1.4 Multiply and Divide Integers

Learning objectives

Learning objectives

• Multiply integers

• Divide integers

• Simplify expressions with integers

• Evaluate variable expressions with integers

• Translate English phrases to algebraic expressions

• Use integers in applications

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our 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. Here, we will use the model just to help us discover the pattern.

We remember that $a\cdot{}b$ means add a, b times.

The next two examples are more interesting.

What does it mean to multiply 5 by $-3?$ It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

In summary:

Notice that for multiplication of two signed numbers, when the:

• signs are the same, the product is positive.

• signs are different, the product is negative.

We’ll put this all together in the chart below.

Example 1.46

Solution

Table

1.8

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

Each time we multiply a number by $-1,$ we get its opposite!

Example 1.47

Solution

Table

1.9

Divide Integers

What about division? Division is the inverse operation of multiplication. So, $15\mathrm{÷}3=5$ because $5\cdot{}3=15.$ In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

• signs are the same, the quotient is positive.

• signs are different, the quotient is negative.

And remember that we can always check the answer of a division problem by multiplying.

Example 1.48

Solution

Table

1.10

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

Example 1.49

Solution

Table

1.11

Example 1.50

Solution

Table

1.12

Notice the difference in parts ⓐ and ⓑ. In part ⓐ , the exponent means to raise what is in the parentheses, the $(-2)$ to the ${4}^{\mathrm{th}}$ power. In part ⓑ , the exponent means to raise just the 2 to the ${4}^{\mathrm{th}}$ power and then take the opposite.

The next example reminds us to simplify inside parentheses first.

Example 1.51

Solution

Table

1.13

Example 1.52

Solution

Table

1.14

Example 1.53

Solution

Table

1.15

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Example 1.54

Solution

ⓐ

ⓑ

Example 1.55

Solution

Example 1.56

Solution

ⓐ

ⓑ

Example 1.57

Solution

Substitute $4\,\mathrm{for}\,x.$ Use parentheses to show multiplication.

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Example 1.58

Solution

Table

1.16

When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

Be careful to get a and b in the right order!

Example 1.59

Solution

Table

1.17

Once again, our prior work translating English 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 1.60

Solution

Table

1.18

Example 1.61

Solution

Table

1.19

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Example 1.62

Example 1.63

Solution

Table

1.20

Section 1.4 Exercises

Section 1.4 Exercises

Multiply Integers

In the following exercises, multiply.

Divide Integers

In the following exercises, divide.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate English Phrases to Algebraic Expressions

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

Use Integers in Applications

In the following exercises, solve.

Everyday Math

Writing Exercises

ⓐ 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?

1. Multiply: ⓐ $-6\cdot{}8$ ⓑ $-4(-7)$ ⓒ $9(-7)$ ⓓ $5\cdot{}12.$

2. Multiply: ⓐ $-8\cdot{}7$ ⓑ $-6(-9)$ ⓒ $7(-4)$ ⓓ $3\cdot{}13.$

3. Multiply: ⓐ $-1\cdot{}9$ ⓑ $-1\cdot(-17).$

4. Multiply: ⓐ $-1\cdot{}8$ ⓑ $-1\cdot(-16).$

1. Divide: ⓐ $-42\mathrm{÷}6$ ⓑ $-117\mathrm{÷}(-3).$

2. Divide: ⓐ $-63\mathrm{÷}7$ ⓑ $-115\mathrm{÷}(-5).$

3. Simplify: $8(-3)+5(-7)-4.$

4. Simplify: $9(-3)+7(-8)-1.$

1. Simplify: ⓐ ${(-3)}^{4}$ ⓑ $-{3}^{4}.$

2. Simplify: ⓐ ${(-7)}^{2}$ ⓑ $-{7}^{2}.$

3. When $n=-8,$ evaluate ⓐ $n+2$ ⓑ $-n+2.$

4. When $y=-9,$ evaluate ⓐ $y+8$ ⓑ $-y+8.$

1. Evaluate: $17-k$ when ⓐ $k=19$ and ⓑ $k=-19.$

2. Evaluate: $-5-b$ when ⓐ $b=14$ and ⓑ $b=-14.$

3. Translate and simplify the sum of 9 and $-16,$ increased by 4.

4. Translate and simplify the sum of $-8$ and $-12,$ increased by 7.

1. Translate and simplify ⓐ the difference of 14 and $-23$ ⓑ subtract 21 from $-17.$

2. Translate and simplify ⓐ the difference of 11 and $-19$ ⓑ subtract 18 from $-11.$

3. Translate to an algebraic expression and simplify if possible: the product of $-5$ and 12.

4. Translate to an algebraic expression and simplify if possible: the product of 8 and $-13.$

1. Translate to an algebraic expression and simplify if possible: the quotient of $-63$ and $-9.$

2. Translate to an algebraic expression and simplify if possible: the quotient of $-72$ and $-9.$

3. In the morning, the temperature in Anchorage, Alaska was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

4. The temperature in Denver was $-6$ degrees at lunchtime. By sunset the temperature had dropped to $-15$ degrees. What was the difference in the lunchtime and sunset temperatures?

1. $-1\cdot{}6$

2. $-24\mathrm{÷}6$

3. $-180\mathrm{÷}15$

4. $5(-6)+7(-2)-3$

1. $-{4}^{2}$

2. $65\mathrm{÷}(-5)+(-28)\mathrm{÷}(-7)$

3. $9-2[3-8(-2)]$

4. $y+(-14)$ when ⓐ $y=-33$ ⓑ $y=30$

1. 1. ⓐ $a+3$ when $a=-7$

2. ⓑ $-a+3$ when $a=-7$

2. $m+n$ when$m=-15,n=7$

3. $r+s$ when $r=-9,s=-7$

4. $-2x+17$ when

1. ⓐ $x=8$

2. ⓑ $x=-8$

1. $10-3m$ whenⓐ $m=5$ⓑ $m=-5$

2. $9a-2b-8$ when$a=-6\,\mathrm{and}\,b=-3$

3. the sum of 3 and $-15,$ increased by 7

4. the difference of 10 and $-18$

1. the difference of $-5$ and $-30$

2. the product of $\mathrm{-3 and 15}$

3. the quotient of $-60$ and $-20$

4. the quotient of $-6$ and the sum of a and b

1. the product of $-10$ and the difference of $p\,\mathrm{and}\,q$

2. Temperature On January $15,$ the high temperature in Anaheim, California, was $84^{\circ}.$ That same day, the high temperature in Embarrass, Minnesota was $-12^{\circ}.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

3. Checking Account Mayra has $124 in her checking account. She writes a check for$152. What is the new balance in her checking account?

4. Checking Account Diontre has a balance of $-\$38$ in his checking account. He deposits $225 to the account. What is the new balance?

1. The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

2. Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

3. $-4\cdot{}8$

4. $9(-7)$

1. Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio?

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

3. Why is $-{2}^{4}\ne{(-2)}^{4}?$