 2 ⋅ 8 − 3 ⋅ 5146. 3 ⋅ 4 − 6 ⋅ 7−3047. 3 ⋅ 5 − 2 ⋅ 7148. 6 ⋅ 10 − 5 ⋅ 20−4049. 5 ⋅ 9 − 2 ⋅ 3 − 6 ⋅ 22750. 4 ⋅ 3 − 7 ⋅ 1 − 9 ⋅ 4−3151. 3 ⋅ 8 − 2 ⋅ 4 − 6 ⋅ 7−2652. 5 ⋅ 9 − 3 ⋅ 8 − 4 ⋅ 5153. 2 ⋅ 3 2 − 5 ⋅ 2 2−254. 3 ⋅ 7 2 − 2 ⋅ 8 21955. 4 ⋅ 3 3 − 5 ⋅ 2 36856. 3 ⋅ 6 2 − 2 ⋅ 3 2 − 8 ⋅ 6 2−198Subtract.57. −3.4 − 7.9 −11.3 58. −3.5 − 2.3 −5.8 59. 3.3 − 6.9 −3.6 60. 2.2 − 7.5 −5.3

1.4 Problem Set43Solve.61. Find the value of x + y − 4 when62. Find the value of x − y − 3 when_c. x = − ​ 3 5 ​and y = ​8 _5 ​ −3 c. x = −5 and y = −2 −6a. x = −3 and y = −2 −9a. x = −4 and y = 1 −8b. x = −9 and y = 3 −10b. x = 2 and y = −1 0C Rewrite each of the following phrases as an equivalent expression in symbols, and then simplify. [Examples 8–12]63. Subtract 4 from −7.−7 − 4 = −1164. Subtract 5 from −19.−19 − 5 = −2465. Subtract −8 from 12.12 − (−8) = 2066. Subtract −2 from 10.10 − (−2) = 1267. Subtract −7 from −5.−5 − (−7) = 268. Subtract −9 from −3.−3 − (−9) = 669. Subtract 17 from the sum of 4 and −5.[4 + (−5)] − 17 = −1870. Subtract −6 from the sum of 6 and −3.[6 + (−3)] − (−6) = 9C Recall that the word difference indicates subtraction. The difference of a and b is a − b, in that order. Write a numericalexpression that is equivalent to each of the following phrases, and then simplify. [Example 12]71. The difference of 8 and 58 − 5 = 372. The difference of 5 and 85 − 8 = −373. The difference of −8 and 5−8 − 5 = −1374. The difference of −5 and 8−5 − 8 = −1375. The difference of 8 and −58 − (−5) = 1376. The difference of 5 and −85 − (−8) = 13CAnswer the following questions.77. What number do you subtract from 8 to get −2?1078. What number do you subtract from 1 to get −5?679. What number do you subtract from 8 to get 10?−280. What number do you subtract from 1 to get 5?−4

44Chapter 1 The BasicsApplying the Concepts81. Pitchers The chart shows the number of strikeouts foractive starting pitchers with the most career strikeouts.How many more strikeouts does Randy Johnsonhave than Greg Maddux?1,418 strikeouts82. Sound The low-frequency pulses made by blue whaleshave been measured up to 188 decibels (dB) making itthe loudest sound emitted by any living source. Howmuch louder is a blue whale than normal conversation?128 dBKing of the HillSounds Around UsRandy Johnson4,789Normal Conversation60 dBRodger ClemensGreg Maddux3,3714,672Football Stadium117 dBPedro Martinez3,117Blue Whale188 dBSource: www.mlb.com, November 2008Source: www.4to40.com83. Savings Account Balance A man with $1,500 in a savingsaccount makes a withdrawal of $730. Write an expressionusing subtraction that describes this situation.1,500 − 730First BankAccount No. 12345Date Withdrawals Deposits Balance1/1/08 1,5002/2/08 73084. Temperature Change The temperature inside a space shuttleis 73°F before reentry. During reentry the temperatureinside the craft increases 10°. On landing it drops 8°F.Write an expression using the numbers 73, 10, and 8 todescribe this situation. What is the temperature insidethe shuttle on landing?73 + 10 − 8, 75°F85. Gambling A man who has lost $35 playing roulette inLas Vegas wins $15 playing blackjack. He then loses$20 playing the wheel of fortune. Write an expressionusing the numbers −35, 15, and 20 to describe thissituation and then simplify it.−35 + 15 − 20 = −$4086. Altitude Change An airplane flying at 10,000 feet lowersits altitude by 1,500 feet to avoid other air traffic. Thenit increases its altitude by 3,000 feet to clear a mountainrange. Write an expression that describes this situationand then simplify it.10,000 − 1,500 + 3,000 = 11,500 feet

1.4 Problem Set4587. Checkbook Balance Bob has $98 in his checking accountwhen he writes a check for $65 and then another checkfor $53. Write a subtraction problem that gives the newbalance in Bob’s checkbook. What is his new balance?98 − 65 − 53 = −$2088. Temperature Change The temperature at noon is 23°F. Sixhours later it has dropped 19°F, and by midnight it hasdropped another 10°F. Write a subtraction problem thatgives the temperature at midnight. What is the temperatureat midnight?23 − 19 − 10 = −6°F89. Depreciation Stacey buys a used car for $4,500. Witheach year that passes, the car drops $550 in value.Write a sequence of numbers that gives the value ofthe car at the beginning of each of the first 5 years sheowns it. Can this sequence be considered an arithmeticsequence?$4,500, $3,950, $3,400, $2,850, $2,300; yes90. Depreciation Wade buys a computer system for $6,575.Each year after that he finds that the system is worth$1,250 less than it was the year before. Write a sequenceof numbers that gives the value of the computer systemat the beginning of each of the first four years heowns it. Can this sequence be considered an arithmeticsequence?$6,575, $5,325, $4,075, $2,825; yesDrag Racing In the sport of drag racing, two cars at the starting line race to the finish line ​_ 1 ​mile away. The car that crosses4the finish line first wins the race. Jim Rizzoli owns and races an alcohol dragster. On board the dragster is a computer thatrecords data during each of Jim’s races. Table 1 gives some of the data from a race Jim was in. In addition to showing thetime and speed of Jim Rizzoli’s dragster during a race, it also shows the distance past the starting line that his dragster hastraveled. Use the information in the table shown here to answer the following questions.91. Find the difference in the distance traveled by the dragsterafter 5 seconds and after 2 seconds.769 feetSpeed and Distance for a Race CarTime in Speed in Distance TraveledSeconds Miles/Hour in Feet0 0 092. How much faster is he traveling after 4 seconds than heis after 2 seconds?62.3 miles per hour1 72.7 692 129.9 2313 162.8 4394 192.2 7285 212.4 1,00093. How far from the starting line is he after 3 seconds?439 feet6 228.1 1,37394. How far from the starting line is he when his speed is192.2 miles per hour?728 feet95. How many seconds have gone by between the time hisspeed is 162.8 miles per hour and the time at which hehas traveled 1,000 feet?2 seconds96. How many seconds have gone by between the time atwhich he has traveled 231 feet and the time at which hisspeed is 228.1 miles per hour?4 seconds

46Chapter 1 The BasicsD Find x in each of the following diagrams. [Example 13]97.98.x 45°55°x35°45°99.100.x120°x150°30°60°101. Grass Growth The bar chart below shows the growthof a certain species of grass over a period of 10 days.a. Use the chart to fill in the missing entries in thetable.b. How much higher is the grass after 8 days thanafter 3 days? 11.5 inches102. Wireless Phone Costs The bar chart below shows the projectedcost of wireless phone use through 2003.a. Use the chart to fill in the missing entries in thetable.b. What is the difference in cost between 1998 and1999? 5 cents per minutePlant HeightDay (inches)0 01 0.52 13 1.54 35 4Plant HeightDay (inches)6 67 98 139 1810 23Year Cents/Minute1998 331999 282000 252001 232002 222003 20Height (in.)252015105000Overall Plant Height23181390.5 1 1.5 63 41 2 3 4 5 6 7 8 9 10Days after germinationCost (cents per minute)35302520151050Wireless Talk33282523 22201998 1999 2000 2001 2002 2003Year

Properties of Real Numbers 1.5In this section we will list all the facts (properties) that you know from past experienceare true about numbers in general. We will give each property a name sowe can refer to it later in this book. Mathematics is very much like a game. Thegame involves numbers. The rules of the game are the properties and rules weare developing in this chapter. The goal of the game is to extend the basic rules toas many situations as possible.You know from past experience with numbers that it makes no difference in whichorder you add two numbers; that is, 3 + 5 is the same as 5 + 3. This fact about numbersis called the commutative property of addition. We say addition is a commutativeoperation. Changing the order of the numbers does not change the answer.There is one other basic operation that is commutative. Because 3(5) is thesame as 5(3), we say multiplication is a commutative operation. Changing theorder of the two numbers you are multiplying does not change the answer.ObjectivesA Rewrite expressions using thecommutative and associativeproperties.B Multiply using the distributiveproperty.C Identify properties used to rewritean expression.Examples now playing atMathTV.com/booksAThe Commutative and Associative PropertiesFor all properties listed in this section, a, b, and c represent real numbers.Commutative Property of AdditionIn symbols: a + b = b + aIn words:Changing the order of the numbers in a sum will not change theresult.Commutative Property of MultiplicationIn symbols: a ⋅ b = b ⋅ aIn words:Changing the order of the numbers in a product will not changethe result.Examples1. The statement 5 + 8 = 8 + 5 is an example of the commutative property ofaddition.2. The statement 2 ⋅ y = y ⋅ 2 is an example of the commutative property ofmultiplication.3. The expression 5 + x + 3 can be simplified using the commutative property ofaddition:5 + x + 3 = x + 5 + 3 Commutative property of addition= x + 8 AdditionThe other two basic operations, subtraction and division, are not commutative.The order in which we subtract or divide two numbers makes a difference in theanswer.Another property of numbers that you have used many times has to do withgrouping. You know that when we add three numbers it makes no difference whichtwo we add first. When adding 3 + 5 + 7, we can add the 3 and 5 first and then the7, or we can add the 5 and 7 first and then the 3. Mathematically, it looks like this:(3 + 5) + 7 = 3 + (5 + 7). This property is true of multiplication as well. Operationsthat behave in this manner are called associative operations. The answer will notchange when we change the association (or grouping) of the numbers.1.5 Properties of Real NumbersPractice Problems1. Give an example of the commutativeproperty of addition using2 and y.2. Give an example of the commutativeproperty of multiplicationusing the numbers 5 and 8.3. Use the commutative propertyof addition to simplify 4 + x + 7.Answers1. 2 + y = y + 2 2. 5(8) = 8(5)3. x + 1147

48Chapter 1 The BasicsAssociative Property of AdditionIn symbols: a + (b + c) = (a + b) + cIn words:Changing the grouping of the numbers in a sum will not changethe result.Associative Property of MultiplicationIn symbols: a(bc) = (ab)cIn words:Changing the grouping of the numbers in a product will not changethe result.The following examples illustrate how the associative properties can be used tosimplify expressions that involve both numbers and variables.Simplify.4. 3 + (7 + x)Example 4Simplify 4 + (5 + x).Solution 4 + (5 + x) = (4 + 5) + x Associative property of addition= 9 + x Addition5. 3(4x)Example 5Simplify 5(2x).Solution 5(2x) = (5 ⋅ 2)x Associative property of multiplication= 10x Multiplication6. ​_1 ​( 3 x)3Example 6Simplify ​_1 ​( 5 x).5Solution​_1 5 ​( 5 x) = ​ _ ​ 1 5 ​⋅ 5 ​x Associative property of multiplication= 1x Multiplication= x7. 4 ​ ​ 1 _4 ​x ​Example 7Simplify 3 ​ ​ 1 _3 ​x ​.Solution3 _​ ​ 1 3 ​x ​= ​ 3 ⋅ ​_1 3 ​ ​x Associative property of multiplication= 1x Multiplication= x8. 15 ​ ​ 3 _5 ​x ​Example 8Simplify 12 ​ ​ 2 _3 ​x ​.Solution12 ​ ​ 2 _3 ​x ​= ​ 12 ⋅ ​ 2 _3 ​ ​x Associative property of multiplication= 8x MultiplicationAnswers4. 10 + x 5. 12x 6. x 7. x8. 9xB The Distributive PropertyThe associative and commutative properties apply to problems that are either allmultiplication or all addition. There is a third basic property that involves bothaddition and multiplication. It is called the distributive property and looks like this.

1.5 Properties of Real Numbers49Distributive propertyIn symbols: a(b + c) = ab + acIn words: Multiplication distributes over addition.You will see as we progress through the book that the distributive property isused very frequently in algebra. We can give a visual justification to the distributiveproperty by finding the areas of rectangles. Figure 1 shows a large rectanglethat is made up of two smaller rectangles. We can find the area of the large rectangletwo different ways.NoteBecause subtractionis defi ned interms of addition,it is also true that the distributiveproperty applies to subtractionas well as addition; that is,a(b − c) = ab − ac for any threereal numbers a, b, and c.mEtHOD 1 We can calculate the area of the large rectangle directly by finding itslength and width. The width is 5 inches, and the length is (3 + 4) inches.III5 inchesArea of large rectangle = 5(3 + 4)= 5(7)= 35 square inchesmEtHOD 2 Because the area of the large rectangle is the sum of the areas of thetwo smaller rectangles, we find the area of each small rectangle and then add tofind the area of the large rectangle.3 inches 4 inchesFIGuRE 1Area of large rectangle = Area of rectangle I + Area of rectangle II= 5(3) + 5(4)= 15 + 20= 35 square inchesIn both cases the result is 35 square inches. Because the results are the same,the two original expressions must be equal. Stated mathematically, 5(3 + 4) =5(3) + 5(4). We can either add the 3 and 4 first and then multiply that sum by 5,or we can multiply the 3 and the 4 separately by 5 and then add the products. Ineither case we get the same answer.Here are some examples that illustrate how we use the distributive property.ExAmplE 9Apply the distributive property to the expression 2(x + 3),and then simplify the result.Apply the distributive property toeach expression, and then simplifythe result.9. 3(x + 2)SOlutiON 2(x + 3) = 2(x) + 2(3) Distributive property= 2x + 6 MultiplicationExAmplE 10Apply the distributive property to the expression 5(2x − 8),and then simplify the result.10. 4(2x − 5)SOlutiON 5(2x − 8) = 5(2x) − 5(8) Distributive property= 10x − 40 MultiplicationNotice in this example that multiplication distributes over subtraction as well asaddition.ExAmplE 11Apply the distributive property to the expression 4(x + y),and then simplify the result.11. 5(x + y)SOlutiON 4(x + y) = 4x + 4y Distributive propertyAnswers9. 3x + 6 10. 8x − 2011. 5x + 5y

50Chapter 1 The Basics12. 3(7x − 6y)13. ​_1 ​( 3 x + 6)3Example 12Apply the distributive property to the expression 5(2x + 4y),and then simplify the result.Solution 5(2x + 4y) = 5(2x) + 5(4y) Distributive property= 10x + 20y MultiplicationExample 13Apply the distributive property to the expression ​ 1 _and then simplify the result.Solution​_1 2 ​( 3 x + 6) = ​1 _2 ​( 3 x) + ​1 _​(6) Distributive property2​( 3 x + 6),2= ​_3 ​x + 3 Multiplication214. 5(7a − 3) + 2Example 14Apply the distributive property to the expression4(2a + 3) + 8, and then simplify the result.Solution 4(2a + 3) + 8 = 4(2a) + 4(3) + 8 Distributive property= 8a + 12 + 8 Multiplication= 8a + 20 Addition15. x ​ 2 + ​ 2 _x ​ ​16. 4 ​ ​ 1 _4 ​x − 3 ​Example 15Apply the distributive property to the expression a ​ 1 + ​ 1 _a ​ ​,and then simplify the result.Solutiona ​ 1 + ​ 1 _a ​ ​= a ⋅ 1 + a ⋅ ​ 1 _a ​Distributive property= a + 1 MultiplicationExample 16Apply the distributive property to the expression 3 ​ ​ 1 _3 ​x + 5 ​,and then simplify the result.Solution3 _​ ​ 1 3 ​x + 5 ​= 3 ⋅ ​_1 ​x + 3 ⋅ 53Distributive property= x + 15 Multiplication17. 8 ​ ​ 3 _4 ​x + ​1 _2 ​y ​Example 17Apply the distributive property to the expression12 ​ ​ 2 _3 ​x + ​1 _2 ​y ​, and then simplify the result.Solution12 ​ ​ 2 _3 ​x + ​1 _2 ​y ​= 12 ⋅ ​ 2 _3 ​x + 12 ⋅ ​1 _​y Distributive property2= 8x + 6y MultiplicationSpecial NumbersIn addition to the three properties mentioned so far, we want to include in our listtwo special numbers that have unique properties. They are the numbers zero andone.Additive Identity PropertyThere exists a unique number 0 such thatAnswers12. 21x − 18y 13. x + 214. 35a − 13 15. 2x + 216. x − 12 17. 6x + 4yIn symbols:In words:a + 0 = a and 0 + a = aZero preserves identities under addition. (The identity of thenumber is unchanged after addition with 0.)

1.5 Properties of Real Numbers51Multiplicative Identity PropertyThere exists a unique number 1 such thatIn symbols:In words:a(1) = a and 1(a) = aThe number 1 preserves identities under multiplication. (Theidentity of the number is unchanged after multiplication by 1.)Additive Inverse PropertyFor each real number a, there exists a unique number −a such thatIn symbols: a + (−a) = 0In words: Opposites add to 0.Multiplicative Inverse PropertyFor every real number a, except 0, there exists a unique real number ​ 1 _a ​suchthatIn symbols:a ​ ​ 1 _a ​ ​= 1In words: Reciprocals multiply to 1.CIdentifying PropertiesOf all the basic properties listed, the commutative, associative, and distributiveproperties are the ones we will use most often. They are important because theywill be used as justifications or reasons for many of the things we will do.The following examples illustrate how we use the preceding properties. Eachone contains an algebraic expression that has been changed in some way. Theproperty that justifies the change is written to the right.Example 18State the property that justifies x + 5 = 5 + x.Solution x + 5 = 5 + x Commutative property of additionState the property that justifies thegiven statement.18. 5 ⋅ 4 = 4 ⋅ 5Example 19State the property that justifies (2 + x) + y = 2 + (x + y).19. x + 7 = 7 + xSolution (2 + x) + y = 2 + (x + y) Associative property of additionExample 20State the property that justifies 6(x + 3) = 6x + 18.20. (3 + a) + b = 3 + (a + b)Solution 6(x + 3) = 6x + 18 Distributive propertyExample 21State the property that justifies 2 + (−2) = 0.Solution 2 + (−2) = 0 Additive inverse propertyExample 22State the property that justifies 3 ​ ​ 1 _3 ​ ​= 1.Solution3 ​ ​ 1 _3 ​ ​= 1 Multiplicative inverse propertyExample 23State the property that justifies (2 + 0) + 3 = 2 + 3.Solution (2 + 0) + 3 = 2 + 3 Additive identity property21. 6(x + 5) = 6x + 3022. 4 ​ ​ 1 _4 ​ ​= 123. 8 + (−8) = 0Answers18. Commutative property ofmultiplication 19. Commutativeproperty of addition 20. Associativeproperty of addition 21. Distributiveproperty 22. Multiplicativeinverse 23. Additive inverse

52Chapter 1 The Basics24. (x + 2) + 3 = x + (2 + 3)Example 24State the property that justifies (2 + 3) + 4 = 3 + (2 + 4).Solution (2 + 3) + 4 = 3 + (2 + 4) Commutative and associative propertiesof addition25. 5(1) = 5Example 25State the property that justifies (x + 2) + y = (x + y) + 2.Solution (x + 2) + y = (x + y) + 2 Commutative and associative propertiesof additionAs a final note on the properties of real numbers, we should mention thatalthough some of the properties are stated for only two or three real numbers,they hold for as many numbers as needed. For example, the distributive propertyholds for expressions like 3(x + y + z + 5 + 2); that is,3(x + y + z + 5 + 2) = 3x + 3y + 3z + 15 + 6It is not important how many numbers are contained in the sum, only that it is asum. Multiplication, you see, distributes over addition, whether there are twonumbers in the sum or two hundred.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. What is the commutative property of addition?2. Do you know from your experience with numbers that the commutativeproperty of addition is true? Explain why.3 . Write the commutative property of multiplication in symbols and words.Answers24. Associative property of addtion25. Multiplicative identity

1.5 Problem Set53Problem Set 1.5C State the property or properties that justify the following. [Examples 18–25]1. 3 + 2 = 2 + 32. 5 + 0 = 5CommutativeAdditive identity3. 4 ​ ​ 1 _4 ​ ​= 1Multiplicative inverse4. 10(0.1) = 1Multiplicative inverse5. 4 + x = x + 4Commutative6. 3(x − 10) = 3x − 30Distributive7. 2(y + 8) = 2y + 16Distributive8. 3 + (4 + 5) = (3 + 4) + 5Associative9. (3 + 1) + 2 = 1 + (3 + 2)Commutative, associative10. (5 + 2) + 9 = (2 + 5) + 9Commutative11. (8 + 9) + 10 = (8 + 10) + 9Commutative, associative12. (7 + 6) + 5 = (5 + 6) + 7Commutative, associative13. 3(x + 2) = 3(2 + x)Commutative14. 2(7y) = (7 ⋅ 2)yCommutative, associative15. x(3y) = 3(xy)Commutative, associative16. a(5b) = 5(ab)Commutative, associative17. 4(xy) = 4(yx)Commutative18. 3[2 + (−2)] = 3(0)Additive inverse19. 8[7 + (−7)] = 8(0)Additive inverse20. 7(1) = 7Multiplicative identityEach of the following problems has a mistake in it. Correct the right-hand side.21. 3(x + 2) = 3x + 23x + 622. 5(4 + x) = 4 + 5x20 + 5x23. 9(a + b) = 9a + b9a + 9b24. 2(y + 1) = 2y + 12y + 225. 3(0) = 3026. 5 ​ ​ 1 _5 ​ ​= 5127. 3 + (−3) = 1028. 8(0) = 8029. 10(1) = 01030. 3 ⋅ ​_1 3 ​= 01Selected exercises available online at www.webassign.net/brookscole

54Chapter 1 The BasicsA Use the associative property to rewrite each of the following expressions, and then simplify the result. [Examples 4–6]31. 4 + (2 + x)(4 + 2) + x = 6 + x32. 5 + (6 + x)(5 + 6) + x = 11 + x33. (x + 2) + 7x + (2 + 7) = x + 934. (x + 8) + 2x + (8 + 2) = x + 1035. 3(5x)(3 ⋅ 5)x = 15x36. 5(3x)(5 ⋅ 3)x = 15x37. 9(6y)(9 ⋅ 6)y = 54y38. 6(9y)(6 ⋅ 9)y = 54y39. ​_1 ​( 3 a)2_​ ​ 1 2 ​ ⋅ 3 ​a = ​_ 340. ​_1 ​( 2 a)32 ​a​ ​ 1 _3 ​ ⋅ 2 ​a = ​ 2 _41. ​_1 ​( 3 x)33 ​ a​ ​ 1 _3 ​ ⋅ 3 ​x = x42. ​_1 ​( 4 x)4_​ ​ 1 4 ​ ⋅ 4 ​x = x43. ​_1 ​( 2 y)2_​ ​ 1 2 ​ ⋅ 2 ​y = y44. ​_1 ​( 7 y)7_​ ​ 1 7 ​ ⋅ 7 ​y = y45. ​ 3 _4 ​​ ​ 4 _3 ​x ​​ ​ 3 _4 ​ ⋅ ​ 4 _3 ​ ​x = x46. ​ 3 _2 ​​ ​ 2 _3 ​x ​​ ​ 3 _2 ​ ⋅ ​ 2 _3 ​ ​x = x47. ​ 6 _5 ​​ ​ 5 _6 ​a ​48. ​ 2 _5 ​​ ​ 5 _2 ​a ​​ ​ 6 _5 ​ ⋅ ​5 _6 ​ ​a = a​ ​ 2 _5 ​ ⋅ 5 _2 ​ ​a = aB Apply the distributive property to each of the following expressions. Simplify when possible. [Examples 9–12]49. 8(x + 2)8x + 1650. 5(x + 3)5x + 1551. 8(x − 2)8x − 1652. 5(x − 3)5x − 1553. 4(y + 1)4y + 454. 4(y − 1)4y − 455. 3(6x + 5)18x + 1556. 3(5x + 6)15x + 1857. 2(3a + 7)6a + 1458. 5(3a + 2)15a + 1059. 9(6y − 8)54y − 7260. 2(7y − 4)14y − 8

1.5 Problem Set55B Apply the distributive property to each of the following expressions. Simplify when possible. [Examples 13–14]61. ​_1 ​( 3 x − 6)262. ​_1 ​( 2 x − 6)363. ​_1 ​( 3 x + 6)3​_ 3 2 ​x − 3 ​_ 2 3 ​x − 2 x + 264. ​_1 ​( 2 x + 4)2x + 265. 3(x + y)66. 2(x − y)67. 8(a − b)68. 7(a + b)3x + 3y2x − 2y8a − 8b7a + 7b69. 6(2x + 3y)70. 8(3x + 2y)71. 4(3a − 2b)72. 5(4a − 8b)12x + 18y24x + 16y12a − 8b20a − 40b73. ​_1 ​( 6 x + 4y)23x + 2y74. ​_1 ​( 6 x + 9y)32x + 3y75. 4(a + 4) + 94a + 2576. 6(a + 2) + 86a + 2077. 2(3x + 5) + 278. 7(2x + 1) + 379. 7(2x + 4) + 1080. 3(5x + 6) + 206x + 1214x + 1014x + 3815x + 38BHere are some problems you will see later in the book. Apply the distributive property and simplify, if possible.[Examples 15–17]81. 0.09(x + 2,000)82. 0.04(x + 7,000)83. 0.05(3x + 1,500)84. 0.08(4x + 3,000)0.09x + 1800.04x + 2800.15x + 750.32x + 24085. 6 ​ ​ 1 _2 ​x − ​1 _3 ​ y ​86. 12 ​ ​ 1 _4 ​x + ​2 _3 ​ y ​87. 12 ​ ​ 1 _3 ​x + ​1 _4 ​ y ​88. 15 ​ ​ 2 _5 ​x − ​1 _3 ​ y ​3x − 2y3x + 8y4x + 3y6x − 5y89. ​_1 ​( 4 x + 2)22x + 190. ​_1 ​( 6 x + 3)32x + 191. ​_3 ​( 8 x − 4)46x − 392. ​_2 ​( 5 x + 10)52x + 493. ​_5 ​( 6 x + 12)65x + 1094. ​_2 ​( 9 x − 3)36x − 295. 10 ​ ​ 3 _5 ​x + ​1 _2 ​ ​6x + 596. 8 ​ ​ 1 _4 ​x − ​5 _8 ​ ​2x − 597. 15 ​ ​ 1 _3 ​x + ​2 _5 ​ ​98. 12 ​ ​ 1 _12 ​m + ​1 _6 ​ ​99. 12 ​ ​ 1 _2 ​m − ​ 5 _12 ​ ​100. 8 ​ ​ 1 _8 ​+ ​1 _2 ​m ​5x + 6m + 26m − 51 + 4m101. 21 ​ ​ 1 _3 ​+ ​1 _7 ​x ​102. 6 ​ ​ 3 _2 ​y + ​1 _3 ​ ​103. 6 ​ ​ 1 _2 ​x − ​1 _3 ​y ​104. 12 ​ ​ 1 _4 ​x − ​2 _3 ​y ​7 + 3x9y + 23x − 2y3x + 8y105. 0.09(x + 2,000)106. 0.04(x + 7,000)107. 0.12(x + 500)108. 0.06(x + 800)0.09x + 1800.04x + 2800.12x + 600.06x + 48109. a ​ 1 + ​ 1 _a ​ ​110. a ​ 1 − ​ 1 _a ​ ​111. a ​ ​ 1 _a ​− 1 ​112. a ​ ​ 1 _a ​+ 1 ​a + 1a − 11 − a1 + a

56Chapter 1 The BasicsApplying the Concepts113. Getting Dressed While getting dressed for work, aman puts on his socks and puts on his shoes. Are thetwo statements “put on your socks” and “put on yourshoes” commutative?No114. Getting Dressed Are the statements “put on your leftshoe” and “put on your right shoe” commutative?Yes115. Skydiving A skydiver flying over the jump area isabout to do two things: jump out of the plane andpull the rip cord. Are the two events “jump out of theplane” and “pull the rip cord” commutative? That is,will changing the order of the events always producethe same result?No, not commutative116. Commutative Property Give an example of two events inyour daily life that are commutative.Brushing your teeth and brushing your hair117. Solar and Wind Energy The chart shows the cost toinstall either solar panels or a wind turbine. A homeownerbuys 3 solar modules, and each module is $100off the original price. Use the distributive property tocalculate the total cost.3(6,200 − 100) = 18,600 − 300 = $18,300Solar Versus Wind Energy CostsEquipment Cost:Modules $6200Fixed Rack $1570Charge Controller $971Cable $440TOTAL $9181Equipment Cost:Turbine $3300Tower $3000Cable $715TOTAL $7015118. Pitchers The chart shows the number of saves for activerelief pitchers. Which property of real numbers can beused to find the sum of the saves of Troy Percival andBilly Wagner in either order?Commutative Property of AdditionKing of the HillTrevor Hoffman554Mariano Rivera482Billy Wagner385Troy Percival352Source: www.mlb.com, November 2008Source: a Limited 2006119. Division Give an example that shows that division isnot a commutative operation; that is, find two numbersfor which changing the order of division givestwo different answers.8 ÷ 4 ≠ 4 ÷ 8120. Subtraction Simplify the expression 10 − (5 − 2) and theexpression (10 − 5) − 2 to show that subtraction is notan associative operation.10 − (5 − 2) = 10 − 3 = 7(10 − 5) − 2 = 5 − 2 = 3121. Take-Home Pay Jose works at a winery. His monthlysalary is $2,400. To cover his taxes and retirement,the winery withholds $480 from each check. Calculatehis yearly “take-home” pay using the numbers2,400, 480, and 12. Do the calculation two differentways so that the results give further justification forthe distributive property.12(2,400 − 480) = $23,040;12(2,400) − 12(480) = $23,040122. Hours Worked Carlo works as a waiter. He works doubleshifts 4 days a week. The lunch shift is 2 hours and thedinner shift is 3 hours. Find the total number of hourshe works per week using the numbers 2, 3, and 4. Dothe calculation two different ways so that the resultsgive further justification for the distributive property.4(2 + 3) = 20(4 ⋅ 2) + (4 ⋅ 3) = 20

Multiplication of Real NumbersSuppose that you own 5 shares of a stock and the price per share drops $3. Howmuch money have you lost? Intuitively, we know the loss is $15. Because it isa loss, we can express it as −$15. To describe this situation with numbers, wewould write5 shares each lose $3 for a total of $158888n5(−3) = −15Reasoning in this manner, we conclude that the product of a positive number witha negative number is a negative number. Let’s look at multiplication in more detail.A Multiplication of Real Numbers88nFrom our experience with counting numbers, we know that multiplication is simplyrepeated addition; that is, 3(5) = 5 + 5 + 5. We will use this fact, along withour knowledge of negative numbers, to develop the rule for multiplication of anytwo real numbers. The following examples illustrate multiplication with all of thepossible combinations of positive and negative numbers.ExAmplESMultiply.1. Two positives: 3(5) = 5 + 5 + 5= 15 Positive answer2. One positive: 3(−5) = −5 + (−5) + (−5)= −15 Negative answer3. One negative: −3(5) = 5(−3) Commutative property= −3 + (−3) + (−3) + (−3) + (−3)= −15 Negative answer4. Two negatives: −3(−5) = ?With two negatives, −3(−5), it is not possible to work the problem in terms ofrepeated addition. (It doesn’t “make sense” to write −5 down a −3 number oftimes.) The answer is probably +15 (that’s just a guess), but we need some justificationfor saying so. We will solve a different problem and in so doing get theanswer to the problem (−3)(−5).Here is a problem to which we know the answer. We will work it two differentways.−3[5 + (−5)] = −3(0) = 0The answer is zero. We also can work the problem using the distributive property.−3[5 + (−5)] = −3(5) + (−3)(−5) Distributive property= −15 + ?Because the answer to the problem is 0, our ? must be +15. (What else could weadd to −15 to get 0? Only +15.)8888n1.6ObjectivesA Multiply any combination of positiveand negative numbers.B Simplify expressions using the rulefor order of operations.C Multiply fractions.D Multiply using the associativeproperty.EfMultiply using the distributiveproperty.Extend a geometric sequence.pRACtiCE pROBlEmSMultiply.1. 4(2)2. 4(−2)3. −4(2)4. −4(−2)NoteExamples now playing atMathTV.com/booksYou may have to readthe explanation forExample 4 severaltimes before you understand itcompletely. The purpose of theexplanation in Example 4 is simplyto justify the fact that the productof two negative numbers is a positivenumber. If you have no troublebelieving that, then it is not soimportant that you understandeverything in the explanation.Here is a summary of the results we have obtained from the first four examples.Original Numbers Havethe Answer isthe same sign 3(5) = 15 positivedifferent signs 3(−5) = −15 negativedifferent signs −3(5) = −15 negativethe same sign −3(−5) = 15 positive1.6 Multiplication of Real NumbersAnswers1. 8 2. −8 3. −8 4. 857

58Chapter 1 The BasicsNoteSome students havetrouble with theexpression −8(−3)because they want to subtractrather than multiply. Because weare very precise with the notationwe use in algebra, the expression−8(−3) has only one meaning—multiplication. A subtraction problemthat uses the same numbersis −8 − 3. Compare the two followinglists.All Multiplication No Multiplication5(4) 5 + 4−5(4) −5 + 45(−4) 5 − 4−5(−4) −5 − 4By examining Examples 1 through 4 and the preceding table, we can use theinformation there to write the following rule. This rule tells us how to multiplyany two real numbers.RuleTo multiply any two real numbers, simply multiply their absolute values. Thesign of the answer is1. Positive if both numbers have the same sign (both + or both −).2. Negative if the numbers have opposite signs (one +, the other −).The following examples illustrate how we use the preceding rule to multiplyreal numbers.ExAmplESMultiply.Multiply.5. −4(−9)6. −11(−12)7. −2(−7)8. 4(−3)9. −2(10)5. −8(−3) = 246. −10(−5) = 507. −4(−7) = 288. 5(−7) = −359. −4(8) = −321 0 . −6(10) = −60}}If the two numbers in the product have the samesign, the answer is positiveIf the two numbers in the product have differentsigns, the answer is negative10. −3(13)Simplify as much as possible.11. −7(−6)(−1)B using Order of OperationsIn the following examples, we combine the rule for order of operations with therule for multiplication to simplify expressions. Remember, the rule for order ofoperations specifi es that we are to work inside the parentheses fi rst and thensimplify numbers containing exponents. After this, we multiply and divide, left toright. The last step is to add and subtract, left to right.ExAmplE 11Simplify −5(−3)(−4) as much as possible.SOlutiON−5(−3)(−4) = 15(−4)= −6012. 2(−8) + 3(−7) − 413. (−3) 514. −5(−2) 3 − 7(−3) 2Answers5. 36 6. 132 7. 14 8. −129. −20 10. −39 11. −4212. −41 13. −243 14. −23ExAmplE 12SOlutiONExAmplE 13Simplify 4(−3) + 6(−5) − 10 as much as possible.4(−3) + 6(−5) − 10 = −12 + (−30) − 10 Multiply= −42 − 10 Add= −52 SubtractSimplify (−2) 3 as much as possible.SOlutiON (−2) 3 = (−2)(−2)(−2) Definition of exponents= −8 Multiply, left to rightExAmplE 14Simplify −3(−2) 3 − 5(−4) 2 as much as possible.SOlutiON −3(−2) 3 − 5(−4) 2 = −3(−8) − 5(16) Exponents first= 24 − 80 Multiply= −56 Subtract

1.6 Multiplication of Real Numbers59Example 15Simplify 6 − 4(7 − 2) as much as possible.Solution 6 − 4(7 − 2) = 6 − 4(5) Inside parentheses first= 6 − 20 Multiply= −14 SubtractCMultiplying FractionsPreviously, we mentioned that to multiply two fractions we multiply numeratorsand multiply denominators. We can apply the rule for multiplication of positiveand negative numbers to fractions in the same way we apply it to other numbers.We multiply absolute values: The product is positive if both fractions have thesame sign and negative if they have different signs. Here are some examples.Example 16SolutionMultiply − ​ 3 _4 ​ ​ 5 _7 ​ ​._− ​ 3 4 ​ ​_5 7 ​ _​= − ​ 3 ⋅ 5 ​ Different signs give a negative answer4 ⋅ 7_= − ​ 1528 ​15. 7 − 3(4 − 9)Multiply.16. − ​ 2 _3 ​ ​ 5 _9 ​ ​Example 17SolutionMultiply −6​ ​ 1 _2 ​ ​.−6​ ​ 1 _2 ​ ​= − ​ 6 _1 ​​ ​ 1 _2 ​ ​ Different signs give a negative answer= − ​ 6 _2 ​= −317. −8 ​ ​ 1 _2 ​ ​Example 18Multiply − ​ 2 _3 ​ − ​ 3 _2 ​ ​.SolutionExample 19_− ​ 2 3 ​ _− ​ 3 2 ​ ​= ​_2 ⋅ 3 ​ Same signs give a positive answer3 ⋅ 2= ​ 6 _6 ​= 1Figure 1 gives the calories that are burned in 1 hour for avariety of forms of exercise by a person weighing 150 pounds. Figure 2 gives thecalories that are consumed by eating some popular fast foods. Find the netchange in calories for a 150-pound person playing handball for 2 hours and theneating a Whopper.18. − ​ 3 _4 ​ − ​ 4 _3 ​ ​19. Find the net change in caloriesfor a 150-pound person joggingfor 2 hours then eating aBig Mac.Calories Burned in 1 Hour by a 150-Pound Person700600680 680Calories5004003003745442002651000Bicycling Bowling HandballActivityFigure 1JoggingSkiingAnswers_15. 22 16. − ​ 10 ​ 17. −4 18. 127

60Chapter 1 The Basics700Calories in Fast FoodCalories600500400300200270 260 2805106301000McDonald's Burger KingHamburger HamburgerJack in theBoxHamburgerMcDonald'sBig MacBurger KingWhopperFigure 2Solution The net change in calories will be the difference of the caloriesgained from eating and the calories lost from exercise.Net change in calories = 630 − 2(680) = −730 caloriesApply the associative property, andthen simplify.20. −5(3x)D Using the Associative PropertyWe can use the rule for multiplication of real numbers, along with the associativeproperty, to multiply expressions that contain numbers and variables.Example 20Apply the associative property, and then multiply.Solution −3(2x) = (−3 ⋅ 2)x Associative property= −6x Multiplication21. 4(−8y)22. −3 ​ − ​ 1 _3 ​x ​Example 21Apply the associative property, and then multiply.Solution 6(−5y) = [6(−5)]y Associative property= −30y MultiplicationExample 22SolutionApply the associative property, and then multiply.−2 ​ − ​ 1 _2 ​x ​= ​ (−2)​ − ​ 1 _2 ​ ​ ​x Associative property= 1x Multiplication= x MultiplicationE Using the Distributive PropertyThe following examples show how we can use both the distributive property andmultiplication with real numbers.Multiply by applying the distributiveproperty.23. −3(a + 5)Example 23Apply the distributive property to −2(a + 3).Solution −2(a + 3) = −2a + (−2)(3) Distributive property= −2a + (−6) Multiplication= −2a − 6Answers19. −850 calories 20. −15x21. −32y 22. x 23. −3a − 15

1.6 Multiplication of Real Numbers61Example 24Apply the distributive property to −3(2x + 1).Solution −3(2x + 1) = −3(2x) + (−3)(1) Distributive property= −6x + (−3) MultiplicationExample 25SolutionDistributive property= −6x − 3_Apply the distributive property to − ​ 1 ​( 2 x − 6).3− ​ 1 _3 ​( 2 x − 6) = − ​1 _3 ​( 2 x) − ​ − ​ 1 _3 ​ ​(6)24. −5(2x + 1)_25. − ​ 1 ​( 4 x − 12)2_= − ​ 2 ​x − (−2) Multiplication3= − ​ 2 _3 ​x + 2Example 26Apply the distributive property to −4(3x − 5) − 8.Solution −4(3x − 5) − 8 = −4(3x) − (−4)(5) − 8 Distributive property= −12x − (−20) − 8 Multiplication= −12x + 20 − 8 Definition of subtraction= −12x + 12 Subtraction26. −3(2x − 6) − 7The next examples continue the work we did previously with finding the value ofan algebraic expression for given values of the variable or variables.Example 27 _Find the value of − ​ 2 ​x − 4 when3Solution_a. x = 0 b. x = 3 c. x = − ​ 9 2 ​Substituting the values of x into our expression one at a time we have_a. − ​ 2 ​(0) − 4 = 0 − 4 = −4327. Find the value of − ​ 3 _4 ​x + 2whena. x = 0b. x = 4c. x = − ​ 8 _3 ​_b. − ​ 2 ​(3) − 4 = −2 − 4 = −63c. − ​ 2 _3 ​ − ​ 9 _2 ​ ​− 4 = 3 − 4 = −1Example 28SolutionFFind the value of 5x − 4y whena. x = 4 and y = 0b. x = 0 and y = −8c. x = −2 and y = 3Substituting the values of x into our expression one at a time we havea. 5(4) − 4(0) = 20 − 0 = 20b. 5(0) − 4(−8) = 0 + 32 = 32c. 5(−2) − 4(3) = −10 − 12 = −22Geometric SequencesA geometric sequence is a sequence of numbers in which each number (after thefirst number) comes from the number before it by multiplying by the same amounteach time. For example, the sequence2, 6, 18, 54, . . .28. Find the value of 3x − 5y whena. x = 4 and y = 0b. x = 0 and y = −2c. x = −2 and y = 3Answers24. −10x − 5 25. −2x + 626. −6x + 1127. a. 2 b. −1 c. 428. a. 12 b. 10 c. −21

62Chapter 1 The Basicsis a geometric sequence because each number is obtained by multiplying thenumber before it by 3.29. Find the next number in eachof the following geometricsequences.a. 4, 8, 16, . . .b. 2, −6, 18, . . .c. ​_1 9 ​, ​1 _​, 1, . . .3Example 29Each sequence below is a geometric sequence. Find thenext number in each sequence.Solutiona. 5, 10, 20, . . . b. 3, −15, 75, . . . c. ​ 1 _8 ​, ​1 _4 ​, ​1 _2 ​, . . .Because each sequence is a geometric sequence, we know that eachterm is obtained from the previous term by multiplying by the same number eachtime.a. 5, 10, 20, . . . : The sequence starts with 5. After that, each number isobtained from the previous number by multiplying by 2 each time. Thenext number will be 20 ⋅ 2 = 40.b. 3, −15, 75, . . . : The sequence starts with 3. After that, each numberis obtained by multiplying by −5 each time. The next number will be75(−5) = −375.c. ​_ 1 8 ​, ​1 _4 ​, ​1 _2 ​, . . . : This sequence starts with ​1 _​. Multiplying each number in the8sequence by 2 produces the next number in the sequence. To extend thesequence, we multiply ​_ 1 2 ​by 2: ​1 _​⋅ 2 = 1.2The next number in the sequence is 1.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. How do you multiply two negative numbers?2. How do you multiply two numbers with different signs?3. Explain how some multiplication problems can be thought of asrepeated addition.4. What is a geometric sequence?Answer29. a. 32 b. −54 c. 3

1.6 Problem Set63Problem Set 1.6A Use the rule for multiplying two real numbers to find each of the following products. [Examples 1–10]1. 7(−6)2. 8(−4)3. −8(2)4. −16(3)−42−32−16−485. −3(−1)36. −7(−1)77. −11(−11)1218. −12(−12)144B Use the rule for order of operations to simplify each expression as much as possible. [Examples 11–15]9. −3(2)(−1)610. −2(3)(−4)2411. −3(−4)(−5)−6012. −5(−6)(−7)−21013. −2(−4)(−3)(−1)2414. −1(−3)(−2)(−1)615. (−7) 24916. (−8) 26417. (−3) 3−2718. (−2) 41619. −2(2 − 5)620. −3(3 − 7)1221. −5(8 − 10)1022. −4(6 − 12)2423. (4 − 7)(6 − 9)924. (3 − 10)(2 − 6)2825. (−3 − 2)(−5 − 4)4526. (−3 − 6)(−2 − 8)9027. −3(−6) + 4(−1)1428. −4(−5) + 8(−2)429. 2(3) − 3(−4) + 4(−5)−230. 5(4) − 2(−1) + 5(6)5231. 4(−3) 2 + 5(−6) 221632. 2(−5) 2 + 4(−3) 286Selected exercises available online at www.webassign.net/brookscole

64Chapter 1 The Basics33. 7(−2) 3 − 2(−3) 3−234. 10(−2) 3 − 5(−2) 4−16035. 6 − 4(8 − 2)−1836. 7 − 2(6 − 3)137. 9 − 4(3 − 8)2938. 8 − 5(2 − 7)3339. −4(3 − 8) − 6(2 − 5)3840. −8(2 − 7) − 9(3 − 5)5841. 7 − 2[−6 − 4(−3)]−542. 6 − 3[−5 − 3(−1)]1243. 7 − 3[2(−4 − 4) − 3(−1 − 1)]3744. 5 − 3[7(−2 − 2) − 3(−3 + 1)]7145. Simplify each expression.a. 5(−4)(−3) 60b. 5(−4) − 3 −23c. 5 − 4(−3) 17d. 5 − 4 − 3 −246. Simplify each expression.a. −2(−3)(−5) −30b. −2(−3) − 5 1c. −2 − 3(−5) 13d. −2 − 3 − 5 −10C Multiply the following fractions. [Examples 16–18]47. − ​ 2 _3 ​⋅ ​5 _7 ​−​ 10 _21 ​ 48. −​ 6 _5 ​⋅ ​2 _7 ​ 49. −8 _​ ​ 1 2 ​ ​_−​ 12−435 ​50. −12 ​ ​ 1 _3 ​ ​−451. ​ _−​ 3 24 ​ ​ ​ 52. ​ −​ 2 2_5 ​ ​ ​​_ 916 ​ ​_ 425 ​53. Simplify each expression.54. Simplify each expression.a. ​_5 8 ​(24) + ​3 _​(28) 27a. ​_57 6 ​(18) + ​3 _​(15) 245b. ​_5 8 ​(24) − ​3 _7 ​(28) 3b. ​_5 6 ​(18) − ​3 _5 ​(15) 6c. ​_5 8 ​( −24) + ​3 _​( −28) −27c. ​_57 6 ​( −18) + ​3 _​( −15) −245_d. −​ 5 8 ​(24) − ​3 _7 ​(28) −27 _d. −​ 5 6 ​(18) − ​3 _​(15) −245Simplify.55. ​ _​ 1 22 ​⋅ 6 ​ ​ 9 56. ​ _​ 1 22 ​⋅ 10 ​ ​ 25 57. ​ _​ 1 22 ​⋅ 5 ​ ​ ​_254 ​ 58. ​ ​_1 22 ​(0.8) ​ ​ 0.1659. ​ ​_1 22 ​(−4) ​ ​ 4 60. ​ ​_1 22 ​(−12) ​ ​ 36 61. ​ ​_1 22 ​(−3) ​ ​​_ 9 4 ​ 62. ​ ​_1 22 ​(−0.8) ​ ​ 0.16

1.6 Problem Set65D Find the following products. [Examples 20–22]63. −2(4x)−8x64. −8(7x)−56x65. −7(−6x)42x66. −8(−9x)72x_67. −​ 1 ​( −3x)3x_68. −​ 1 ​( −5x)5xE Apply the distributive property to each expression, and then simplify the result. [Examples 23–28]69. −4(a + 2)−4a − 870. −7(a + 6)−7a − 42_71. −​ 1 ​( 3 x − 6)2_−​ 32 ​x + 3 72. −​ 1 _​( 2 x − 4)4−​ 1 _2 ​x + 173. −3(2x − 5) − 7−6x + 874. −4(3x − 1) − 8−12x − 475. −5(3x + 4) − 10−15x − 3076. −3(4x + 5) − 20−12x − 3577. −4(3x + 5y)−12x − 20y78. 5(5x + 4y)25x + 20y79. −2(3x + 5y)−6x − 10y80. −2(2x − y)−4x + 2y81. ​_1 ​( −3x + 6)282. ​_1 ​( 5 x − 20)483. ​_1 ​( −2x + 6)384. ​_1_−​ 3 2 ​x + 3 ​_ 5 4 ​x − 5 _−​ 2 3 ​x + 2​( −4x + 20)5−​ 4 _5 ​x + 4_85. −​ 1 ​( −2x + 6)3_86. −​ 1 ​( −2x + 6)2​_ 2 3 ​x − 2 x − 387. 8 ​ −​ 1 _4 ​x + 1 _8 ​y ​−2x + y88. 9 ​ −​ 1 _9 ​x + ​1 _3 ​y ​−x + 3y_89. Find the value of −​ 1 ​x + 2 when3a. x = 0 2b. x = 3 1c. x = −3 3_90. Find the value of −​ 2 ​x + 1 when3a. x = 0 1b. x = 3 −1c. x = −3 391. Find the value of 2x + y when92. Find the value of 2x − 5y whena. x = 2 and y = −1 3a. x = 2 and y = 3 −11b. x = 0 and y = 3 3b. x = 0 and y = −2 10c. x = ​_3 2 ​and y = −7 −4 c. x = ​_5 2 ​and y = 1 093. Find the value of 2x 2 − 5x when94. Find the value of 49a 2 − 16 whena. x = 4 12a. a = ​_4_b. x = − ​ 3 7 ​ 02 ​ 12 b. a = −​ _ 4 7 ​ 0

66Chapter 1 The Basics95. Find the value of y(2y + 3) when96. Find the value of x(13 − x) whena. y = 4 44a. x = 5 40_b. y = − ​ 112 ​ 44 b. x = 8 4097. Five added to the product of 3 and −10 is whatnumber?−2598. If the product of −8 and −2 is decreased by 4, whatnumber results?1299. Write an expression for twice the product of −4 and x,and then simplify it.2(−4x) = −8x100. Write an expression for twice the product of −2 and 3x,and then simplify it.2[−2(3x)] = −12x101. What number results if 8 is subtracted from theproduct of −9 and 2?−26102. What number results if −8 is subtracted from theproduct of −9 and 2?−10F Each of the following is a geometric sequence. In each case, find the next number in the sequence. [Example 29]103. 1, 2, 4, . . .104. 1, 5, 25, . . .105. 10, −20, 40, . . .106. 10, −30, 90, . . .8125−80−270107. 1, ​_1 2 ​, ​1 _4 ​, . . .108. 1, ​_1 3 ​, ​1 _9 ​, . . .109. 3, −6, 12, . . .​_ 1 8 ​ ​_ 1 −2427 ​110. −3, 6, −12, . . .24Here are some problems you will see later in the book. Simplify.111. 3(x − 5) + 4112. 5(x − 3) + 2113. 2(3) − 4 − 3(−4)114. 2(3) + 4(5) − 5(2)3x − 115x − 131416115. ​ _​ 1 22 ​⋅ 18 ​ ​81116. ​ ​_1 22 ​( −10) ​ ​25117. ​ _​ 1 22 ​⋅ 3 ​ ​118. ​ _​ 1​_ 9 4 ​ ​_ 254 ​22 ​⋅ 5 ​ ​

1.6 Problem Set67Applying the Concepts119. picture messaging The graph shows the number ofpicture messages sent each of the first nine monthsof the year. Jan got a new phone in March thatallowed picture messaging. If she gets charged $0.25per message, how much did she get charged in April?$2.50.120. google Earth The Google Earth map shows YellowstoneNational Park. The park covers about 3,472 squaremiles. If there is an average of 2.3 moose per squaremile, about how many moose live in Yellowstone?Round to the nearest moose.7,986 mooseA Picture’s Worth 1,000 Words504030201031021 212632410JanFebMarAprMayJunJulAugSepImage © 2008 DigitalGlobe121. Stock Value Suppose you own 20 shares of a stock. Ifthe price per share drops $3, how much money haveyou lost?$60123. temperature Change The temperature is 25°F at 5:00in the afternoon. If the temperature drops 6°F everyhour after that, what is the temperature at 9:00 in theevening?1°F125. Nursing problem A patient’s prescription requires himto take two 25 mg tablets twice a day. What is thetotal dosage for the day?100 mg127. Reading Charts Refer to Figures 3 and 4 to find thenet change in calories for a 150-pound person whobowls for 3 hours and then eats 2 Whoppers.465 caloriesCalories Burned in 1 Hour by a 150-Pound Person700Calories600680 680500544400300 3742002651000Bicycling Bowling Handball Jogging SkiingActivityFIGuRE 3122. Stock Value Imagine that you purchase 50 shares of astock at a price of $18 per share. If the stock is sellingfor $11 a share a week after you purchased it, howmuch money have you lost?$350124. temperature Change The temperature is −5°F at 6:00 inthe evening. If the temperature drops 3°F every hourafter that, what is the temperature at midnight?−23°F126. Nursing problem A patient takes three 50 mg capsules aday. How many milligrams is he taking daily?150 mg128. Reading Charts Refer to Figures 3 and 4 to find the netchange in calories for a 150-pound person who eats aBig Mac and then skis for 3 hours.−1,122 caloriesCalories7006005004003002001000McDonald's Burger KingHamburger HamburgerCalories in Fast Food270 260 280Jack in theBoxHamburgerFIGuRE 4510McDonald'sBig Mac630Burger KingWhopper

Division of Real Numbers 1.7Suppose that you and four friends bought equal shares of an investment for atotal of $15,000 and then sold it later for only $13,000. How much did each personlose? Because the total amount of money that was lost can be representedby −$2,000, and there are 5 people with equal shares, we can represent eachperson’s loss with division:_ −$2,000= −$4005From this discussion it seems reasonable to say that a negative number dividedby a positive number is a negative number. Here is a more detailed discussion ofdivision with positive and negative numbers.The last of the four basic operations is division. We will use the same approachto defi ne division as we used for subtraction; that is, we will defi ne division interms of rules we already know.Recall that we developed the rule for subtraction of real numbers by definingsubtraction in terms of addition. We changed our subtraction problems to additionproblems and then added to get our answers. Because we already have arule for multiplication of real numbers, and division is the inverse operation ofmultiplication, we will simply define division in terms of multiplication._We know that division by the number 2 is the same as multiplication by 1 2 ; that_is, 6 divided by 2 is 3, which is the same as 6 times 1 . Similarly, dividing a numberby 5 gives the same result as multiplying by 1 . We can extend this idea to all2_5real numbers with the following rule.ADividing Positive and Negative NumbersDivision by a number is the same as multiplication by its reciprocal. Becauseevery division problem can be written as a multiplication problem and becausewe already know the rule for multiplication of two real numbers, we do not haveto write a new rule for division of real numbers. We will simply replace our divisionproblem with multiplication and use the rule we already have.ExAmplESWrite each division problem as an equivalent multiplicationproblem, and then multiply.1. 6 _2 = 6 1 _2 = 3 The product of two positives is positive2.RuleIf a and b represent any two real numbers (b cannot be 0), then it is alwaystrue that6_−2 = 6 − 1 _2 = −33. −6 _2 = −6 1 _2 = −3a ÷ b = a _b = a 1 _b}Theproduct of a positive and a negative is a negative4. −6 _−2 = −6 − 1 _2 = 3 The product of two negatives is positiveObjectivesA Divide any combination of positiveand negative numbers.B Divide fractions.C Simplify expressions using the rulefor order of operations.NoteWe are defi ning divisionthis way simplyso that we can usewhat we already know about multiplicationto do division problems.We actually want as few rules aspossible. Defi ning division in termsof multiplication allows us to avoidwriting a separate rule for division.pRACtiCE pROBlEmSChange each problem to an equivalentmultiplication problem, andthen multiply to get your answer.1.2.3.4.12 _412 _−4−12 _4−12 _−4Examples now playing atMathTV.com/booksThe second step in the previous examples is used only to show that we can write_divi sion in terms of multiplication. [In actual practice we wouldn’t write 6 2 as 6 _ 1 . ]21.7 Division of Real NumbersAnswers1. 3 2. −3 3. −3 4. 369

70Chapter 1 The BasicsNoteWhat we are sayinghere is that the workshown in Examples 1through 4 is shown simply to justifythe answers we obtain. In thefuture we won’t show the middlestep in these kinds of problems.Even so, we need to know thatdivision is defi ned to be multiplicationby the reciprocal.Divide.The answers, therefore, follow from the rule for multiplication; that is, like signsproduce a positive answer, and unlike signs produce a negative answer.Here are some examples. This time we will not show division as multiplicationby the reciprocal. We will simply divide. If the original numbers have the samesigns, the answer will be positive. If the original numbers have different signs, theanswer will be negative.ExAmplE 5Divide 12 _6 .SOlutiON_ 12= 2 Like signs give a positive answer65.6.18 _318 _−3ExAmplE 6Divide 12 _−6 .SOlutiON12 _−6 = −2unlike signs give a negative answer7.−18 _3ExAmplE 7Divide −12 _6 .SOlutiON_ −12= −2 unlike signs give a negative answer68.−18 _−3ExAmplE 8Divide −12 _−6 .SOlutiON_ −12= 2 Like signs give a positive answer−69.30 _−10ExAmplE 9Divide 15 _−3 .SOlutiON15 _−3 = −5unlike signs give a negative answer10. −50 _−25ExAmplE 10SOlutiONDivide −40 _−5 ._ −40= 8 Like signs give a positive answer−511. −21 _3ExAmplE 11Divide −14 _2 .SOlutiONBDivision with Fractions_ −14= −7 unlike signs give a negative answer2Answers5. 6 6. −6 7. −6 8. 6 9. −310. 2 11. −7We can apply the definition of division to fractions. Because dividing by a fractionis equivalent to multiplying by its reciprocal, we can divide a number by the fraction3 4 by multiplying it by the reciprocal of _ 3 4 , which is _ 4 . For example,_32_5 ÷ _ 3 4 = _ 2 5 ⋅ _ 4 3 = _ 815You may have learned this rule in previous math classes. In some math classes,multiplication by the reciprocal is referred to as “inverting the divisor and mul-

1.7 Division of Real Numbers71tiplying.” No matter how you say it, division by any number (except 0) is alwaysequivalent to multiplication by its reciprocal. Here are additional examples thatinvolve division by fractions.Example 12SolutionExample 13Divide ​ 2 _3 ​÷ ​5 _7 ​.​_2 3 ​÷ ​5 _7 ​= ​2 _3 ​⋅ ​7 _​ Rewrite as multiplication by the reciprocal5= ​ 14 _15 ​ MultiplyDivide − ​ 3 _4 ​÷ ​7 _9 ​.Divide.12. ​ 3 _4 ​÷ ​5 _7 ​13. − ​ 7 _8 ​÷ ​3 _5 ​Solution_− ​ 3 4 ​÷ ​7 _9 ​= − ​3 _4 ​⋅ ​9 _​ Rewrite as multiplication by the reciprocal7= − ​ 27 _28 ​ MultiplyExample 14Divide 8 ÷ ​ − ​ 4 _5 ​ ​.14. 10 ÷ ​ − ​ 5 _6 ​ ​Solution8 ÷ ​ − ​ 4 _5 ​ ​ = ​ 8 _1 ​​ − ​ 5 _4 ​ ​ Rewrite as multiplication by the reciprocal= − ​ 40 _4 ​ Multiply= −10 Divide 40 by 4CUsing Order of OperationsThe last step in each of the following examples involves reducing a fraction tolowest terms. To reduce a fraction to lowest terms, we divide the numerator anddenominator by the largest number that divides each of them exactly. For example,to reduce ​_1520 ​to lowest terms, we divide 15 and 20 by 5 to get ​3 _4 ​.Example 15Simplify as much as possible: ​_−4(5) ​6Solution ​_−4(5) ​= ​_−20 ​ Simplify numerator6 6= − ​ 10 _3 ​Reduce to lowest terms by dividingnumerator and denominator by 2Example 16Simplify as much as possible: ​_ 30−4 − 5 ​Simplify.15. ​_−6(5) ​91016. ​_−7 − 1 ​Solution ​30_​= ​30_−4 − 5 −9 ​= − ​ 10 _3 ​Simplify denominatorReduce to lowest terms by dividingnumerator and denominator by 3In the examples that follow, the numerators and denominators contain expressionsthat are somewhat more complicated than those we have seen thus far. Toapply the rule for order of operations to these examples, we treat fraction barsthe same way we treat grouping symbols; that is, fraction bars separate numeratorsand denominators so that each will be simplified separately.Answers12. ​_21 _ 35​ 13. − ​ ​ 14. −1220 24_15. − ​ 103 ​ 16. − ​ _ 5 4 ​

72Chapter 1 The BasicsSimplify.17. ​ −6 − 6 _−2 − 4 ​18. ​__3(−4) + 9 ​66(−2) + 5(−3)19. ​__​5(4) − 11Example 17Simplify: ​ −8 − 8 _−5 − 3 ​Solution ​_−8 − 8 ​= ​−16_−5 − 3 −8 ​= 2 DivisionExample 18Simplify: ​__2(−3) + 4Solution ​__2(−3) + 4 ​= ​_−6 + 4 ​12 12Example 19SolutionSimplify numerator and denominator separately12= ​ −2 _12 ​ Addition​In the numerator, we multiply before we add_= − ​ 1 to lowest terms by dividing​ Reduce6 numerator and denominator by 25(−4) + 6(−1)Simplify: ​__​2(3) − 4(1)5(−4) + 6(−1)​__​= ​__−20 + (−6)​2(3) − 4(1) 6 − 4Multiplication before addition= ​_−26 ​ Simplify numerator and denominator2= −13 Divide −26 by 2We must be careful when we are working with expressions such as (−5) 2 and−5 2 that we include the negative sign with the base only when parentheses indicatewe are to do so.Unless there are parentheses to indicate otherwise, we consider the base tobe only the number directly below and to the left of the exponent. If we want toinclude a negative sign with the base, we must use parentheses.To simplify a more complicated expression, we follow the same rule. For example,The bases are 7 and 3; the sign between7 2 − 3 2 = 49 − 9the two terms is a subtraction signFor another example,5 3 − 3 4 = 125 − 81 We simplify exponents first, then subtractSimplify.20. ​ 42 − 2 2_−4 + 2 ​Example 20Simplify: ​ 52 − 3 2_−5 + 3 ​Solution ​_52 − 3 2−5 + 3 ​= ​25 _ − 9 ​ Simplify numerator and denominator separately−2= ​ 16 _−2 ​= −8(4 + 3)221. ​_−4 2 − 3 2​Answers_17. 2 18. − ​ 1 ​ 19. −3 20. −62_21. − ​ 4925 ​Example 21 (3 + 2)2Simplify: ​_−3 2 − 2 2​(3 + 2)2Solution ​_−3 2 − 2 2​= ​ _ 5 2−9 − 4 ​= ​ 25 _−13 ​= − ​ 25 _13 ​Simplify numerator and denominator separately

1.7 Division of Real Numbers73We can combine our knowledge of the properties of multiplication with our definitionof division to simplify more expressions involving fractions. Here are two examples:Example 22Simplify 10 ​ ​ x _2 ​ ​.​as if they were undefined also. We will treat problems like ​_ 0 0_22. Simplify 8 ​ ​ x 4 ​ ​.Solution_10 ​ ​ x 2 ​ ​= 10 _​ ​ 1 2 ​x ​ Dividing by 2 is the same as multiplying by ​_1 2 ​= ​ 10 ⋅ ​_1 2 ​ ​x Associative property= 5x MultiplicationExample 23Simplify a ​ _​ 3 a ​− 4 ​.Solution a _​ ​ 3 a ​− 4 ​= a ⋅ ​_3 a ​− a ⋅ 4 Distributive property= 3 − 4a MultiplicationDivision with the Number 0For every division problem there is an associated multiplication problem involvingthe same numbers. For example, the following two problems say the samething about the numbers 2, 3, and 6:Division Multiplication_​ 6 ​= 236 = 2(3)We can use this relationship between division and multiplication to clarify divisioninvolving the number 0.First, dividing 0 by a number other than 0 is allowed and always results in 0. Tosee this, consider dividing 0 by 5. We know the answer is 0 because of the relationshipbetween multiplication and division. This is how we write it:_​ 0 ​= 05because 0 = 0(5)However, dividing a nonzero number by 0 is not allowed in the real numbers.Suppose we were attempting to divide 5 by 0. We don’t know if there is an answerto this problem, but if there is, let’s say the answer is a number that we can representwith the letter n. If 5 divided by 0 is a number n, then_​ 5 ​= n0and 5 = n(0)This is impossible, however, because no matter what number n is, when wemultiply it by 0 the answer must be 0. It can never be 5. In algebra, we say expressionslike ​_ 5 ​are undefined because there is no answer to them; that is, division by00 is not allowed in the real numbers.The only other possibility for division involving the number 0 is 0 divided by 0.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Why do we define division in terms of multiplication?2. What is the reciprocal of a number?3. How do we divide fractions?4 . Why is division by 0 not allowed with real numbers?23. Simplify x ​ ​ 4 _x ​− 5 ​.Answers22. 2x 23. 4 − 5x

1.7 Problem Set75Problem Set 1.7A Find the following quotients (divide). [Examples 1–11]1. ​_8−4 ​2. ​_10−5 ​−2−23. ​ −48 _16 ​−34. ​ −32 _4 ​−85. ​_−721 ​6. ​_−25100 ​7. ​_−39−13 ​_−​ 1 3 ​ _−​ 1 4 ​ 38. ​ −18 _−6 ​39. ​_−6−42 ​10. ​_−4−28 ​11. ​_0−32 ​​_ 1 7 ​ ​_ 1 7 ​ 012. ​ 0 _17 ​0The following problems review all four operations with positive and negative numbers. Perform the indicated operations.13. −3 + 12914. 5 + (−10)−515. −3 − 12−1516. 5 − (−10)1517. −3(12)−3618. 5(−10)−5019. −3 ÷ 1220. 5 ÷ (−10)_−​ 1 4 ​ _−​ 1 2 ​B Divide and reduce all answers to lowest terms. [Examples 12–14]21. ​_4 5 ​÷ ​3 _4 ​22. ​_6 8 ​÷ ​3 _4 ​_23. −​ 5 6 ​÷ ​ _ −​ 5 8 ​ _ ​24. −​ 7​_1615 ​ 1​_ 4 3 ​ ​_ 143 ​9 ​÷ ​ −​ 1 _6 ​ ​25. ​ 10 _13 ​÷ ​ −​ 5 _4 ​ ​12 ​÷ ​ −​ 10 _3 ​ ​_−​ 8 13 ​ _−​ 1 8 ​ −126. ​_5 _27. −​ 56 ​÷ ​5 _6 ​_28. −​ 8 9 ​÷ ​8 _9 ​−129. −​ 3 _4 ​÷ ​ −​ 3 _4 ​ ​130. −​ 6 _7 ​÷ ​ −​ 6 _7 ​ ​1Selected exercises available online at www.webassign.net/brookscole

76Chapter 1 The BasicsC The following problems involve more than one operation. Simplify as much as possible. [Examples 15–23]31. ​_3(−2)−10 ​32. ​_4(−3)24 ​33. ​_−5(−5)−15​34. ​_−7(−3)−35​​_ 3 5 ​ _−​ 1 2 ​ _−​ 5 3 ​ _−​ 3 5 ​35. ​_−8(−7)−28​−236. ​_−3(−9) ​37. ​_27−64 − 13 ​_−​ 9 2 ​ −32738. ​_13 − 4 ​339. ​ 20 − 6 _5 − 5 ​10 − 1240. ​_3 − 3 ​41. ​ −3 + 9 _2 ⋅ 5 − 10 ​42. ​ −4 + 8 _2 ⋅ 4 − 8 ​UndefinedUndefinedUndefinedUndefined15(−5) − 2543. ​__​2(−10)510(−3) − 2044. ​__​5(−2)527 − 2(−4)45. ​__​46. ​__20 − 5(−3)​−3(5)10(−3)_−​ 7 3 ​ _−​ 7 6 ​12 − 6(−2)47. ​__​12(−2)−13(−4) + 5(−6)48. ​__​10 − 649. ​_52 − 2 2−5 + 2 ​_−​ 212 ​ −750. ​ 72 − 4 2_−7 + 4 ​−1151. ​ 82 − 2 2_8 2 + 2 2 ​​_15(5 + 3)2 _−5 2 − 3 2​52. ​_42 − 6 24 2 + 6 ​53. ​ 54. ​ 217 ​ _−​ 5 13 ​ _−​ 3217 ​(7 + 2)2 _−7 2 − 2 2​−​ 81 _53 ​(8 − 4)255. ​_8 2 − 4 ​56. ​_(6 − 2)22 6 2 − 2 ​57. ​__−4 ⋅ 32 − 5 ⋅ 2 2​2 −8(7)​_ 1 3 ​ ​_ 1 2 ​ 158. ​__−2 ⋅ 52 + 3 ⋅ 2 3​−3(13)​_ 2 3 ​

1.7 Problem Set7759. ​__3 ⋅ 102 + 4 ⋅ 10 + 5​345160. ​__5 ⋅ 102 + 6 ⋅ 10 + 7​56717 − [(2 − 3) − 4]61. ​__​−1 − 2 − 3−22 − [(3 − 5) − 8]62. ​__​−3 − 4 − 5−16(−4) − 2(5 − 8)63. ​__​64. ​__3(−4) − 5(9 − 11)​−6 − 3 − 5−9 − 2 − 3​_ 9 7 ​ ​_ 1 7 ​3(−5 − 3) + 4(7 − 9)65. ​___​66. ​___−2(6 − 10) − 3(8 − 5)​5(−2) + 3(−4)6(−3) − 6(−2)​_1611 ​ ​_ 1 6 ​67. ​_| ​ 3 − 9 |​3 − 9 ​−168. ​_| ​ 4 − 7 |​4 − 7 ​−12+ 0.15(10)69. ​__​70. ​__5(5)+ 250​10640(5)_​ 7 20 ​= 0.35 _​ 11128 ​ = 0.0871. ​ 1 − 3 _3 − 1 ​−125 − 1672. ​_16 − 25 ​−173. Simplify.a. ​ 5 − 2 _3 − 1​ 3 _2 ​b. ​ 2 − 5 _1 − 3 ​ ​3 _a. ​ 6 − 2 _3 − 5 ​ −2b. ​ 2 − 6 _5 − 3 ​ −22 ​ 74. Simplify.75. Simplify.a. ​ −4 − 1 _5 − (−2) ​ −​5 _7 ​b. ​ 1 − (−4) _(−2) − 5 ​ −​5 _7 ​ 76. Simplify.a. ​ −6 − 1 _4 − (−5) ​ −​7 _9 ​b. ​ 1 − (−6) _−5 − 4 ​ −​7 _9 ​77. Simplify.a. ​_3 + 2.236 ​ 2.6182b. ​_3 − 2.236 ​ 0.3822c. ​_3 + 2.236 ​+ ​_3 − 2.236 ​ 32278. Simplify.a. ​_1 + 1.732 ​ 1.3662b. ​_1 − 1.732 ​ −0.3662c. ​_1 + 1.732 ​+ ​_1 − 1.732 ​ 122

78Chapter 1 The Basics79. Simplify each expression.a. 20 ÷ 4 ⋅ 5 25b. −20 ÷ 4 ⋅ 5 −25c. 20 ÷ (−4) ⋅ 5 −25d. 20 ÷ 4(−5) −25e. −20 ÷ 4(−5) 2580. Simplify each expression.a. 32 ÷ 8 ⋅ 4 16b. −32 ÷ 8 ⋅ 4 −16c. 32 ÷ (−8) ⋅ 4 −16d. 32 ÷ 8(−4) −16e. −32 ÷ 8(−4) 1681. Simplify each expression.a. 8 ÷ ​ 4 _5 ​ 10b. 8 ÷ ​ 4 _5 ​ − 10 0c. 8 ÷ ​_4 ​( −10) −1005d. 8 ÷ ​ −​ 4 _5 ​ ​ − 10 −2082. Simplify each expression.a. 10 ÷ ​ 5 _6 ​ 12b. 10 ÷ ​ 5 _6 ​ − 12 0c. 10 ÷ ​_5 ​( −12) −1446d. 10 ÷ ​ −​ 5 _6 ​ ​ − 12 −24Apply the distributive property.83. 10 ​ ​ x _2 ​+ ​3 _5 ​ ​84. 6 ​ ​ x _3 ​+ ​5 _2 ​ ​85. 15 ​ ​ x _5 ​+ ​4 _3 ​ ​86. 6 ​ ​ x _3 ​+ ​1 _2 ​ ​5x + 62x + 153x + 202x + 387. x ​ ​ 3 _x ​+ 1 ​88. x ​ ​ 4 _x ​+ 3 ​89. 21 ​ ​ x _7 ​− ​y _3 ​ ​90. 36 ​ ​ x _4 ​− ​y _9 ​ ​3 + x4 + 3x3x − 7y9x − 4y91. a ​ ​ 3 _a ​− ​2 _a ​ ​92. a ​ ​ 7 _a ​+ ​1 _a ​ ​93. 2y ​ ​ 1 _y ​− ​1 _2 ​ ​94. 5y ​ ​ 3 _y ​− ​4 _5 ​ ​182 − y15 − 4yAnswer the following questions.95. What is the quotient of −12 and −4?396. The quotient of −4 and −12 is what number?​_ 1 3 ​97. What number do we divide by −5 to get 2?−1098. What number do we divide by −3 to get 4?−1299. Twenty-seven divided by what number is −9?−3100. Fifteen divided by what number is −3?−5101. If the quotient of −20 and 4 is decreased by 3, whatnumber results?−8102. If −4 is added to the quotient of 24 and −8, what numberresults?−7

1.7 Problem Set79Applying the Concepts103. Golf The chart shows the lengths of the longest golfcourses used for the U.S. open. If the length of theBethpage course is 7,214 yards, what is the averagelength of each hole on the 18-hole course?401 yards104. Broadway The chart shows the number of plays performedby the longest running Broadway musicals. IfLes Miserables ran for 16 years, what was the meannumber of shows per year? Round to the nearest show.418 shows per yearLongest Courses in U.S. Open HistoryIt’s on Broadway7,214 yds7,214 ydsBethpage State Park, Black Course (2002)Pinehurst, No. 2 Course (2005)CatsThe Phantom of the Opera7,485 shows7,061 shows7,213 ydsCongressional C.C. Blue Course (1997)Les Miserables6,680 shows7,195 ydsMedinah C.C. (1990)A Chorus Line6,137 shows7,191 ydsBellerive C.C. St. Lous (1965)Beauty and the Beast4,386 showsSource: http://www.usopen.comSource: The League of American Theaters and Producers105. Investment Suppose that you and 3 friends boughtequal shares of an investment for a total of $15,000and then sold it later for only $13,600. How much dideach person lose?$350107. Temperature Change Suppose that the temperatureoutside is dropping at a constant rate. If the temperatureis 75°F at noon and drops to 61°F by 4:00 in theafternoon, by how much did the temperature changeeach hour?Drops 3.5°F each hour109. Nursing Problem A patient is required to take 75 mg ofa drug. If the capsule strength is 25 mg, how manycapsules should she take?3 capsules111. Nursing Problem A patient is required to take 1.2 mgof drug and is told to take 4 tablets. What is thestrength of each tablet?0.3 mg113. Internet Mailing Lists A company sells products onthe Internet through an email list. They predict thatthey sell one $50 product for every 25 people on theirmailing list.a. What is their projected revenue if their list contains10,000 email addresses? $20,000b. What is their projected revenue if their list contains25,000 email addresses? $50,000c. They can purchase a list of 5,000 email addressesfor $5,000. Is this a wise purchase?Yes, the projected revenue for 5,000 email addresses is $10,000,which is $5,000 more than the list costs.106. Investment If 8 people invest $500 each in a stamp collectionand after a year the collection is worth $3,800,how much did each person lose?$25108. Temperature Change In a chemistry class, a thermometeris placed in a beaker of hot water. The initial temperatureof the water is 165°F. After 10 minutes the waterhas cooled to 72°F. If the water temperature drops at aconstant rate, by how much does the water temperaturechange each minute?Drops 9.3°F each minute110. Nursing Problem A patient is given prescribed a dosageof 25 mg, and the strength of each tablet is 50 mg.How many tablets should he take?½ tablet112. Nursing Problem A patient is required to take two tabletsfor a 3.6 mg dosage. What is the dosage strength ofeach tablet?1.8 mg114. Internet Mailing Lists A new band has a following on theInternet. They sell their CDs through an email list. Theypredict that they sell one $15 CD for every10 people on their mailing list.a. What is their projected revenue if their list contains5,000 email addresses? $7,500b. What is their projected revenue if their list contains20,000 email addresses? $30,000c. If they need to make $45,000, how many people dothey need on their email list?30,000 people

Subsets of the Real Numbers 1.8In Section 1.2 we introduced the real numbers and defined them as the numbersassociated with points on the real number line. At that time, we said thereal numbers include whole numbers, fractions, and decimals, as well as othernumbers that are not as familiar to us as these numbers. In this section we take amore detailed look at the kinds of numbers that make up the set of real numbers.The numbers that make up the set of real numbers can be classified as countingnumbers, whole numbers, integers, rational numbers, and irrational numbers; eachis said to be a subset of the real numbers.DefinitionSet A is called a subset of set B if set A is contained in set B, that is, if eachand every element in set A is also a member of set B.ObjectivesA Associate numbers with subsets ofthe real numbers.B Factor whole numbers into theproduct of prime factors.C Reduce fractions to lowest terms.Examples now playing atMathTV.com/booksASubsets of Real NumbersHere is a detailed description of the major subsets of the real numbers.The counting numbers are the numbers with which we count. They are the numbers1, 2, 3, and so on. The notation we use to specify a group of numbers like thisis set notation. We use the symbols { and } to enclose the members of the set.Counting numbers = {1, 2, 3, . . . }Example 1Which of the numbers in the following set are not countingnumbers?1. Is 5 a counting number?Practice Problems​ −3, 0, ​ 1 _2 ​, 1, 1.5, 3 ​SolutionThe numbers −3, 0, ​_ 1 ​, and 1.5 are not counting numbers.2The whole numbers include the counting numbers and the number 0.Whole numbers = {0, 1, 2, . . . }The set of integers includes the whole numbers and the opposites of all thecounting numbers.Integers = { . . . , −3, −2, −1, 0, 1, 2, 3, . . . }When we refer to positive integers, we are referring to the numbers 1, 2, 3, . . . .Likewise, the negative integers are −1, −2, −3, . . . . The number 0 is neither positivenor negative.Example 2Which of the numbers in the following set are not integers?2. Is −7 a whole number?​ −5, −1.75, 0, ​ 2 _3 ​, 1, π, 3 ​Solutionand π.The only numbers in the set that are not integers are −1.75, ​ 2 _3 ​,The set of rational numbers is the set of numbers commonly called “fractions”together with the integers. The set of rational numbers is difficult to listAnswers1. Yes 2. No1.8 Subsets of the Real Numbers81

82Chapter 1 The Basicsin the same way we have listed the other sets, so we will use a different kind ofnotation:Rational numbers = ​​ a _b ​∙ a and b are integers (b ≠ 0) ​This notation is read “The set of elements ​_a ​such that a and b are integers (andbb is not 0).” If a number can be put in the form ​_a ​, where a and b are both frombthe set of integers, then it is called a rational number.Rational numbers include any number that can be written as the ratio of twointegers; that is, rational numbers are numbers that can be put in the form​_integerinteger ​3. a. Is ​_3 ​a rational number?4b. Is 7 a rational number?Example 3Show why each of the numbers in the following set is arational number.Solutionthe ratio of −3 to 1; that is,​ −3, − ​ 2 _3 ​, 0, 0.333 . . . , 0.75 ​The number −3 is a rational number because it can be written as−3 = ​ −3 _1 ​_Similarly, the number − ​ 2 ​can be thought of as the ratio of −2 to 3, whereas the3number 0 can be thought of as the ratio of 0 to 1.Any repeating decimal, such as 0.333 . . . (the dots indicate that the 3's repeatforever), can be written as the ratio of two integers. In this case 0.333 . . . is thesame as the fraction ​ 1 _3 ​.Finally, any decimal that terminates after a certain number of digits can be writtenas the ratio of two integers. The number 0.75 is equal to the fraction ​_ 3 ​and is4therefore a rational number.Still other numbers exist, each of which is associated with a point on the realnumber line, that cannot be written as the ratio of two integers. In decimal formthey never terminate and never repeat a sequence of digits indefinitely. They arecalled irrational numbers (because they are not rational):Irrational numbers = {nonrational numbers}= {nonrepeating, nonterminating decimals}Answer3. a. Yes b. YesWe cannot write any irrational number in a form that is familiar to us becausethey are all nonterminating, nonrepeating decimals. Because they are not rational,they cannot be written as the ratio of two integers. They have to be represented inother ways. One irrational number you have probably seen before is π. It is not 3.14.Rather, 3.14 is an approximation to π. It cannot be written as a terminating decimalnumber. Other representations for irrational numbers are ​√ — 2 ​, ​√ — 3 ​, ​√ — 5 ​, ​√ — 6 ​, and, ingeneral, the square root of any number that is not itself a perfect square. (If you arenot familiar with square roots, you will be after Chapter 8.) Right now it is enough toknow that some numbers on the number line cannot be written as the ratio of twointegers or in decimal form. We call them irrational numbers.The set of real numbers is the set of numbers that are either rational or irrational;that is, a real number is either rational or irrational.Real numbers = {all rational numbers and all irrational numbers}

1.8 Subsets of the Real Numbers83BPrime Numbers and FactoringThe following diagram shows the relationship between multiplication andfactoring:MultiplicationFactors 3 ⋅ 4 = 12 ProductFactoringWhen we read the problem from left to right, we say the product of 3 and 4is 12. Or we multiply 3 and 4 to get 12. When we read the problem in the otherdirection, from right to left, we say we have factored 12 into 3 times 4, or 3 and 4are factors of 12.The number 12 can be factored still further:12 = 4 ⋅ 3= 2 ⋅ 2 ⋅ 3= 2 2 ⋅ 3The numbers 2 and 3 are called prime factors of 12 because neither of them canbe factored any further.DefinitionIf a and b represent integers, then a is said to be a factor (or divisor) of b if adivides b evenly, that is, if a divides b with no remainder.DefinitionA prime number is any positive integer larger than 1 whose only positivefactors (divisors) are itself and 1.Here is a list of the first few prime numbers.Prime numbers = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, . . . }When a number is not prime, we can factor it into the product of prime numbers.To factor a number into the product of primes, we simply factor it until it cannotbe factored further.NoteThe number 15 isnot a prime numberbecause it has factorsof 3 and 5; that is, 15 = 3 ⋅ 5.When a whole number largerthan 1 is not prime, it is said to becomposite.ExAmplE 4Factor the number 60 into the product of prime numbers.SOlutiON We begin by writing 60 as the product of any two positive integerswhose product is 60, like 6 and 10:4. Factor 90 into the product ofprime numbers.60 = 6 ⋅ 10We then factor these numbers:60 = 6 ⋅ 10= (2 ⋅ 3) ⋅ (2 ⋅ 5)= 2 ⋅ 2 ⋅ 3 ⋅ 5= 2 2 ⋅ 3 ⋅ 5NoteIt is customary to writeprime factors in orderfrom smallest to largest.Answer4. 2 ⋅ 3 2 ⋅ 5

84Chapter 1 The Basics5. Factor 420 into the product ofprimes.ExAmplE 5Factor the number 630 into the product of primes.SOlutiON Let’s begin by writing 630 as the product of 63 and 10:630 = 63 ⋅ 10= (7 ⋅ 9) ⋅ (2 ⋅ 5)NoteThere are some“tricks” to fi nding thedivisors of a number.For instance, if a number ends in0 or 5, then it is divisible by 5. If anumber ends in an even number(0, 2, 4, 6, or 8), then it is divisibleby 2. A number is divisible by 3 ifthe sum of its digits is divisible by3. For example, 921 is divisible by3 because the sum of its digits is9 + 2 + 1 = 12, which is divisibleby 3.= 7 ⋅ 3 ⋅ 3 ⋅ 2 ⋅ 5= 2 ⋅ 3 2 ⋅ 5 ⋅ 7It makes no difference which two numbers we start with, as long as their productis 630. We always will get the same result because a number has only one set ofprime factors.630 = 18 ⋅ 35= 3 ⋅ 6 ⋅ 5 ⋅ 7= 3 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 7= 2 ⋅ 3 2 ⋅ 5 ⋅ 7If we factor 210 into its prime factors, we have 210 = 2 ⋅ 3 ⋅ 5 ⋅ 7, which meansthat 2, 3, 5, and 7 divide 210, as well as any combination of products of 2, 3, 5,and 7; that is, because 3 and 7 divide 210, then so does their product 21. Because3, 5, and 7 each divide 210, then so does their product 105:21 divides 210210 = 2 ⋅ 3 ⋅ 5 ⋅ 7C Reducing to Lowest Terms105 divides 210Recall that we reduce fractions to lowest terms by dividing the numerator anddenominator by the same number. We can use the prime factorization of numbersto help us reduce fractions with large numerators and denominators._6. Reduce 154 to lowest terms.1,155NoteThe small lines wehave drawn throughthe factors that arecommon to the numerator anddenominator are used to indicatethat we have divided the numeratorand denominator by thosefactors.Answers5. 2 2 ⋅ 3 ⋅ 5 ⋅ 7 6. 2 _15ExAmplE 6Reduce 210 _SOlutiONto lowest terms.231First we factor 210 and 231 into the product of prime factors. Thenwe reduce to lowest terms by dividing the numerator and denominator by anyfactors they have in common._ 210231 = __2 ⋅ 3 ⋅ 5 ⋅ 7 Factor the numerator and denominator completely3 ⋅ 7 ⋅ 112 _ ⋅ 3∙ ⋅ 5 ⋅ 7∙= Divide the numerator and denominator by 3 ⋅ 73∙ ⋅ 7∙ ⋅ 11_= 2 ⋅ 511 = _ 1011Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. What is a whole number?2. How are factoring and multiplication related?3. Is every integer also a rational number? Explain.4 . What is a prime number?

1.8 Problem Set85Problem Set 1.8AGiven the numbers in the set {−3, −2.5, 0, 1, ​ 3 _2 ​, ​√— 15 ​}: [Examples 1–3]1. List all the whole numbers.0, 12. List all the integers.−3, 0, 13. List all the rational numbers.−3, −2.5, 0, 1, ​_ 3 2 ​4. List all the irrational numbers.​√ — 15 ​5. List all the real numbers.AllA Given the numbers in the set {−10, −8, −0.333 . . . , −2, 9, ​_25 ​, π}:36. List all the whole numbers.7. List all the integers.9−10, −8, −2, 98. List all the rational numbers.−10, −8, −0.333 . . . , −2, 9, ​_ 253 ​9. List all the irrational numbers.π10. List all the real numbers.AllIdentify the following statements as either true or false.11. Every whole number is also an integer.T13. A number can be both rational and irrational.F15. Some whole numbers are also negative integers.F12. The set of whole numbers is a subset of the set ofintegers.T14. The set of rational numbers and the set of irrationalnumbers have some elements in common.F16. Every rational number is also a real number.T17. All integers are also rational numbers.T18. The set of integers is a subset of the set of rationalnumbers.TBLabel each of the following numbers as prime or composite. If a number is composite, then factor it completely.19. 48Composite, 2 4 ⋅ 320. 72Composite, 2 3 ⋅ 3 2 21. 37Prime22. 23Prime23. 1,023Composite, 3 ⋅ 11 ⋅ 3124. 543Composite, 3 ⋅ 181Selected exercises available online at www.webassign.net/brookscole

86Chapter 1 The BasicsB Factor the following into the product of primes. When the number has been factored completely, write its prime factorsfrom smallest to largest. [Examples 4–5]25. 14426. 28827. 382 4 ⋅ 3 2 2 5 ⋅ 3 2 2 ⋅ 1928. 633 2 ⋅ 729. 1053 ⋅ 5 ⋅ 730. 2102 ⋅ 3 ⋅ 5 ⋅ 731. 1802 2 ⋅ 3 2 ⋅ 532. 9002 2 ⋅ 3 2 ⋅ 5 233. 3855 ⋅ 7 ⋅ 1134. 1,9255 2 ⋅ 7 ⋅ 1135. 12111 2 36. 5462 ⋅ 3 ⋅ 7 ⋅ 1337. 4202 2 ⋅ 3 ⋅ 5 ⋅ 738. 5982 ⋅ 13 ⋅ 2339. 6202 2 ⋅ 5 ⋅ 3140. 2,3102 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11C Reduce each fraction to lowest terms by first factoring the numerator and denominator into the product of prime factorsand then dividing out any factors they have in common. [Examples 6]41. ​_105165 ​42. ​_165385 ​43. ​_525735 ​44. ​_550735 ​​_711 ​ ​_ 3 7 ​ ​_ 5 7 ​ ​_ 110147 ​45. ​ 385 _455 ​46. ​_385735 ​47. ​_322345 ​48. ​_266285 ​13 ​ ​_ 1121 ​ ​_ 1415 ​ ​_ 1415 ​​_1149. ​_205369 ​50. ​_111185 ​51. ​_215344 ​52. ​_279310 ​​_ 5 9 ​ ​_ 3 5 ​ ​_ 5 8 ​ ​_ 910 ​

1.8 Problem Set87The next two problems are intended to give you practice reading, and paying attention to, the instructions that accompanythe problems you are working. You will see a number of problems like this throughout the book. Working these problemsis an excellent way to get ready for a test or a quiz.53. Work each problem according to the instructions given. 54. Work each problem according to the instructions given.d. Divide: ​_50−80 ​ −​5 _8 ​ d. Divide: ​_−2.57.5 ​ −​1 _3 ​(Note that each of these instructions could be replacedwith the instruction Simplify.)(Note that each of these instructions could be replacedwith the instruction Simplify.)a. Add: 50 + (−80) −30a. Add: −2.5 + 7.5 5b. Subtract: 50 − (−80) 130b. Subtract: −2.5 − 7.5 −10c. Multiply: 50(−80) −4,000c. Multiply: −2.5(7.5) −18.75Simplify each expression without using a calculator.55. ​_6.289(3.14) ​ ​2 _​ 56. ​_12.56​ 1 57. ​_9.429 4(3.14) 2(3.14) ​ ​3 _​ 58. ​_12.562 2(3.14) ​ 259. ​_320.5 ​ 64 60. ​16 _0.5 ​ 32 61. ​5,599 _​ 509 62. ​840_11 80 ​ 10.563. Find the value of ​_2 + 0.15x ​for each of the values of xxgiven below. Write your answers as decimals, to thenearest hundreth.a. x = 10 0.35b. x = 15 0.28c. x = 20 0.255x + 25064. Find the value of ​_​for each of the values of x640xgiven below. Write your answers as decimals, to thenearest thousandth.a. x = 10 0.047b. x = 15 0.034c. x = 20 0.02765. Factor 6 3 into the product of prime factors by first factoring6 and then raising each of its factors to the third toring 12 and then raising each of its factors to the sec-66. Factor 12 2 into the product of prime factors by first fac-power.ond power.6 3 = (2 ⋅ 3) 3 = 2 3 ⋅ 3 3 12 2 = (2 2 ⋅ 3) 2 = (2 2 ) 2 (3) 2 = 2 4 ⋅ 3 268. Factor 10 2 ⋅ 12 3 into the product of prime factors by first67. Factor 9 4 ⋅ 16 2 into the product of prime factors by first9 4 ⋅ 16 2 = (3 2 ) 4 (2 4 ) 2 = 2 8 ⋅ 3 8 10 2 ⋅ 12 3 = (2 ⋅ 5) 2 (2 2 ⋅ 3) 3 = 2 8 ⋅ 3 3 ⋅ 5 2factoring 9 and 16 completely.factoring 10 and 12 completely.

88Chapter 1 The Basics69. Simplify the expression 3 ⋅ 8 + 3 ⋅ 7 + 3 ⋅ 5, and thenfactor the result into the product of primes. (Noticeone of the factors of the answer is 3.)3 ⋅ 8 + 3 ⋅ 7 + 3 ⋅ 5 = 24 + 21 + 15 = 60 = 2 2 ⋅ 3 ⋅ 570. Simplify the expression 5 ⋅ 4 + 5 ⋅ 9 + 5 ⋅ 3, and then factorthe result into the product of primes.5 ⋅ 4 + 5 ⋅ 9 + 5 ⋅ 3 = 20 + 45 + 15 = 80 = 2 4 ⋅ 5Applying the Concepts71. Cars The chart shows the fastest cars in America.Factor the speed of the Saleen S 7 Twin Turbo into theproduct of primes.2 2 ⋅ 5 ⋅ 13Ready for the Races72. Classroom Energy The chart shows how much energy iswasted in the classroom by leaving appliances on. Writethe energy used by a ceiling fan over the energy used bythe television and then reduce to lowest terms.​_ 2526 ​Energy EstimatesAll units given as watts per hour.Ford GT 205 mphEvans 487 210 mphSaleen S7 Twin Turbo 260 mphSSC Ultimate Aero 273 mphCeiling fanStereoTelevisionVCR/DVD playerPrinterPhotocopierCoffee maker201251304004004001000Source: Forbes.comSource: dosomething.org 2008Recall the Fibonacci sequence we introduced earlier in this chapter.Any number in the Fibonacci sequence is a Fibonacci number.Fibonacci sequence = 1, 1, 2, 3, 5, 8, . . .73. The Fibonacci numbers are not a subset of which ofthe following sets: real numbers, rational numbers,irrational numbers, whole numbers?Irrational numbers74. Name three Fibonacci numbers that are primenumbers.2, 3, 575. Name three Fibonacci numbers that are compositenumbers.8, 21, 3476. Is the sequence of odd numbers a subset of theFibonacci numbers?No

Addition and Subtraction with Fractions 1.9You may recall from previous math classes that to add two fractions with thesame denominator, you simply add their numerators and put the result over thecommon denominator:3_4 + 2 _4 = _ 3 + 24= 5 _4The reason we add numerators but do not add denominators is that we must_follow the distributive property. To see this, you first have to recall that 3 can be4_written as 3 ⋅ 1 4 , and _ 2 4 can be written as 2 ⋅ _ 1 (dividing by 4 is equivalent to multiplyingby 1 ). Here is the addition problem again, this time showing the use of4_4the distributive property:ObjectivesA Add or subtract two or morefractions with the samedenominator.B Find the least common denominatorfor a set of fractions.C Add or subtract fractions withdifferent denominators.D Extend a sequence of numberscontaining fractions.3_4 + _ 2 4 = 3 ⋅ _ 1 4 + 2 ⋅ _ 1 4= (3 + 2) ⋅ 1 _4Distributive propertyExamples now playing atMathTV.com/booksA= 5 ⋅ 1 _4= 5 _4Adding and Subtracting with the Same DenominatorWhat we have here is the sum of the numerators placed over the commondenominator. In symbols we have the following.Addition and Subtraction of fractionsIf a, b, and c are integers and c is not equal to 0, thena_c + b _c = a + b _cThis rule holds for subtraction as well; that is,a_c − b _c = a − b _cNoteMost people whohave done any workwith adding fractionsknow that you add fractions thathave the same denominator byadding their numerators but nottheir denominators. However,most people don’t know why thisworks. The reason why we addnumerators but not denominatorsis because of the distributiveproperty. That is what the discussionat the left is all about. If youreally want to understand additionof fractions, pay close attention tothis discussion.In Examples 1–4, find the sum or difference. (Add or subtract as indicated.)Reduce all answers to lowest terms. (Assume all variables represent nonzeronumbers.)ExAmplE 1Add 3 _8 + 1 _8 .SOlutiON3_8 + _ 1 8 = _ 3 + 18= 4 _8Add numerators; keep the same denominatorThe sum of 3 and 1 is 4pRACtiCE pROBlEmSFind the sum or difference. Reduceall answers to lowest terms.(Assume all variables representnonzero numbers.)1.3_10 + 1 _10= 1 _2Reduce to lowest termsExAmplE 2 _Subtract: a + 5 _− 3 8 8 .SOlutiON_ a + 5 _− 3 8 8 = _ a + 5 − 38= a + 2 _8Combine numerators; keep the same denominator1.9 Addition and Subtraction with Fractions2.Answers1. 2 _5_ a − 5 _+ 3 12 122. a − 2 _1289

90Chapter 1 The Basics3.8_x − _ 5 xExAmplE 3Subtract: 9 _x − 3 _x .SOlutiON9_x − _ 3 x = _ 9 − 3xSubtract numerators; keep the same denominator= 6 _xThe difference of 9 and 3 is 64.5_9 − _ 8 9 + _ 5 9ExAmplE 4Simplify: 3 _7 + 2 _7 − 9 _7 .SOlutiON3_7 + _ 2 7 − _ 9 7 = _ 3 + 2 − 97= −4 _7= − 4 _7unlike signs give a negative answerBFinding the Least Common DenominatorWe will now turn our attention to the process of adding fractions that have differentdenominators. To get started, we need the following definition.DefinitionThe least common denominator (LCD) for a set of denominators is thesmallest number that is exactly divisible by each denominator, also calledthe least common multiple.In other words, all the denominators of the fractions in a problem mustdivide into the least common denominator without a remainder.5. Find the LCD for the fractions5 18 and _ 3 _14 .NoteThe ability to fi nd leastcommon denominatorsis very importantin mathematics. The discussionhere is a detailed explanation ofhow to do it.ExAmplE 5Find the LCD for the fractions 5 _12 and 7 _18 .SOlutiON The least common denominator for the denominators 12 and 18must be the smallest number divisible by both 12 and 18. We can factor 12 and 18completely and then build the LCD from these factors. Factoring 12 and 18 completelygives us12 = 2 ⋅ 2 ⋅ 3 18 = 2 ⋅ 3 ⋅ 3Now, if 12 is going to divide the LCD exactly, then the LCD must have factors of2 ⋅ 2 ⋅ 3. If 18 is to divide it exactly, it must have factors of 2 ⋅ 3 ⋅ 3. We don’t needto repeat the factors that 12 and 18 have in common:12 divides the LCD12 = 2 ⋅ 2 ⋅ 318 = 2 ⋅ 3 ⋅ 3 }LCD = 2 ⋅ 2 ⋅ 3 ⋅ 3 = 3618 divides the LCDIn other words, first we write down the factors of 12, then we attach the factors of18 that do not already appear as factors of 12.The LCD for 12 and 18 is 36. It is the smallest number that is divisible by both12 and 18; 12 divides it exactly three times, and 18 divides it exactly two times.Answers3. 3 _x4. 2 _95. 126

1.9 Addition and Subtraction with Fractions91CAdding and Subtracting with Different DenominatorsWe can use the results of Example 5 to find the sum of the fractions ​ 5 _12 ​and ​ 7 _18 ​.Example 6Add ​ 5 _12 ​+ ​ 7 _18 ​.SolutionWe can add fractions only when they have the same denominators.In Example 5 we found the LCD for ​_512 ​and ​ _ 718 ​to be 36. We change ​ _ 512 ​and ​ _ 718 ​ t oequivalent fractions that each have 36 for a denominator by applying Property 1,on page 18 for fractions:_​ 5 12 ​= ​ _ 5 ⋅ 312 ⋅ 3 ​= ​15 _5_​ is equivalent to ​_ 153612 36 ​The fraction ​ 15 __​ 7 18 ​= ​ _ 7 ⋅ 218 ⋅ 2 ​= ​14 _3636 ​is equivalent to ​ _ 5the numerator and denominator by 3. Likewise, ​ 146. Add ​_57_14​ is equivalent to ​_18 36 ​​, because it was obtained by multiplying both12_36 ​is equivalent to ​ _ 7 ​because it18​. Let’s write the complete solution again3612 ⋅ 3 ​+ ​ _ 7 ⋅ 2 ​ Rewriteeach fraction as an equivalent18 ⋅ 2 fraction with denominator 36_​+ ​14_36 36 ​ Add numerators; keep thecommon denominator_​ Rewrite each fraction as an equivalent fractionAnswers6 ⋅ 2 with denominator 126. ​_3112 ​ Add numerators; keep the same denominator 63was obtained by multiplying the numerator and denominator by 2. All we haveleft to do is to add numerators:_​ 1536 ​+ ​14 _​= ​29_36 36 ​The sum of ​_512 ​and ​ _ 718 ​is the fraction ​29 _step-by-step.​_512 ​+ ​ _ 718 ​= ​ _ 5 ⋅ 3= ​ 15= ​ 29 _36 ​ Example 7Find the LCD for ​_3 4 ​and ​1 _6 ​.Solution We factor 4 and 6 into products of prime factors and build the LCDfrom these factors:4 = 2 ⋅ 2} 6 = 2 ⋅ 3LCD = 2 ⋅ 2 ⋅ 3 = 12The LCD is 12. Both denominators divide it exactly; 4 divides 12 exactly threetimes, and 6 divides 12 exactly two times.Example 8Add ​ 3 _4 ​+ ​1 _6 ​.SolutionIn Example 7 we found that the LCD for these two fractions is 12. Webegin by changing ​_ 3 4 ​and ​1 _​to equivalent fractions with denominator 12:6​_3 ​= ​3_ ⋅ 34 4 ⋅ 3 ​= ​ _ 912 ​ ​3 _4 ​is equivalent to ​ _ 912 ​​ 1 _6​= ​1⋅ 2 _6 ⋅ 2 ​= ​ 2 _12 ​ ​1 _6 ​is equivalent to ​ 2 _12 ​To complete the problem, we add numerators:_​ 9 12 ​+ ​ _ 212​= ​11_12 ​The sum of ​_ 3 4 ​and ​1 _6 ​ i s ​11 _12​_3 4 ​+ ​1 _6 ​= ​3 _ ⋅ 34 ⋅ 3= ​ 9 _12 ​+ ​ 2 _​. Here is how the complete problem looks:​+ ​1 ⋅ 212 ​ = ​11 _18 ​+ ​ 3 _14 ​.7. Find the LCD for ​ 2 _9 ​and ​ 4 _15 ​.8. Add ​ 2 _9 ​+ ​ 4 _15 ​ .​ 7. 45 8. ​22_45 ​

92Chapter 1 The Basics9. Subtract ​ 8 _25 ​− ​ 3 _20 ​.Example 9Subtract ​_715 ​− ​ _ 310 ​.Solution Let’s factor 15 and 10 completely and use these factors to build theLCD:15 divides the LCD15 = 3 ⋅ 510 = 2 ⋅ 5 }LCD = 2 ⋅ 3 ⋅ 5 = 3010 divides the LCDChanging to equivalent fractions and subtracting, we have​_715 ​− ​ _ 310 ​= ​ _ 7 ⋅ 215 ⋅ 2 ​− ​ _ 3 ⋅ 3 ​ Rewriteas equivalent fractions with the LCD for10 ⋅ 3denominator= ​_1430 ​− ​ _ 930 ​= ​_5 ​ Subtract numerators; keep the LCD30= ​_1 ​ Reduce to lowest terms6As a summary of what we have done so far and as a guide to working otherproblems, we will now list the steps involved in adding and subtracting fractionswith different denominators.Strategy To Add or Subtract Any Two Fractionsthe denominators in the problem.)number.the sum or difference is the LCD.already in lowest terms.Example 10Add ​_1 6 ​+ ​1 _8 ​+ ​1 _4 ​.Solutionas usual:100 ​10. Add ​ 1 _9 ​+ ​1 _4 ​+ ​1 _6 ​.Answers9. ​ 17 _Step 1: Factor each denominator completely and use the factors to build theLCD. (Remember, the LCD is the smallest number divisible by each ofStep 2: Rewrite each fraction as an equivalent fraction that has the LCD forits denominator. This is done by multiplying both the numerator anddenominator of the fraction in question by the appropriate wholeStep 3: Add or subtract the numerators of the fractions produced in step 2.This is the numerator of the sum or difference. The denominator ofStep 4: Reduce the fraction produced in step 3 to lowest terms if it is notThe idea behind adding or subtracting fractions is really very simple. We canadd or subtract only fractions that have the same denominators. If the fractionswe are trying to add or subtract do not have the same denominators, we rewriteeach of them as an equivalent fraction with the LCD for a denominator.Here are some further examples of sums and differences of fractions.We begin by factoring the denominators completely and building theLCD from the factors that result. We then change to equivalent fractions and add

1.9 Addition and Subtraction with Fractions936 = 2 ⋅ 38 = 2 ⋅ 2 ⋅ 2}4 = 2 ⋅ 28 divides the LCDLCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 = 244 divides the LCD 6 divides the LCD1_6 ​+ ​1 _8 ​+ ​1 _4 ​= ​1 _ ⋅ 46 ⋅ 4 ​+ ​1 _ ⋅ 3 ​+ ​1_ ⋅ 68 ⋅ 3 4 ⋅ 6 ​= ​ 4 _24 ​+ ​ 3 _24 ​+ ​ 6 _24 ​= ​ 13 _24 ​Example 11Subtract 3 − ​_5 6 ​.Solution The denominators are 1 ​ because 3 = ​_ 3 1 ​ ​and 6. The smallest numberdivisible by both 1 and 6 is 6.11. Subtract 2 − ​ 3 _4 ​.3 − ​ 5 _6 ​= ​3 _1 ​− ​5 _6 ​= ​ 3 ⋅ 6 _1 ⋅ 6 ​− ​5 _6 ​= ​ 18 _6 ​− ​5 _6 ​= ​ 13 _6 ​DSequences Containing FractionsExample 12SolutionFind the next number in each sequence.a. ​ 1 _2 ​, 0, − ​1 _2 ​, . . . b. ​1 _2 ​, 1, ​3 _2 ​, . . . c. ​1 _2 ​, ​1 _4 ​, ​1 _8 ​, . . .a. ​_ 1 2 ​, 0, − ​ _ 1 2 ​, . . . : Adding − ​ _ 1 ​to each term produces the next term.2_The fourth term will be − ​ 1 2 ​+ ​ _ − ​ 1 2 ​ ​= −1. This is an arithmeticsequence.b. ​_ 1 2 ​, 1, ​3 _​, . . . : Each term comes from the term before it by adding​2_ 1 2 ​. The fourth term will be ​3 _2 ​+ ​1 _​= 2. This sequence is also2an arithmetic sequence.12. Find the next number in eachsequencea. ​ 1 _3 ​, 0, − ​ 1 _3 ​, . . .b. ​_1 3 ​, ​2 _​, 1, . . .3c. 1, ​ 1 _3 ​, ​1 _9 ​, . . .c. ​_ 1 2 ​, ​1 _4 ​, ​1 _​, . . . : This is a geometric sequence in which each term8comes from the term before it by multiplying by ​_ 1 ​each time.2The next term will be ​_ 1 8 ​⋅ ​1 _2 ​= ​ _ 116 ​.Example 13SolutionSubtract: ​_x 5 ​− ​1 _6 ​The LCD for 5 and 6 is their product, 30. We begin by rewriting eachfraction with this common denominator:x_5 ​− ​1 _6 ​= ​x _ ⋅ 6 ​− ​1_ ⋅ 55 ⋅ 6 6 ⋅ 5 ​= ​ 6x _30 ​− ​ 5 _30 ​= ​_6x − 5 ​3013. Subtract: ​ x _4 ​− ​1 _5 ​Answers10. ​_1936 ​ 11. ​5 _4 ​12. a. − ​ 2 _3 ​ b. ​4 _3 ​ c. ​ 1 _27 ​13. ​_5x − 4 ​20

94Chapter 1 The Basics_14. Add: 5 x + _ 2 3NoteIn Example 14, it isunderstood that xcannot be 0. Do youknow why?ExAmplE 14SOlutiON_Add: 4 x + _ 2 3The LCD for x and 3 is 3x. We multiply the numerator and thedenominator of the first fraction by 3 and the numerator and the denominator ofthe second fraction by x to get two fractions with the same denominator. We thenadd the numerators:4_x + _ 2 3 = _ 4 ⋅ 3x ⋅ 3 + _ 2 ⋅ x Change to equivalent fractions3 ⋅ x_= 123x + _ 2x3x=12 + 2x _3xAdd the numeratorsWhen we are working with fractions, we can change the form of a fractionwithout changing its value. There will be times when one form is easier to workwith than another form. Look over the material below and be sure you see thatthe pairs of expressions are equal._The expressions x 2 and _ 1 x are equal.2_The expressions 3a4 and _ 3 a are equal.4_The expressions 7y3 and _ 7 y are equal.315. Simplify: 1 _2 x + 3 _ExAmplE 15SOlutiON_Add: x 3 + _ 5x6 .We can do the problem two ways. One way probably seems easier,but both ways are valid methods of finding this sum. You should understand bothof them.x_mEtHOD 1:3 + _ 5x6 = _ 2 ⋅ x2 ⋅ 3 + _ 5xLCD6mEtHOD 2:_ 2x + 5x= Add numerators6_= 7x6x_3 + _ 5x6 = _ 13 x + _ 5 6 x4 x Getting Ready for Class= _ 1 3 + _ 5 6 x Distributive property= _ 2 ⋅ 12 ⋅ 3 + _ 5 6 x= _ 2 6 + _ 5 6 x=_ 7 6 xAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answer_ 15 + 2x14.3x15. 5 _4 x1. How do we add two fractions that have the same denominator?2. What is a least common denominator?3. What is the first step in adding two fractions that have differentdenominators?4. What is the last thing you do when adding two fractions?

1.9 Problem Set95Problem Set 1.9A Find the following sums and differences, and reduce to lowest terms. Add and subtract as indicated. Assume allvariables represent nonzero numbers. [Examples 1–4]1. ​ 3 _6 ​+ ​1 _6 ​2. ​_2​_ 2 3 ​ 15 ​+ ​3 _5 ​3. ​ 3 _8 ​− ​5 _8 ​4. ​_1_−​ 1 4 ​7 ​− ​6 _7 ​−​ 5 _7 ​_5. −​ 1 4 ​+ ​3 _4 ​_6. −​ 4 9 ​+ ​7 _9 ​7. ​_x 3 ​− ​1 _3 ​​_ 1 2 ​ ​_ 1 3 ​ ​_ x − 1 ​38. ​ x _8 ​− ​1 _8 ​​_ x − 1 ​89. ​_1 4 ​+ ​2 _4 ​+ ​3 _4 ​10. ​_2 5 ​+ ​3 _5 ​+ ​4 _5 ​11. ​_x + 72 ​− ​1 _2 ​​_ 3 2 ​ ​_ 9 5 ​ ​_ x + 6 ​212. ​ x + 5 _4 ​− ​3 _4 ​​_ x + 2 ​413. ​ 1 _10 ​− ​ 3 _10 ​− ​ 4 _10 ​_−​ 3 5 ​ _−​ 1 10 ​14. ​_320 ​− _ 120 ​− ​ _ 420 ​15. ​_1a ​+ ​4 _a ​+ ​5 _a ​​_ 10a ​16. ​ 5 _a ​+ ​4 _a ​+ ​3 _a ​​ 12 _a ​Selected exercises available online at www.webassign.net/brookscole

96Chapter 1 The BasicsB C Find the LCD for each of the following; then use the methods developed in this section to add and subtract asindicated. [Examples 5–11]17. ​_1 8 ​+ ​3 _4 ​18. ​_1 6 ​+ ​2 _3 ​19. ​_310 ​− ​1 _5 ​20. ​_5 6 ​− ​ _ 112 ​​_ 7 8 ​ ​_ 5 6 ​ ​_ 110 ​ ​_ 3 4 ​21. ​_4 9 ​+ ​1 _3 ​22. ​_1 2 ​+ ​1 _4 ​23. 2 + ​_1 3 ​24. 3 + ​_1 2 ​​_ 7 9 ​ ​_ 3 4 ​ ​_ 7 3 ​ ​_ 7 2 ​_25. − ​ 3 4 ​+ 1_26. −​ 3 4 ​+ 227. ​_1 2 ​+ ​2 _3 ​28. ​_2​_ 1 4 ​ ​_ 5 4 ​ ​_ 7 6 ​3 ​+ ​1 _4 ​​_ 1112 ​29. ​ 5 _12 ​− ​ −​ 3 _8 ​ ​30. ​_916 ​− ​ _ −​ 7 12 ​ _ ​31. −​ 1 20 ​+ _ 830 ​_32. −​ 124 ​ ​_ 5548 ​ ​_ 1360 ​ ​_ 23120 ​​_1930 ​+ ​ 9 _40 ​33. ​_17 ​+ ​11_30 42 ​34. ​_19 ​+ ​13_42 70 ​35. ​_25 ​+ ​41_84 90 ​36. ​_23 ​+ ​29_70 84 ​​_2935 ​ ​_ 67105 ​ ​_ 9491,260 ​ ​_ 283420 ​

1.9 Problem Set9737. ​_13 ​− ​_13126 180 ​38. ​_17 ​− ​17_84 90 ​39. ​_3 4 ​+ ​1 _8 ​+ ​5 _6 ​40. ​_3​_13420 ​ ​_ 171,260 ​ ​_ 4124 ​8 ​+ ​2 _5 ​+ ​1 _4 ​​_ 4140 ​41. ​ 1 _2 ​+ ​1 _3 ​+ ​1 _4 ​+ ​1 _6 ​42. ​_1​_ 5 4 ​8 ​+ ​1 _4 ​+ ​1 _5 ​+ ​ 1 _10 ​​_ 2740 ​43. 1 − ​_5 2 ​44. 1 − ​_5 3 ​_−​ 3 2 ​ _−​ 2 3 ​45. 1 + ​_1 2 ​46. 1 + ​_2 3 ​​_ 3 2 ​ ​_ 5 3 ​D Find the fourth term in each sequence. [Example 12]47. ​ 1 _3 ​, 0, −​ 1 _3 ​, . . ._−​ 2 3 ​ _−​ 4 3 ​ ​_ 7 3 ​48. ​_2 3 ​, 0, −​2 _3 ​, . . .49. ​_13 ​, 1, ​5 _3 ​, . . .50. 1, ​_3 ​, 2, . . .251. 1, ​_1 5 ​, ​ _ 125 ​, . . ._52. 1, −​ 1 2 ​, ​1 _4 ​, . . .​_ 5 2 ​ ​_ 1125 ​ _−​ 1 8 ​

98Chapter 1 The BasicsUse the rule for order of operations to simplify each expression.53. 9 − 3 ​ ​ 5 _3 ​ ​54. 6 − 4 ​ ​ 7 _2 ​ ​55. −​ 1 _2 ​ + 2 ​ −​ 3 _4 ​ ​56. ​ 5 _4 ​ − 3 ​ ​ 7 _12 ​ ​4−8−2−​ 1 _2 ​57. ​_3 5 ​( −10) + ​4 _​( −21)758. −​ 3 _5 ​(10) − ​4 _7 ​(21)59. 16 ​ _−​ 1 22 ​ ​ ​ − 125 _​ −​ 2 25 ​ ​ ​60. 16 ​ _−​ 1 32 ​ ​ ​ − 125 _​ −​ 2 35 ​ ​ ​−18−18−16661. − ​ 4 _3 ​ ÷ 2 ⋅ 3−2_62. −​ 8 7 ​ ÷ 4 ⋅ 2_63. −​ 4 3 ​ ÷ 2(−3)_−​ 4 7 ​ 264. −​ 8 _7 ​ ÷ 4(−2)​ 4 _7 ​65. −6 ÷ ​ 1 _2 ​ ⋅ 1266. −6 ÷ ​ −​ 1 _2 ​ ​ ⋅ 1267. −15 ÷ ​ 5 _3 ​ ⋅ 1868. −15 ÷ ​ −​ 5 _3 ​ ​ ⋅ 18−144144−162162Add or subtract the following fractions. (Assume all variables represent nonzero numbers. [Examples 13–15]69. ​ x _4 ​+ ​1 _5 ​​_5x + 4 ​2070. ​ x _3 ​+ ​1 _5 ​​_ 5x + 3 ​1571. ​ 1 _3 ​+ ​ a _12 ​​_ a + 412 ​72. ​ 1 _8 ​+ ​ a _32 ​​_ a + 432 ​ 73. ​ x _2 ​+ ​1 _3 ​+ ​x _4 ​​_ 9x + 4 ​1274. ​ x _3 ​+ ​1 _4 ​+ ​x _5 ​32x + 15​_​6075. ​ 2 _x ​+ ​3 _5 ​10 + 3x​_​5x76. ​ 3 _x ​− ​2 _5 ​15 − 2x​_​5x77. ​ 3 _7 ​+ ​4 _y ​3y + 28​_​7y78. ​ 2 _9 ​+ ​5 _y ​2y + 45​_​9y79. ​ 3 _a ​+ ​3 _4 ​+ ​1 _5 ​60 + 19a​_​20a80. ​ 4 _a ​+ ​2 _3 ​+ ​1 _2 ​24 + 7a​_​6a

1.9 Problem Set9981. ​_1 2 ​x + ​1 _6 ​x82. ​_2 3 ​x + ​5 _6 ​x83. ​_1 2 ​x − ​3 _4 ​x84. ​_2​_ 2 3 ​x ​_ 3 2 ​x _−​ 1 4 ​x3 ​x − ​5 _6 ​x−​ 1 _6 ​x85. ​ 1 _3 ​x + ​3 _5 ​x86. ​_2 3 ​x − ​3 _5 ​x87. ​_3x4 ​+ ​x _6 ​88. ​_3x415 ​x ​_ 115 ​x ​_ 1112 ​x ​_ 112 ​x​_14​− ​2x_3 ​89. ​_2x ​+ ​5x_5 8 ​​_4140 ​x 90. ​ 3x_5​_ 940 ​x​− ​3x_8 ​Use the rule for order of operations to simplify each expression.91. 1 − ​_1 x ​ ​x _ − 1 ​ 92. 1 + ​_1xx ​ ​x _ + 1 ​xThe next two problems are intended to give you practice reading, and paying attention to, the instructions that accompanythe problems you are working. As we mentioned previously, working these problems is an excellent way to get readyfor a test or a quiz.93. Work each problem according to the instructions given. 94. Work each problem according to the instructions given.(Note that each of these instructions could be replaced_a. Add: −​ 5 with the instruction Simplify.)8 ​+ ​ _ −​ 1 2 ​ _ ​ −​ 9 8 ​a. Add: ​_3 4 ​+ ​ _−​ 1 2 ​ ​ ​_ 1 4 ​_b. Subtract: −​ 5 8 ​− ​ _ −​ 1 2 ​ _ ​ −​ 1 8 ​b. Subtract: ​_3 4 ​− ​ _−​ 1 2 ​ ​ ​_ 5 4 ​_c. Multiply: −​ 5 8 ​ ​ _ −​ 1 2 ​ 5_ ​16 ​c. Multiply: ​_3 4 ​ ​ _−​ 1 2 ​ _ ​ −​ 3 8 ​_d. Divide: −​ 5 8 ​÷ ​ _ −​ 1 2 ​ ​ ​_ 5 4 ​2 ​d. Divide: ​ 3 _4 ​÷ ​ −​ 1 _2 ​ ​ −​ 3 _

100Chapter 1 The BasicsSimplify.95. ​ 1 − ​ 1 _2 ​ ​ 1 − ​ 1 _3 ​ ​96. ​ 1 + ​_1​_ 1 3 ​ 22 ​ ​ 1 + ​ 1 _3 ​ ​97. ​ 1 + ​ 1 _2 ​ ​ 1 − ​ 1 _2 ​ ​98. ​ 1 + ​_1​_ 3 4 ​ ​_ 8 9 ​3 ​ ​ 1 − ​ 1 _3 ​ ​99. Find the value of 1 + ​_1 ​when x isx​ c. 4 ​_ 5 4c. 4 ​_ 3 4 ​100. Find the value of 1 − ​_1a. 2 ​_ 3 2 ​a. 2 ​_ 1 2 ​b. 3 ​_ 4 3 ​b. 3 ​_ 2 3 ​​when x isx101. Find the value of 2x + ​_6 ​when x isxa. 1 8b. 2 7c. 3 8102. Find the value of x + ​_4 ​when x isxa. 1 5b. 2 4c. 3 ​ 13 _3 ​Applying the ConceptsRainfall The chart shows the average monthly rainfall for Death Valley. Use the chart to answer Problems 103 and 104.103. What is the total rainfall for the months of December,October, and February?Death Valley Rainfall​_4760 ​ in.1211 340104. How much more rain was there in August than in June?​_120 ​in.inches325 1201101414Feb Apr Jun Aug Oct Dec

Chapter 1 SummaryThe number(s) in brackets next to each heading indicates the section(s) in whichthat topic is discussed.Symbols [1.1]a = ba ≠ ba < ba ≮ ba > ba ≯ ba ≥ ba ≤ ba is equal to ba is not equal to ba is less than ba is not less than ba is greater than ba is not greater than ba is greater than or equal to ba is less than or equal to bNoteWe will use the marginsin the chaptersummaries to giveexamples that correspond to thetopic being reviewed whenever itis appropriate.Exponents [1.1]Exponents are notation used to indicate repeated multiplication. In the expression3 4 , 3 is the base and 4 is the exponent.3 4 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 81ExAmplES1. 2 5 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 325 2 = 5 ⋅ 5 = 2510 3 = 10 ⋅ 10 ⋅ 10 = 1,0001 4 = 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1Order of Operations [1.1]When evaluating a mathematical expression, we will perform the operations inthe following order, beginning with the expression in the innermost parenthesesor brackets and working our way out.1. Simplify all numbers with exponents, working from left to right if more thanone of these numbers is present.2. 10 + (2 ⋅ 3 2 − 4 ⋅ 2)= 10 + (2 ⋅ 9 − 4 ⋅ 2)= 10 + (18 − 8)= 10 + 10= 202. Then do all multiplications and divisions left to right.3. Finally, perform all additions and subtractions left to right.Absolute Value [1.2]The absolute value of a real number is its distance from zero on the real numberline. Absolute value is never negative.3. | 5 | = 5| −5 | = 5Opposites [1.2]Any two real numbers the same distance from zero on the number line but inopposite directions from zero are called opposites. Opposites always add to zero.4. The numbers 3 and −3 areopposites; their sum is 0:3 + (−3) = 0Chapter 1Summary101

102Chapter 1 The BasicsReciprocals [1.2]5. The numbers 2 and ​_ 1 ​are reciprocals;their product is21:2 ​ ​ 1 _2 ​ ​= 1Any two real numbers whose product is 1 are called reciprocals. Every real numberhas a reciprocal except 0.Addition of Real Numbers [1.3]6. Add all combinations of positiveand negative 10 and 13.10 + 13 = 2310 + (−13) = −3−10 + 13 = 3−10 + (−13) = −23To add two real numbers with1. The same sign: Simply add their absolute values and use the common sign.2. Different signs: Subtract the smaller absolute value from the larger absolutevalue. The answer has the same sign as the number with the larger absolutevalue.Subtraction of Real Numbers [1.4]7. Subtracting 2 is the same asadding −2:7 − 2 = 7 + (−2) = 5To subtract one number from another, simply add the opposite of the number youare subtracting; that is, if a and b represent real numbers, thena − b = a + (−b)Properties of Real Numbers [1.5]For AdditionFor MultiplicationCommutative: a + b = b + a a ⋅ b = b ⋅ aAssociative: a + (b + c) = (a + b) + c a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ cIdentity: a + 0 = a a ⋅ 1 = aInverse: a + (−a) = 0 a ​ ​ 1 _a ​ ​= 1Distributive:a(b + c) = ab + ac8. 3(5) = 153(−5) = −15−3(5) = −15−3(−5) = 15Multiplication of Real Numbers [1.6]To multiply two real numbers, simply multiply their absolute values. Like signsgive a positive answer. Unlike signs give a negative answer.Division of Real Numbers [1.7]9. − ​ 6 _2 ​= −6 ​ ​ 1 _2 ​ ​= −3​ −6 _−2 ​= −6 ​ − ​ 1 _2 ​ ​= 3Division by a number is the same as multiplication by its reciprocal. Like signsgive a positive answer. Unlike signs give a negative answer.

Chapter 1 Summary103Subsets of the Real Numbers [1.8]Counting numbers: {1, 2, 3, . . . }Whole numbers: {0, 1, 2, 3, . . . }Integers: { . . . , −3, −2, −1, 0, 1, 2, 3, . . . }Rational numbers: {all numbers that can be expressed as the ratio of twointegers}Irrational numbers: {all numbers on the number line that cannot beexpressed as the ratio of two integers}Real numbers: {all numbers that are either rational or irrational}10. a. 7 and 100 are countingnumbers, but 0 and −2 arenot.b. 0 and 241 are whole numbers,but −4 and ​_ 1 ​are not.2c. −15, 0, and 20 are integers._d. −4, − ​ 1 ​, 0.75, and20.666 . . . are rationalnumbers.e. −π, ​√ — 3 ​, and π are irrationalnumbers.f. All the numbers listedabove are real numbers.Factoring [1.8]Factoring is the reverse of multiplication.MultiplicationFactors 0 3 ⋅ 5 = 15 9 ProductFactoring11. The number 150 can be factoredinto the product of primenumbers:150 = 15 ⋅ 10= (3 ⋅ 5)(2 ⋅ 5)= 2 ⋅ 3 ⋅ 5 2Least Common Denominator (LCD) [1.9]The least common denominator (LCD) for a set of denominators is the smallestnumber that is exactly divisible by each denominator.12. The LCD for ​_512 ​and ​ _ 7 ​is 36.18Addition and Subtraction of Fractions [1.9]To add (or subtract) two fractions with a common denominator, add (or subtract)numerators and use the common denominator._​ a c ​+ ​b _c ​= ​a _ + b ​ and ​_acc ​− ​b _c ​= ​a _ − b ​c13. ​ 5 _12 ​+ ​ 7 _18 ​= ​ 5 _12 ​⋅ ​3 _3 ​+ ​ 7 _18 ​⋅ ​2 _2 ​= ​_15 ​+ ​14_36 36 ​= ​ 29 _36 ​Common Mistakes1. Interpreting absolute value as changing the sign of the number inside theabsolute value symbols. ​| −5 |​ = +5, ​| +5 |​ = −5. (The first expression is correct;the second one is not.) To avoid this mistake, remember: Absolutevalue is a distance and distance is always measured in positive units.2. Using the phrase “two negatives make a positive.” This works only withmultiplication and division. With addition, two negative numbers producea negative answer. It is best not to use the phrase “two negatives make apositive” at all.

Chapter 1 ReviewWrite the numerical expression that is equivalent to each phrase, and then simplify. [1.3, 1.4, 1.6, 1.7]1. The sum of −7 and −10 −7 + (−10); −17 2. Five added to the sum of −7 and 4 (−7 + 4) + 5 = 23. The sum of −3 and 12 increased by 5 (−3 + 12) + 5; 14 4. The difference of 4 and 9 4 − 9 = −55. The difference of 9 and −3 9 − (−3) = 12 6. The difference of −7 and −9 −7 − (−9) = 27. The product of −3 and −7 decreased by 6(−3)(−7) − 6 = 158. Ten added to the product of 5 and −6 5(−6) + 10 = −209. Twice the product of −8 and 3x 2[(−8)(3x)] = −48x 10. The quotient of −25 and −5 ​ −25 _−5 ​ = 5Simplify. [1.2]11. ​| −1.8 |​ 1.8 12. − ​| −10 |​ −10For each number, give the opposite and the reciprocal. [1.2]13. 6 −6, ​_ 1 _​ 14. −​126 5 ​ ​12 _5 ​, −​5 _12 ​Multiply. [1.2, 1.6]15. ​ 1 _2 ​( −10) −5 16. ​ − ​ 4 _Add. [1.3]5 ​ ​​ _​ 2516 ​ _ ​ −​ 5 4 ​17. −9 + 12 3 18. −18 + (−20) −3819. −2 + (−8) + [−9 + (−6)] −25 20. (−21) + 40 + (−23) + 5 1Subtract. [1.4]21. 6 − 9 −3 22. 14 − (−8) 22 23. −12 − (−8) −4 24. 4 − 9 − 15 −20Find the products. [1.6]25. (−5)(6) −30 26. 4(−3) −12 27. −2(3)(4) −24 28. (−1)(−3)(−1)(−4) 12Find the following quotients. [1.7]29. ​ 12 _−3 ​ −4 30. − ​8 _9 ​÷ ​4 _3 ​ −​2 _3 ​Chapter 1Review105

106Chapter 1 The BasicsSimplify. [1.1, 1.6, 1.7]31. 4 ⋅ 5 + 3 23 32. 9 ⋅ 3 + 4 ⋅ 5 4733. 2 3 − 4 ⋅ 3 2 + 5 2 −3 34. 12 − 3(2 ⋅ 5 + 7) + 4 −3535. 20 + 8 ÷ 4 + 2 ⋅ 5 32 36. 2(3 − 5) − (2 − 8) 237. 30 ÷ 3 ⋅ 2 20 38. (−2)(3) − (4)(−3) − 9 −339. 3(4 − 7) 2 − 5(3 − 8) 2 −98 40. (−5 − 2)(−3 − 7) 7041. ​_4(−3)−6 ​ 2 42. ​ _ 32 + 5 2​17_(3 − 5) 2​ 2 ​15 − 1043. ​_​ Undefined 44. ​2(−7)__+ (−11)(−4)​ 36 − 6 7 − (−3)State the property or properties that justify the following. [1.5]45. 9(3y) = (9 ⋅ 3)y Associative 46. 8(1) = 8 Multiplicative identity47. (4 + y) + 2 = (y + 4) + 2 Commutative 48. 5 + (−5) = 0 Additive inverse49. (4 + 2) + y = (4 + y) + 2 Commutative, associative 50. 5(w − 6) = 5w − 30 DistributiveUse the associative property to rewrite each expression, and then simplify the result. [1.5]51. 7 + (5 + x) 12 + x 52. 4(7a) 28a 53. ​ 1 _9 ​( 9 x) x 54. ​4 _5 ​ ​ 5 _4 ​y ​ yApply the distributive property to each of the following expressions. Simplify when possible. [1.5, 1.6]55. 7(2x + 3) 14x + 21 56. 3(2a − 4) 6a − 12 57. ​ 1 _2 ​( 5 x − 6) ​5 _2 ​x − 3 58. −​1 _2 ​( 3 x − 6) −​3 _2 ​x + 3For the set {​√ — _7 ​, − ​ 1 3 ​, 0, 5, −4.5, ​2 _​, π, −3} list all the [1.8]559. rational numbers 60. whole numbers_−​ 1 3 ​, 0, 5, −4.5, ​2 _5 ​, −3 0, 561. irrational numbers​√ — 7 ​, π62. integers0, 5, −3Factor into the product of primes. [1.8]63. 90 2 ⋅ 3 2 ⋅ 5 64. 840 2 3 ⋅ 3 ⋅ 5 ⋅ 7Combine. [1.9]65. ​_18 ​+ ​13_35 42 ​ ​173 _210 ​ 66. ​x _6 ​+ ​ _ 712 ​ ​2x _ + 7 ​12Find the next number in each sequence. [1.1, 1.2, 1.3, 1.6, 1.9]67. 10, 7, 4, 1, . . . −2 68. 10, −30, 90, −270, . . . 81069. 1, 1, 2, 3, 5, . . . 8 70. 4, 6, 8, 10, . . . 1271. 1, ​ 1 _2 ​, 0, −​ 1 _2 ​, . . . −1 72. 1, − ​1 _2 ​, ​1 _4 ​, −​1 _8 ​, . . . ​ 1 _16 ​

Chapter 1 TestTranslate into symbols. [1.1]1. The sum of x and 3 is 8. x + 3 = 82. The product of 5 and y is 15. 5y = 15Simplify according to the rule for order of operations. [1.1]3. 5 2 + 3(9 − 7) + 3 2 40 4. 10 − 6 ÷ 3 + 2 3 16For each number, name the opposite, reciprocal, and absolutevalue. [1.2]5. −4 4, −​ 1_ 4 ​, 4 6. ​3 _4 ​ −​3_ 4 ​, ​4_ 3 ​, ​3_ 4 ​Multiply by applying the distributive property. [1.5, 1.6]_27. 2(3x + 5) 6x + 10 28. −​ 1 ​( 4 x − 2) −2x + 12From the set of numbers {−8, ​ 3_ 4 ​, 1, ​√— 2 ​, 1.5} list all the elementsthat are in the following sets. [1.8]29. Integers −8, 1 30. Rational numbers−8, ​ 3_ 4 ​, 1, 1.531. Irrational numbers​√ — 2 ​32. Real numbers−8, ​ 3_ 4 ​, 1, ​√— 2 ​, 1.5Factor into the product of primes. [1.8]33. 592 2 4 ⋅ 37 34. 1,340 2 2 ⋅ 5 ⋅ 67Add. [1.3]7. 3 + (−7) −4 8. ​| −9 + (−6) |​ + ​| −3 + 5 |​Subtract. [1.4]9. −4 − 8 −12 10. 9 − (7 − 2) − 4 0Match each expression below with the letter of the propertythat justifies it. [1.5]11. (x + y) + z = x + ( y + z) 12. 3(x + 5) = 3x + 15 ec13. 5(3x) = (5 ⋅ 3)x d 14. (x + 5) + 7 = 7 + (x + 5)aa. Commutative property of additionb. Commutative property of multiplicationc. Associative property of additiond. Associative property of multiplicatione. Distributive propertyMultiply. [1.6]15. −3(7) −21 16. −4(8)(−2) 6417. 8 ​ −​ 1 _4 ​ ​ −2 18. ​ −​ 2 _1733 ​ ​ ​−​ 8_ 27 ​Simplify using the rule for order of operations. [1.1, 1.6, 1.7]19. −3(−4) − 8 4 20. 5(−6) 2 − 3(−2) 3 20421. 7 − 3(2 − 8) 2522. 4 − 2[−3(−1 + 5) + 4(−3)] 52__4(−5) − 2(7) ___2(−3 − 1) + 4(−5 + 2)23. ​ ​ 2 24. ​ ​−10 − 7−3(2) − 42Apply the associative property, and then simplify. [1.5, 1.6]25. 3 + (5 + 2x) 8 + 2x 26. −2(−5x) 10xCombine. [1.9]_35. ​ 5 _​+ ​1115 42 ​ _ ​25 42 ​ 36. ​5 _x ​+ ​3 _x ​Write an expression in symbols that is equivalent to eachEnglish phrase, and then simplify it.​8 _x ​37. The sum of 8 and −3 [1.1, 1.3] 8 + (−3) = 538. The difference of −24 and 2 [1.1, 1.4] −24 − 2 = −2639. The product of −5 and −4 [1.1, 1.6] (−5)(−4) = 2040. The quotient of −24 and −2 [1.1, 1.7] ​ −24 _−2 ​ = 12Find the next number in each sequence. [1.1, 1.2, 1.3, 1.6,1.9]41. −8, −3, 2, 7, . . . 12 42. 8, −4, 2, −1, . . . ​ 1_ 2 ​Use the illustration below to answer the following questions.[1.1, 1.3, 1.4]How Do You Say?BengaliArabicRussianSpanishHindustaniEnglishMandarinChinese2152562754254965140 200 400 600 800 1000(speakers in millions)Source: www.infoplease.comç107543. How many people speak Spanish? 425 million44. What is the total number of people who speak MandarinChinese or Russian? 1.35 billion120045. How many more people speak English thanHindustani? 18 millioníñChapter 1 Test107

Chapter 1 ProjectsThe Basicsgroup PROJECTBinary NumbersStudents and Instructors: The end of each chapter in this book will contain a section like thisone containing two projects. The group project is intended to be done in class. The researchprojects are to be completed outside of class. They can be done in groups or individually. In myclasses, I use the research projects for extra credit. I require all research projects to be done on aword processor and to be free of spelling errors.Number of PeopleTime NeededEquipmentBackground2 or 310 minutesPaper and pencilOur decimal number system is a base-10 numbersystem. We have 10 digits—0, 1, 2, 3, 4, 5,6, 7, 8, and 9—which we use to write all thenumbers in our number system. The number10 is the first number that is written with acombination of digits. Although our numbersystem is very useful, there are other numbersystems that are more appropriate for somedisciplines. For example, computers and programmersuse both the binary number system,which is base 2, and the hexadecimal numbersystem, which is base 16. The binary numbersystem has only digits 0 and 1, which are usedto write all the other numbers. Every numberin our base 10 number system can be writtenin the base 2 number system as well.ProcedureTo become familiar with the binary numbersystem, we first learn to count in base 2. Imaginethat the odometer on your car had only 0'sand 1's. Here is what the odometer would looklike for the first 6 miles the car was driven.Continue the table at left to show the odometerreading for the first 32 miles the car is driven.At 32 miles, the odometer should read1 0 0 0 0 0Odometer Readingmileage0 0 0 0 0 000 0 0 0 0 110 0 0 0 1 020 0 0 0 1 130 0 0 1 0 040 0 0 1 0 150 0 0 1 1 06Chapter 1Projects109

RESEARCH PROJECTSophie GermainThe photograph at the right shows the streetsign in Paris named for the French mathematicianSophie Germain (1776–1831). Among hercontributions to mathematics is her work withprime numbers. In this chapter we had an introductorylook at some of the classifications fornumbers, including the prime numbers. Withinthe prime numbers themselves, there are stillfurther classifications. In fact, a Sophie Germainprime is a prime number P, for which both Pand 2P + 1 are primes. For example, the primenumber 2 is the first Sophie Germain primebecause both 2 and 2 ⋅ 2 + 1 = 5 are primenumbers. The next Germain prime is 3 becauseboth 3 and 2 ⋅ 3 + 1 = 7 are primes.Sophie Germain was born on April 1, 1776,in Paris, France. She taught herself mathematicsby reading the books in her father’s libraryat home. Today she is recognized most for herwork in number theory, which includes her workwith prime numbers. Research the life of SophieGermain. Write a short essay that includes informationon her work with prime numbers andhow her results contributed to solving Fermat’sLast Theorem almost 200 years later.Cheryl Slaughter110Chapter 1 The Basics