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Selected Prealgebra Topics

1-1. Natural Numbers and Number'systems1-2 signed Numbers1-w Numerical Expressions andI quations1-4 Order of Operations1-5 symbols of Grouping1-6 Double Meaning of + and -1-7Nhsolute Value of a Signed Num bert:ombining Signed Numbers1-9Nelational Operators1-10 qultiplying with Signed Numbers1-11 Dividing with Signed Numbers

PERFORWIA CE OBJECTIVESse order , )1' onvrations to solve numerical expressions.

Solve nowt i-",t expressions that include symbols of grouping. Form sigtiol riumbers.

,oltite value of a n umber.i't iii/v ,( .:It olator to combine, multiply, and divideignot tittint), I.s.

Construct compound inequalities using relationaloperators.Static electricity from the Van de Graaf generator spikes thehair of the three youngsters. (Courtesy of Pearson Learning)


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2N CHAPTER 1 Selected Prealgebra Topics

For some, this chapter serves as a review of prealgebra concepts and principles; for others, this chapter is the beginning of an understanding of the fundamentals of algebra.For all, this chapter is a preparation for the study of algebra.Whole numbers are used to introduce you to several concepts used in algebra, including signed numbers, symbols of grouping, and operators.Some abbreviations have been used to add clarity to the examples. The symbolswithin boxes are used to indicate calculator operations. The o ther symb ols are used to

indicate manual operations. The following have been used in this chapter and are usedthroughout the book:and A addand S subtractIx ( and M multiply

and D divideCHS change sign*

therefore*Note.. +/ is also used to change sign.The following symbols are listed here for your reference (a complete list may befound in Appendix A): is equal to # is not equal tois greater than ?_ is greater than or equal to< is less than is less than or equal toields I I take absolute valueIt is assumed that you have read the introduction and that you w ill be using a calculator to aid in your calculations. If you have no t read the introduction, go back and readit before continuing this chapter.

1-1 NATURAL NUMBERS AND NUMBER SYSTEMSOur first introduction to mathematics was counting. The n umbers used w ere one, two,three, and so forth, and are called the counting numbers or natural numbers. Sincezero is not used in counting, zero is not a natural number. This is why our current calendar started with year 1, not year 0.Number SystemsThe system of numbers used by most people in Eu rope, the Americas, Africa, Australia,and New Zealand is the decimal system. This system is a base-ten system; that is, tensymbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are used in this system. These ten symbols are calleddigits. Because the decimal system is a positional number system, only ten digits areneeded to express any num ber. This means that the value of a digit depends on the po-


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CHAPTER 1 Selected Prealgebra Topics

sition of the digit. In contrast, Rom an numerals use new sym bols for larger denominations. For example, the number three hundred thirty-three is written as CCCX XX III iRoman numerals. The same num ber is written as 333 in the decimal system.There are many other number systems in use today. Since you are study ng electronics, you will probably need to learn about the number systems in use by computers. Tablet-1 is a summ ary of some of these compu ter number systems. No tice that thnumber of symbols used in the number system is the same as the base of the system

base two (binary), 2 symbols; base eight (octal), 8 symbols; base ten (decimal), 1symbols; base sixteen (hexadecimal), 16 symbols. Notice also that each system startswith zero.

te u1Name hols ti mhols t d ;4anie o SvnahlBinary Tw o 01 Binary digits or bitsOctal Eight 01234567 Octal digitsDecimal Ten 0123456789 DigitsHexadecimal Sixteen 0 1 2 3 4 5 6 7 8 9 Hex digitsABCDEFWh ile the binary and octal systems use the fam iliar decimal digits, the hexadecimasystem needs six extra digits. Instead of inventing new symbo ls, the first six capital leters (A, B, C, D. E, and F) of the alphabet are used as the six extra digits. Thcomputer number systems of Table 1-1 are covered in greater detail in Chapter 31.

1-2 SIGNED NUMBERSTo solve the wide variety of problems found in electronics, more is needed than the naural numbers. Signed num bers and zero are needed. Numbers preceded by a + sign acalled positive numbers; those preceded by a sign are called negative numbers.plus sign (+) may be written before a number to show that it is a positive number, the plus sign may be left off. A negative sign ( ) is always written before a num ber tshow that it is a negative number. If a number has no sign before it, it has an implieplus sign and is a positive number.Look at the thermometer pictured in Figure 1-1(a). Notice that two kinds of numbers are shown. The positive numbers are measuring the distance above the zero. Thnegative numbers are measuring the distance below the zero. Thus, zero is a referencpoint from which positive numbers are greater and negative numbers are less. Zero ineither positive nor negative.In the following exercise many of the problems can be do ne mentally, but some require calculations. Use a calculator when you need it.


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FIGURE 1-1 Use of signed numbers: (a) temperature measuredabove (+) and below () zero;(b) voltage measured above (+) and below ( ) zero reference voltage.

(a) (b)

EXERCISE 1-11. If +3 stands for a gain of $3, what num ber would be u sed to stand for a loss of $7?2. If an increase of 15% in the cost of resistors is shown by +15, how would a decrease of 3% be shown?3. If 8 means 8 meters below sea level, how can 20 meters above sea level be represented?4. In the diagram below, d istance to the right of the starting point is positive and distanceto the left is negative. Complete the following chart for po ints B, C, E, I and G.

6tartI IEG BD APoints A B C D E F G Distance +2 1 GG5. How w ould you keep score in a game if you were "down 8 points"? Would you use8, +8, or 8 to stand for "down 8"?6. Which is colder, 13 or 18?7. Below are 1 2 test scores for a class of students in an electronics math course. Theaverage grade on the test is 65. Show by signed nu mbers the amount each score isabove or below the average score of 65. The first grade is 55, which is 10 belowthe average. So a 10 is written below the 55. The next grade is 65. The differencebetween it and 65 is zero, so 0 is w ritten below the 6 5. Com plete the table for theremaining scores.

Test Score 55 65 90 40 80 50 75 95 30 65 100 45Difference 1 0 0



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8. A manager of an electronics store kept a record of the monthly sales. She compared them w ith the monthly sales of the year before. She used a p lus sign to showan increase in sales and a minus sign to show a loss in sales. Complete the following table:

Monthly Sales Jan. Feb. Mar. Apr. MayThis year 4250 7840 5628 10 112 6339Last year 3020 8118 3782 9286 700:2Change +1230

1-3 NUME RICAL. EXPRESSIONS AND EQUATIONSIn arithmetic, symbols or o perators are used to indicate addition (+), subtraction ( )multiplication (X), and division (+). These same symbols are also used in algebra.

To show that two numbers are to be added, such as 2 and 5, write the numerical epression: 2 + 5. This expression is read "the sum of 2 and 5" or simply "2 plus 5." Thresult of adding 2 and 5 is 7. Seven is the sum of 2 and 5. To state that "2 + 5" and "7are the same numb er, write the numerical equation: 2 + 5 = 7. This equation is rea"the sum of 2 and 5 is equal to 7" or simply "2 plus 5 equals 7." Table 1-2 shows thforming of numerical expressions and equations for the arithmetic operators of +,, and +.

+ 6 2 + 6 = 8 2 6 2 = 4X 6 2 x 6 = 12+ 2 6 + 2 = 3XAMPLE 1-1 Use the pair of num bers 8 and 4 and the arithmetic operators oTable 1-2 to form numerical expressions and equations for eacof the operators.

SolutionNiven 8, 4, and +,, Form expressions: 8 + 4, 8 4, 8 X 4, 8 + 4


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Form equations:8 + 4 = 128 - 4 = 48 X 4 = 328 + 4 = 2

This solution is summarized in Table 1-3.

TABLE 1-3 Summary of Example 1-1+ 4 + 4 = 128 4 4 = 48 X 4 x 4 = 328 + 4 8 + 4 = 2EXERCISE 1-2

1. Use the four operators of Table 1-2 and the following pairs of numbers to form numerical expressions for each operator.(a) 10, 5b) 16, 8c) 20, 4d) 60, 122. Use the four operators of Table 1-2 and the following pairs of numbers to form numerical expressions for each operator.(a) 15, 3b) 21, 7c) 54, 6d) 64, 4

3. Use the pairs of numbers in problem 1 to form numerical equations for each of theoperators of Table 1-2.4. Use the pairs of numbers in problem 2 to form numerical equations for each of theoperators of Table 1-2.5. Write a numerical expression that indicates 3 is subtracted from 5.6. Write a numerical expression that indicates 4 is multiplied by 6.7. Write a numerical equation that indicates the value of 6 divided by 3.8. Write a numerical equation that indicates the value of 18 subtracted from 20.

1-4 ORDER OF OPERATIOSSo far we have worked with numerical expressions that used only one operator. Wehave performed addition, subtraction, multiplication, and division. Now expressionswith more than one operator will be considered. For example, the expression 9 + 6 +3 has both addition and division. This expression represents a number, but what num-


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ber? If 9 and 6 a re added first, the answer is 15, which is then divided by 3 to result i5. On the other hand, if 6 is divided by 3 first, the answer is 2, which is then added t9 to result in 11. Since there is only one exp ression, it can represent only one num berTo avoid having two answers, the order in which the operations are done must be decided. Otherwise, you would never know which answer was correct. Fortunately, matematicians have come to an agreement on the order in which the operations are to bdone. This agreem ent, called a convention, is shown in Convention 1-1.

:. CONVENTION.1OITIOreOperations. :.,..-:-......:.:: In - si..s of:oper at (..-1. ptc:. ion Intl di . t.,:cti c et ( i .it t Xn tell 10 id )1-1 md sttbtr


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SolutionNhe order of operations indicates that the multiplications areperformed before the addition.M: 5 X 35 + 6 x 2M: 6 X 25 + 12A: 15 + 127 5 X 3 + 6 X 2=27

Chain CalculationsScientific calculators have aids for working comp lex arithmetic problems without having to write down intermediate answers. Check your calculator and ow ner's guide forthe following features used to make chain calculations: memory, stack registers,parentheses, and preprogram med o rder of operations. Each of these features can, in itsown way, save an intermediate answer for later use when making a chain calculation.Thus, by chaining from one operation to the next, the need to write down interm ediateanswers is eliminated.

EXERCISE 1-3In each of the following numerical equations, state whether the answer (right-handnum ber) is correct or not. If the answer is correct, write true. If the answer is not correct, write false and then state the correct answer.1. 7 + 3 - 2 = 8. 14 - 3 X 2 = 223. 8 x 2 - 3 = 14. 12 - 12 + 3 = 05. 2 + 6 + 2 = 5. -2 + 4 X 5 = 187. 12 + 3 + 8 = 12. 6 + 2 X 3 = 19. -5 - 2 + 2 = -4 10. -9 x 2 + 3 = -6MVO0000000000000000 Calculator DrillUse your calculator with Convention 1-1 (order of operations) to perform the following chain calculations. Note: You will find all the answers to the calculator drills in Appendix B. Check each of your answers.11. 2 + 4 x 7 12. 3 - 5 X 3 13. 8 + 9 + 314. -6+12+4 15. 6 - 2 x 3 16. 15 x 2 + 5

17. -8 + 2 X 6 18. -16Nx 3 19. 2 X 3 + 4 X 220. 5 x 3 + 7 X 2 21. 3 X 3 - 4 x 5 22. 8 X 4 - 9 X 323. 15 + 3 - 8 X 2 24. 18 - 6 + 4 X 5 25. 9 + 3 - 14 + 726. -10+ 2-35+7 27. 39 - 3 + 64 + 4 28. 27 + 9 + 20 + 1029. 6 + 12 - 4 X 2 30. 5 - 2 - 18 + 6 31. 7 + 2 X 4 + 7


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TER 1 Selected Prealgebra Topics 9

1-5 SYMBOLS OF GROUPINGAs we have just learned, there is an agreed-upon order in solving a numerical expression. Suppose, however, that you need to add before multiplying. How can we say,"Add first, then multiply"? By using symbols of grouping, the normal order of operation can be overridden. W hen a particular operation needs to be performed first, or if aparticular operation needs to be emphasized, then parentheses ( ) are used.

EXAMPLE 1-4ompute the num ber represented by the expression4 X (3 + 2)SolutionNince 3 + 2 is enclosed in parentheses, it is done first.A: (3 + 2) X 5 NI: 4 X 50

X (3 + 2) = 20Besides parentheses ( ), several other symbols are used to show grouping. Theseinclude brackets [ ] and braces { }. When these signs of grouping are used together,they are used in the following order: parentheses ( ) first, then brackets [ ], and finally braces { } . When one pair of grouping symbols is enclosed in another, the operation in the inner symbol is perform ed first.EXAMPLE 1-5Nompute the num ber represented by the expression[2 + (6 3)] X 5SolutionNork within the parentheses first, and then within the bracke ts:S: (6 3) [2 + 3] X 5A: [2 + 3] 5 x 5

M:X 5 25 [2 + (6 3)] X 5 = 25

EXAMPLE 1-6Nompute the num ber represented by the numerical expression3 + [(6 + 1) + (6 + 8 + 4 X 2)1


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Solutionork within each set of parentheses first and then within thebrackets:A: (6 + 1) 3 + [7 + (6 + 8 + 4 X 2)]1): 8 + 4 3 + [7 + (6 + 2 X 2)]M: 2 X 2 3 + [7 + (6 + 4)]A: (6 + 4) 3 + [7 + 10]A: [7+ 10]+17A: 3 + 170 3+ [(6 + I) + (6 + 8 + 4 X 2)] = 20EXERCISE 41. In working with the expression [(6 + 3) (2 X 2)] + 1, either of two operationmay be performed first. What are they?

2. In working with the expression [(9 2) X (5 + 4)] (3 + 2), any of three operations may be performed first. What are they?Calculator DrillCompute the num ber represented by the following numerical expressions. Use youcalculator to chain calculate the answer. Check each of your answers.

0C,0442c,tasaoc)oo.61413. (6 + 3) + 9 4. (17 2) X 2 + 35. 5 X (12 10) + 5 6. 80 + (4 + 12)7. (8 + 9 5) + 3 8. (2 + 3) + (5 + 4)

9. 55 6 X (3 + 4) 10. (7 + 10 + 8) + 511. 12+5 X (6 + 7) 12. 30 2 X (11 7)13. 6 [(4 + 1) (3 + 2)] 14. 12 + [6 + 3 + (7 4)]15. 21 x (16 + 2) + 26 16. 1040 + (47 39) 12317. [17 + (14 X 3)] (3 X 19)18. [(5 + 4) X (10 + 5)] + (54 + 27)19. [(9 + 3) + 3 (18 + 9)] + 120. [(6 1) + 5 + (4 X 4)] [(3 X 3) + 1]21. [[(12 X 2) + (8 X 2 + 4)] + 31 (3 + 3) + 222. { [(9 + 7) X 2 + 2 + (8 + 2)] 416

1-6 DOUBLE MEANING OF + AND Before learning to add algebraically, you must first have additional understanding othe meaning of the two symbols, plus (+) and minus (). In arithmetic, + is used asthe sign (operator) for addition, while is used as the sign (operator) for subtraction

CHAPTER 1 Selected Prealgebra Topic


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CHAPTER 1 Selected Prealgebra Topics 1

In algebra, each of these symbols has two uses. The + may be used as the operator foaddition or as the sign for a positive number. The - may be used as the operator forsubtraction or as the sign for a negative number. In the expression -3, the minus indicates a negative number, while the minus in 5 - 3 indicates subtraction. In the expression (-3) + (-7), the plus means to add and the minuses indicate negative numbersWhen double signs occur in an expression, they may be simplified to a single signTable 1-4 shows how to combine double signs into a single sign.

TABLE 1-4 CombinIni Double S1gn8Double Signingie Double Sign. Single SignEXAMPLE 1- Combine the double sign in (-3) + (-7) to a single sign.

t Remove the parentheses:-3+-7

Replace + - with -:-3-7

(-3) + (-7) -3-7XAMPLE 1-3 Simplify 6 - (-8). utlonemove the parentheses: 6 - -8 Replace - - with + : 6 + 8 6-(-8)6+8

:EXERCISE 5Rewrite each of the following expressions with all the double signs combined into sigle signs. Use Table 1-4.

-. 3 + (-2) 2. 16 + (+3)3. -8 - (-2) 4. (+4) - (+3)


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5. 2 - 3 + (-7) 6. (-9) - (+3) - (-5)7. 1 + (+2) - (+6) 8. 2 + 3 - +89. 7 - -1 + (-4) 10. -5-+7--3

11. -4 + (+6) - (+7) 12. + (+8) + (-6) - (-5)13. - (+9) - (-3) + (+6) 14. + (-5) + (+3) + (-7)15. - (+7.3) + (+6.1) - -3 16. - (-3.1) + (-6.2) + (+2.4)17. + (-1.4) + (+5.3) - (+6.7) 18. - (9.3) + (+18.1) + (-6.4)19. - (-4.3) + (-8.5) - (+27) 20. + (+16) - (-21) - (+13)

1-7 ABSOLUTE VALUE OF A SIGNED NUMBERWhen you work w ith signed numbers, it is sometimes easier to manipulate the numberwithout the sign. The absolute value of a signed number is just the number without thesign. The absolute value of a number is indicated by placing a vertical bar on either sideof the number. Th e following example will show y ou how to indicate that the absolutevalue is being taken.NXAMPLE 1-9 Indicate the absolute value of -5 and +7.SolutionNbsolute value of -5 is indicated by 1-51.Absolute value of +7 is indicated by (+7(.As yo u have seen, the absolute value of a num ber is indicated by placing a verticalbar on either side of the number. The solution to the operation of taking the absolutevalue is the number without any sign. Therefore, the absolute value of a number is understood to be positive.EXAMPLE 1-10Nake the absolute value of -5 and +7.SolutionNndicate the absolute value of -5:1-51Indicate the absolute value of +7:1+71

EXERCISE -6Take the absolute value of each of the following signed numbers, combining doublesigns first.

1. +3 2. -4 3. -9 4. +2 5. -86. +(+3) 7. -(+7) 8. -(-6) 9. -(+1) 10 . +( -5)



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1-8 COMBINING SIGNED NUMBERSWh en we add two num bers together, a su m is formed. The numbers used to form thsum are called the terms of the sum. For example, in the numerical equation 2 + 5 = the terms of the sum are 2 and 5. The addition and subtraction of signed numbers idone by applying three rules. Each of these rules will be presented w ith examples to ilustrate their application.

RUle Addingtike Signed NuM To :add twc):: nu m bers:With theign: 1d their absolute Values, the Li i he sEXAMPLE 1-11Ndd 5 and 7.

SolutionNvaluate 5 + (-7): I 1: 5, 7, 7 A: 5 + 72Assign () to the sum:125 + (-7). 12EXAMPLE 1-12Ndd 3 and 7.

SolutionNvaluate 3 + 7:1: +3, +7 3, 7A: 3 + 7 10 Assign + sign: +103 + 7 = +10Rule 1-2. Adding Opposite Signed Numbers

signed numberrnSubt ct the smalier ahsolutete I a the larger.. :110t t. Give the difference the Si .. term.


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EXAMPLE 1-13dd 8 and 3.Solutionvaluate 8 + (-3):I: +8, 3,3 Subtract the smaller term from the larger: S: 8 3Assign the sign of the larger term (+):

+5 N 8 + (-3) = +5

EXAMPL E 1-14Ndd 9 and 2.Solutionvaluate 9 + 2.1: 9, +2,2 Subtract the smaller term from the larger:

S: 9 2Assign the sign of the larger term (): 7 9 + 2 = 7 Rule 1-3. Subtracting Signed NumbersTo subtract two signed numbers:I. Change the sign of the number to be subtraq2. Add the numbers using either Rule 1-1 or Rule 1

EXAM PLE 1-15Nubtract 3 from 7.SolutionNse Rule 1-3 to evaluate 7 (-3).Change the sign of 3 and restate problem using Rule 1-1:7 + 3 I 1: +7, +3, 3 A: 7 + 30Assign (+) to the sum:+10. (-3) = 10 CHAPTER 1 Selected Prealgebra Topics


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HAPTER 1 Selected Prealgebra Topics

Wh en you see the word subtract, think sign change. In Exaple 1-15 you may see that the statement "subtract 3 from 7is translated into the numerical expression 7 (-3). By appling the methods for simplifying double signs found in Tab1-4, this expression becomes 7 + 3. However, rather than cocern yourself with condensing the double sign, it is easier think, sign change and add.

PL E 1-16ubtract 6 from 2.tono subtract 6 from 2, change the sign of 6 and add. Thus, 2 sutract 6 becomes 2 add 6. Use Rule 1-2 to evaluate2 + (-6)1: 2, 6 2, 6Subtract the smaller term from the larger:S: 6 2Assign the sign of the larger ( ): 42 6 = 4As in Example 1-16, subtraction can result in a negative number. What would sua result mean? We have seen that negative numbers are used to indicate amounts beloa reference value, for instance, the temperature below freezing, the distance below slevel, and the amount by w hich a checking account is overdrawn.XAMPLE 1-17 You have $42.00 in you r checking account. You write checks the following am ounts: $6.00, $18.00, $7.00, and $13.00. H omuch do you have in your account?SoMlort2.00 (6.00 + 18.00 + 7.00 + 13.00)42.00 44.002.00

Your account is overdrawn by $2.00.

From this example we see that $2.00 means that you have insufficient fundsyour accoun t and will no doubt be charged a service charge for the ove rdraft. This cetainly has real meaning for you.


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EXERCISE 1-7Combine the following signed numbers by adding or subtracting as indicated. Use therules of addition and subtraction, and solve without the use of your calculator.Add the following numbers.

1. 7 and 3 2. 3 and 5 3. 3 and +24. 6 and 1 5. 8 and 4 6. 5 and 77. 17 and 6 8. 21 and +14 9. 13 and 39

Subtract the first number from the second in the following problems. 0. 5 from 3 11. 6 from 2 12. 8 from 3 3. 12 from 4 14. 18 from 3 15. 9 from 4 6. 32 from 14 17. 12 from 7 18. 8 from +16Using signed numbers, rewrite the following phrases into numerical expressions andthen compute the numb er represented by the expression.19. Earning $21 and spending $1620. Climbing up 14 stairs and then climbing up 17 more21. A temperature rise of 18 and a fall of 2622. A gain of $3.00 and a loss of $9.0023. Going 12 steps backward and then going 16 steps forward24. A raise of $10 in pay and an increase of $3 in taxesUsing a Calculator with Signed NumbersYour calculator is designed to work with signed num bers. It uses the operator keysand for addition and subtraction. It uses the change sign key +/ or1CHS to enter negative numbers. It has been programmed to perform arithmetic with signed numbers. The following exam ple will help you to become fam iliar with these features. It isrecomm ended that you consult your owner's guide for additional information on working with signed numbers.EXAMPLE 1-18valuate 7 + 8 6.SolutionNse your calculator. Enter 7; then change the sign.

1eH1 778657 + 8 6 = 5


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EXERCISE 1-8 Calculator DrillEvaluate each of the following expressions.

1. 18 + (-2) 2. 47 + (-17) 3. 12 ( 4)4. 18 (-14) 5. 27+8 6. 17 137. 19 17 8. 22 + (-17) 9. 13+910. 25 (-12) 11. 6 + (-3) + 5 12. 120 30 ( 10)

13. 75 15 + (-12) 14. 13 + (-15) + 39 15. 37 + 14 (23)

=K M0000 000000000000

1-9 RELATIONAL OPERATO RSWe have learned that a numerical equation uses the equal sign (=) to say that the numbers or expressions on either side of the equal sign are exactly the same. The equal siis one of the relational operators.

The relationship of inequality may be shown in several ways. Suppose that wwanted to state the relationship between 5 and 7. We cannot say that 5 and 7 are equabut we can say that they are not equal. This is stated as the mathematical senten5 7. The not-equal symbol (#) is a relational operator. We also know that 7 is greathan 5. Using the greater-than relational operator (>), we can write 7 > 5. The statment "7 > 5" is read "seven is greater than five." Finally, by using the less-than reltional operation ( 1 6 is greater than 1relative sizeIs less than Show inequality and 4 < 5is less than 5relative sizeIs greater than Show relative size 7+ 5is greater than or equaor equal to to 1 plus 5Is less than or Show relative size 0 3is less than or equal toequal to 3EXERCISE 1-9

Use the appropriate relational operator in place of the words to express each stateme. 7 is less than 10. 2. 5 is smaller than 8.. 23 is more than 13. 4. 4 is greater than 7.


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5. 3 is the same as 10 plus 13.. 3 plus 8 is not the same as 2.7. The sum of 2 and 7 is greater than 3.8. The sum of 8 and 3 is less than the difference between 25 and 7.Use =, >, or < in place of "?" to make each statement true.9. 7 ? 12 + 5 10 . 8 ? 3 + 411. 1+61 ?1-31 12 . 5 ?1-9113. 3 ? 13 (10 + 3) 14 . 36 + 4 ? 48 1215 . 3 + 4 ? 7 + (-2) 16 . 8 + 2 4 ? 8 + 417 . 102 8 ? 91 89 18 . 68 + 14 ? 59 + 2Compound inequantiesIn electronic applications of relational operators we often write compound inequalitystatements. These statements use two inequalities that have a number in common. Forexample, 5 < 8 and 8 < 10. These statements may be com bined into a single compoundstatement of 5 < 8 < 10. This statement is read "5 is less than 8, which is less than 10."Here is another compound inequality: 63 > 32 > 25. This statement is read "63 isgreater than 32, which is g reater than 25."In a comp ound inequality, both inequalities must be true in order for the compo undinequality to be true. If either inequality is false, the entire statement is false.

PLE 11.-19etermine if (15 3) < 20 < (14 8) is a true statement.Solutionombine and remove parentheses:S: (15 3)2 < 20 < (14 8)S: (14 8)2 < 20 --.Determine if the first inequality is true:

Yes, 12 is less than 20.Determine if the second inequality is true:

No, 20is not less than 6.The com pound inequality is false.

EXERCI$ Determine wh ich of the following comp ound statements are true and which are false.1. 3 < 6 < 10 2. 2 > 5 > 33. 17 7


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HAPTER 1 Selected Prealgebra Topics 1

5. 7 < 3 < 4 6. 4 > 3 > 67. 13 > 18 > 21 8. 5 < 4 < 29. (3 7) > 3 > (-9 + 3) 10 . (-5 3) < 6 < (-2 3)

1 -40 MULTIPLYING WITH SIGNED NUMBERSWhen we multiply two numbers together, we form a product. The numbers usedform the product are called the factors of the product. For example, in the numericequation 2 X 3 = 6, 2 and 3 are the factors of the product 6. Six is the product of thfactors 2 and 3.The multiplication of signed numbers is done by applying two rules. Each of therules will be presented w ith examples to illustrate its application.

Rule 1u! iplyingLibre Signed N um bTo multiply two nu bet:, gn1. 'vitultipl f the absolut2. Give the paxluct a plu

EXAMPLE 1-20Nultiply 3 times 4.SolutionNse Rule 1-4 to evaluate (-3) X (-4):I I: 3, 4, 4M: 3 X 42Assign plus sign (+) to product:+12

-3) X (-4) = 12EXAMPLE 1-21Nultiply 3 times 4.SolutionNse Rule 1-4 to evaluate (+3) X (+4):I 1: +3, +4, 4M: 3 X 42

Assign plus sign (+) to product:-1-12

(+3) X (+4) = 12


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CHAPTER 1 Selected Prealgebra Topics0

Rule 1-5. ffluftiplying Unlike Signed Numberstwo numbers with dgerent sips;

L Multiply ttle 'absolute values.(Mit' the *duct a minus sign ().XAMP Multiply 8 times 3.Solutionse Rule 1-5 to evaluate 8 X 3: I 1: 8, 3, 3 M: 8 X 34 Assign minus sign ( ) to the product: 24

8 X 3 24 EXAMPLE 1-23ultiply 5 times 3.Solutionse Rule 1-5 to evaluate 5 x (-3):5, 3I I: 5, 3: 5 X 3s Assign minus sign () to the product:

155 X (-3) = 15

From the previous examples, observe that the product of two numbers with likesigns is positive, while the product of two numbers with unlike signs is negative. Thus,in multiplication, like signs give a positive result and unlike signs give a negative result. This concept is graphically presented in Table 1-6.

TABLE 1-6 Multiplying Signed Numbers Signs of Factorsign of Product + times + + times - times - times +


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HAPTER 1 Selected Prealgebra Topics

When using your calculator, remember that it is designed to work w ith signed nubers. It has been programmed to perform multiplication with signed numbers according to the rules you have learned.EXAMPLE 1-24Nse your calculator to multiply 15 X (-6).

SolutionNnter 15:x 6 9015 X (-6) = 90ObservationNhe indicated keystrokes are to remind you of the operatioused in the calculation, but they may not be the same strokes athey may not be in the same order that you will use with yocalculator.

EXERCISE 1-11Use the information in Table 1-6 to multiply the following signed numbers. . 2 and 3 2. +8 and 4 3. 5 and 2 . 1 and 2 5. 5 and 7 6. 8 and 3 . 6 and 7 8. +4 and +3 9. +2 and 2 0. 8 and +5 11. 9 and 10 12. 7 and 1Calculator DrillUsing your calculator, compute the number represented by each of the following nmerical expressions.13. 31 x 40 14. 28 X ( 15) 15. +16 X (+12)16. 34 x (-27) 17. 92 X 102 18. 72 X (-13)19. +18 X 13 20. 47 X 52 21. 37 x (-53)22. 49 X (-25) 23. 13 X 94 24. 37 X (-14)

1-11 DIVIDING WITH SIGNED NUMBERSDivision is indicated in arithmetic by the division symbol (+), as in 6 + 2 = 3. In gebra, division is usually indicated as a fraction. A fraction is made up of three paras shown in Figure 1-2: (1) the numerator (the dividend or the number to be divide(2) the denominator (the divisor or the number to do the dividing), and (3) the bar tween the numerator and the denominator. The bar may appear horizontally or diagnally ( or /), as pictured in Figure 1-2(b). Like parentheses, the bar is a symbol grouping. Thus, the bar controls the order of operations. The following example wdemon strate this important property of the bar.


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CHAPTER 1 Selected PrEisigebra Topics2

Dividend umeratorQuotientDivisor Numerator/DenominatorDenominator(a) (b)FIGURE 1-2 Parts of a fraction. (a) The result of dividing the dividend by the divisor is thequotient; (b) Another nam e for dividend is numerator. Another nam e for divisor is denominator. The bar may be horizontal or diagonal.

PLF,Discuss how to evaluate the following exp ression and then showthe solution:2+135Since 2 + 13 is grouped together by the bar, we evaluate 2 + 13first. We then divide by 5. N otice that the usual order of divisionbefore addition has been overridden by the bar, a symbol ofgrouping.

2+13Ltion 5A: 2 + 13 IS/5D:5/5 32 + 13 = 35

The following rules are used when dividing signed numbers. Each of these rules willbe presented w ith examples to illustrate its application.

Rule 1-6. Dividing Like Signed NumbersTo dividetwO numbers with the same sign:

1. :Divide the absolute. values.:Give the result a plus sign t

EXA '1..1-26ivide 8 by 4.Solutionse Rule 1-6 to evaluate 8/-4:I I: 8, 4., 4D:/4


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Assign plus sign (+):+2

8/-4 = 2

Rule 1-7. Iv ding Us like SignettNumbersTo diinto. u 2 EXAMPLE 1-27 Divide 6 by 2.Solution Use Rule 1-7 to evaluate 6/-2:

I I: 6, 2, 2D: 6/2Assign minus sign (): 36/-2 = 3 50 11EXAMPLE 1-28 Use you r calculator to evaluate 13Solution The bar is a sign of grouping, so combine the num bers in the nmerator.Enter 50:1 39 13 CHS3 50 11 313

EXERCISE 1-12Use Rules 1-6 and 1-7 to evaluate the following expressions.

1. 15/3 2. 8/2 3. 6/-24. 12/-6 5. 14/7 6. 10/5


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7. 18/-6 8. 15/-5 9. 16/-410. 20/-101. 25/5 2. 24/8Calculator DrillIn the following, rememb er that the bar is a sign of grouping.13. 27/9 14. 36/4 15. 34/-1716. 64/16 17. 42/-14 18. 75/ 15

7 + 2 8 6N 24 + 3619. 20. 21.3 2 1213-7 19 11 8+222.N 23.N 4.4 5 + 3 3 5 + 15 12 6-625.N 6.N 7.15 + 5 25 13 2 + 73+17 15 + 15 19 + ( 89)28.N 9.N 0.N5 + 15 20 + 15 7 290000an=L:30:3C!Ot 1 ,30

WAVOMMIZO

SELECTED TERMSabsolute value The value of a number without regard to its sign.chain calculation The technique of solving arithmetic problems using a calculatorwithout w riting ou t all the intermediate results.factors A product is formed when two or more numbers are multiplied; each of thenumb ers in the product is called a factor of the product.inequality A statement that two numerical expressions do not have the same value.natural numbers The set of numbers (1, 2, 3, etc.) used in counting; the countingnumbers.numerical equation A statement that declares two numerical expressions have the

same value.numerical expression A number or a list of numbers joined by arithmetic operators;e.g., 9, or 25 8.terms A sum is formed when two or more numbers are added; each of the numbersin the sum is called a term of the sum .

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