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Prealgebra

3.4 Multiply and Divide Integers

Prealgebra3.4 Multiply and Divide Integers
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Multiply integers
  • Divide integers
  • Simplify expressions with integers
  • Evaluate variable expressions with integers
  • Translate word phrases to algebraic expressions
Be Prepared 3.4

Before you get started, take this readiness quiz.

  1. Translate the quotient of 2020 and 1313 into an algebraic expression.
    If you missed this problem, review Example 1.67.
  2. Add: −5+(−5)+(−5).−5+(−5)+(−5).
    If you missed this problem, review Example 3.21.
  3. Evaluaten+4whenn=−7.Evaluaten+4whenn=−7.
    If you missed this problem, review Example 3.23.

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that a·ba·b means add a,ba,b times. Here, we are using the model shown in Figure 3.19 just to help us discover the pattern.

This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.
Figure 3.19

Now consider what it means to multiply 55 by −3.−3. It means subtract 5,35,3 times. Looking at subtraction as taking away, it means to take away 5,35,3 times. But there is nothing to take away, so we start by adding neutral pairs as shown in Figure 3.20.

This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives.  Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.
Figure 3.20

In both cases, we started with 1515 neutral pairs. In the case on the left, we took away 5,35,3 times and the result was 15.15. To multiply (−5)(−3),(−5)(−3), we took away 5,35,3 times and the result was 15.15. So we found that

5·3=15−5(3)=−155(−3)=−15(−5)(−3)=155·3=15−5(3)=−155(−3)=−15(−5)(−3)=15

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

Multiplication of Signed Numbers

The sign of the product of two numbers depends on their signs.

Same signs Product
•Two positives
•Two negatives
Positive
Positive
Different signs Product
•Positive • negative
•Negative • positive
Negative
Negative

Example 3.47

Multiply each of the following:

  1. −9·3 −9·3
  2. −2(−5)−2(−5)
  3. 4(−8)4(−8)
  4. 7·67·6
Try It 3.93

Multiply:

  1. −6·8−6·8
  2. −4(−7)−4(−7)
  3. 9(−7)9(−7)
  4. 5·125·12
Try It 3.94

Multiply:

  1. −8·7−8·7
  2. −6(−9)−6(−9)
  3. 7(−4)7(−4)
  4. 3·133·13

When we multiply a number by 1,1, the result is the same number. What happens when we multiply a number by−1?−1? Let’s multiply a positive number and then a negative number by −1−1 to see what we get.

−1·4−1(−3)−43−4is the opposite of43is the opposite of−3−1·4−1(−3)−43−4is the opposite of43is the opposite of−3

Each time we multiply a number by −1,−1, we get its opposite.

Multiplication by −1−1

Multiplying a number by −1−1 gives its opposite.

−1a=a−1a=a

Example 3.48

Multiply each of the following:

  1. −1·7−1·7
  2. −1(−11)−1(−11)
Try It 3.95

Multiply.

  1. −1·9−1·9
  2. −1·(−17)−1·(−17)
Try It 3.96

Multiply.

  1. −1·8−1·8
  2. −1·(−16)−1·(−16)

Divide Integers

Division is the inverse operation of multiplication. So, 15÷3=515÷3=5 because 5·3=155·3=15 In words, this expression says that 1515 can be divided into 33 groups of 55 each because adding five three times gives 15.15. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

5·3=15so15÷3=5−5(3)=−15so−15÷3=−5(−5)(−3)=15so15÷(−3)=−55(−3)=−15so−15÷−3=55·3=15so15÷3=5−5(3)=−15so−15÷3=−5(−5)(−3)=15so15÷(−3)=−55(−3)=−15so−15÷−3=5

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Same signs Quotient
•Two positives
•Two negatives
Positive
Positive
Different signs Quotient
•Positive & negative
•Negative & positive
Negative
Negative

Remember, you can always check the answer to a division problem by multiplying.

Example 3.49

Divide each of the following:

  1. −27÷3 −27÷3
  2. −100÷(−4)−100÷(−4)
Try It 3.97

Divide:

  1. −42÷6−42÷6
  2. −117÷(−3)−117÷(−3)
Try It 3.98

Divide:

  1. −63÷7−63÷7
  2. −115÷(−5)−115÷(−5)

Just as we saw with multiplication, when we divide a number by 1,1, the result is the same number. What happens when we divide a number by −1?−1? Let’s divide a positive number and then a negative number by −1−1 to see what we get.

8÷(−1)−9÷(−1)−89−8 is the opposite of 89 is the opposite of −98÷(−1)−9÷(−1)−89−8 is the opposite of 89 is the opposite of −9

When we divide a number by, −1−1 we get its opposite.

Division by −1−1

Dividing a number by −1−1 gives its opposite.

a÷(−1)=−aa÷(−1)=−a

Example 3.50

Divide each of the following:

  1. 16÷(−1)16÷(−1)
  2. −20÷(−1)−20÷(−1)
Try It 3.99

Divide:

  1. 6÷(−1) 6÷(−1)
  2. −36÷(−1)−36÷(−1)
Try It 3.100

Divide:

  1. 28÷(−1) 28÷(−1)
  2. −52÷(−1)−52÷(−1)

Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

Example 3.51

Simplify:7(−2)+4(−7)6.Simplify:7(−2)+4(−7)6.

Try It 3.101

Simplify:

8(−3)+5(−7)−48(−3)+5(−7)−4

Try It 3.102

Simplify:

9(−3)+7(−8)19(−3)+7(−8)1

Example 3.52

Simplify:

  1. (−2)4 (−2)4
  2. −24−24
Try It 3.103

Simplify:

  1. (−3)4 (−3)4
  2. −34−34
Try It 3.104

Simplify:

  1. (−7)2 (−7)2
  2. 7272

Example 3.53

Simplify:123(912).Simplify:123(912).

Try It 3.105

Simplify:

174(811)174(811)

Try It 3.106

Simplify:

166(713)166(713)

Example 3.54

Simplify: 8(−9)÷(−2)3.8(−9)÷(−2)3.

Try It 3.107

Simplify:

12(−9)÷(−3)312(−9)÷(−3)3

Try It 3.108

Simplify:

18(−4)÷(−2)318(−4)÷(−2)3

Example 3.55

Simplify:−30÷2+(−3)(−7).Simplify:−30÷2+(−3)(−7).

Try It 3.109

Simplify:

−27÷3+(−5)(−6)−27÷3+(−5)(−6)

Try It 3.110

Simplify:

−32÷4+(−2)(−7)−32÷4+(−2)(−7)

Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

Example 3.56

Evaluate2x23x+8whenx=−4.Evaluate2x23x+8whenx=−4.

Try It 3.111

Evaluate:

3x22x+6whenx=−33x22x+6whenx=−3

Try It 3.112

Evaluate:

4x2x5whenx=−24x2x5whenx=−2

Example 3.57

Evaluate3x+4y6whenx=−1andy=2.Evaluate3x+4y6whenx=−1andy=2.

Try It 3.113

Evaluate:

7x+6y12whenx=−2andy=37x+6y12whenx=−2andy=3

Try It 3.114

Evaluate:

8x6y+13whenx=−3andy=−58x6y+13whenx=−3andy=−5

Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.

Example 3.58

Translate to an algebraic expression and simplify if possible: the product of −2−2 and 14.14.

Try It 3.115

Translate to an algebraic expression and simplify if possible:

the product of −5 and 12the product of −5 and 12

Try It 3.116

Translate to an algebraic expression and simplify if possible:

the product of 8 and −13the product of 8 and −13

Example 3.59

Translate to an algebraic expression and simplify if possible: the quotient of −56−56 and −7.−7.

Try It 3.117

Translate to an algebraic expression and simplify if possible:

the quotient of −63 and −9the quotient of −63 and −9

Try It 3.118

Translate to an algebraic expression and simplify if possible:

the quotient of −72 and −9the quotient of −72 and −9

Section 3.4 Exercises

Practice Makes Perfect

Multiply Integers

In the following exercises, multiply each pair of integers.

211.

−4·8−4·8

212.

−3·9−3·9

213.

−5(7)−5(7)

214.

−8(6)−8(6)

215.

−18(−2)−18(−2)

216.

−10(−6)−10(−6)

217.

9(−7)9(−7)

218.

13(−5)13(−5)

219.

−1·6−1·6

220.

−1·3−1·3

221.

−1(−14)−1(−14)

222.

−1(−19)−1(−19)

Divide Integers

In the following exercises, divide.

223.

−24÷6−24÷6

224.

−28÷7−28÷7

225.

56÷(−7)56÷(−7)

226.

35÷(−7)35÷(−7)

227.

−52÷(−4)−52÷(−4)

228.

−84÷(−6)−84÷(−6)

229.

−180÷15−180÷15

230.

−192÷12−192÷12

231.

49÷(−1)49÷(−1)

232.

62÷(−1)62÷(−1)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

233.

5(−6)+7(−2)−35(−6)+7(−2)−3

234.

8(−4)+5(−4)−68(−4)+5(−4)−6

235.

−8(−2)−3(−9)−8(−2)−3(−9)

236.

−7(−4)−5(−3)−7(−4)−5(−3)

237.

(−5)3(−5)3

238.

(−4)3(−4)3

239.

(−2)6(−2)6

240.

(−3)5(−3)5

241.

4242

242.

6262

243.

−3(−5)(6)−3(−5)(6)

244.

−4(−6)(3)−4(−6)(3)

245.

−4·2·11−4·2·11

246.

−5·3·10−5·3·10

247.

(811)(912)(811)(912)

248.

(611)(813)(611)(813)

249.

263(27)263(27)

250.

232(46)232(46)

251.

−10(−4)÷(−8)−10(−4)÷(−8)

252.

−8(−6)÷(−4)−8(−6)÷(−4)

253.

65÷(−5)+(−28)÷(−7)65÷(−5)+(−28)÷(−7)

254.

52÷(−4)+(−32)÷(−8)52÷(−4)+(−32)÷(−8)

255.

92[38(−2)]92[38(−2)]

256.

113[74(−2)]113[74(−2)]

257.

(−3)2−24÷(82)(−3)2−24÷(82)

258.

(−4)232÷(124)(−4)232÷(124)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

259.

−2x+17when−2x+17when

  1. x=8x=8
  2. x=−8x=−8
260.

−5y+14when−5y+14when

  1. y=9 y=9
  2. y=−9 y=−9
261.

103mwhen 103mwhen

  1. m=5 m=5
  2. m=−5 m=−5
262.

184nwhen184nwhen

  1. n=3 n=3
  2. n=−3n=−3
263.

p25p+5whenp=−1p25p+5whenp=−1

264.

q22q+9q22q+9 when q=−2q=−2

265.

2w23w+72w23w+7 when w=−2w=−2

266.

3u24u+53u24u+5 when u=−3u=−3

267.

6x5y+156x5y+15 when x=3x=3 and y=−1y=−1

268.

3p2q+93p2q+9 when p=8p=8 and q=−2q=−2

269.

9a2b89a2b8 when a=−6a=−6 and b=−3b=−3

270.

7m4n27m4n2 when m=−4m=−4 and n=−9n=−9

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

271.

The product of −3−3 and 15

272.

The product of −4−4 and 1616

273.

The quotient of −60−60 and −20−20

274.

The quotient of −40−40 and −20−20

275.

The quotient of −6−6 and the sum of aa and bb

276.

The quotient of −7−7 and the sum of mm and nn

277.

The product of −10−10 and the difference of pandqpandq

278.

The product of −13−13 and the difference of canddcandd



Everyday Math

279.

Stock market Javier owns 300300 shares of stock in one company. On Tuesday, the stock price dropped $12$12 per share. What was the total effect on Javier’s portfolio?

280.

Weight loss In the first week of a diet program, eight women lost an average of 3 pounds3 pounds each. What was the total weight change for the eight women?

Writing Exercises

281.

In your own words, state the rules for multiplying two integers.

282.

In your own words, state the rules for dividing two integers.

283.

Why is −24(−2)4?−24(−2)4?

284.

Why is −42(−4)2?−42(−4)2?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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