Prealgebra 2e

# 3.4Multiply and Divide Integers

Prealgebra 2e3.4 Multiply and Divide Integers

### 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 3.7

Before you get started, take this readiness quiz.

Translate the quotient of $2020$ and $1313$ into an algebraic expression.
If you missed this problem, review Example 1.67.

Be Prepared 3.8

Add: $−5+(−5)+(−5).−5+(−5)+(−5).$
If you missed this problem, review Example 3.21.

Be Prepared 3.9

$Evaluaten+4whenn=−7.Evaluaten+4whenn=−7.$
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.

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.

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 $−5,3−5,3$ 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. $−72−72$

### Example 3.53

$Simplify:12−3(9−12).Simplify:12−3(9−12).$

Try It 3.105

Simplify:

$17−4(8−11)17−4(8−11)$

Try It 3.106

Simplify:

$16−6(7−13)16−6(7−13)$

### 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

$Evaluate2x2−3x+8whenx=−4.Evaluate2x2−3x+8whenx=−4.$

Try It 3.111

Evaluate:

$3x2−2x+6whenx=−33x2−2x+6whenx=−3$

Try It 3.112

Evaluate:

$4x2−x−5whenx=−24x2−x−5whenx=−2$

### Example 3.57

$Evaluate3x+4y−6whenx=−1andy=2.Evaluate3x+4y−6whenx=−1andy=2.$

Try It 3.113

Evaluate:

$7x+6y−12whenx=−2andy=37x+6y−12whenx=−2andy=3$

Try It 3.114

Evaluate:

$8x−6y+13whenx=−3andy=−58x−6y+13whenx=−3andy=−5$

### 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.

$−42−42$

242.

$−62−62$

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.

$(8−11)(9−12)(8−11)(9−12)$

248.

$(6−11)(8−13)(6−11)(8−13)$

249.

$26−3(2−7)26−3(2−7)$

250.

$23−2(4−6)23−2(4−6)$

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.

$9−2[3−8(−2)]9−2[3−8(−2)]$

256.

$11−3[7−4(−2)]11−3[7−4(−2)]$

257.

$(−3)2−24÷(8−2)(−3)2−24÷(8−2)$

258.

$(−4)2−32÷(12−4)(−4)2−32÷(12−4)$

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.

$10−3mwhen 10−3mwhen$

1. $m=5 m=5$
2. $m=−5 m=−5$
262.

$18−4nwhen18−4nwhen$

1. $n=3 n=3$
2. $n=−3n=−3$
263.

$p2−5p+5whenp=−1p2−5p+5whenp=−1$

264.

$q2−2q+9q2−2q+9$ when $q=−2q=−2$

265.

$2w2−3w+72w2−3w+7$ when $w=−2w=−2$

266.

$3u2−4u+53u2−4u+5$ when $u=−3u=−3$

267.

$6x−5y+156x−5y+15$ when $x=3x=3$ and $y=−1y=−1$

268.

$3p−2q+93p−2q+9$ when $p=8p=8$ and $q=−2q=−2$

269.

$9a−2b−89a−2b−8$ when $a=−6a=−6$ and $b=−3b=−3$

270.

$7m−4n−27m−4n−2$ 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 $1212$ 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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