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
A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.
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 $a\xb7b$ means add a, b times. Here, we are using the model just to help us discover the pattern.
The next two examples are more interesting.
What does it mean to multiply 5 by $\mathrm{-3}?$ 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.
Multiplication of Signed Numbers
For multiplication of two signed numbers:
Same signs | Product | Example |
---|---|---|
Two positives Two negatives |
Positive Positive |
$\begin{array}{ccc}\hfill 7\xb74& =\hfill & 28\hfill \\ \hfill \mathrm{-8}\left(\mathrm{-6}\right)& =\hfill & 48\hfill \end{array}$ |
Different signs | Product | Example |
---|---|---|
Positive · negative Negative · positive |
Negative Negative |
$\begin{array}{ccc}\hfill 7\left(\mathrm{-9}\right)& =\hfill & \mathrm{-63}\hfill \\ \hfill \mathrm{-5}\xb710& =\hfill & \mathrm{-50}\hfill \end{array}$ |
Example 1.46
Multiply: ⓐ $\mathrm{-9}\xb73$ ⓑ $\mathrm{-2}\left(\mathrm{-5}\right)$ ⓒ $4\left(\mathrm{-8}\right)$ ⓓ $7\xb76.$
ⓐ Multiply, noting that the signs are different so the product is negative. |
$\begin{array}{c}\mathrm{-9}\xb73\\ \mathrm{-27}\end{array}$ |
ⓑ Multiply, noting that the signs are the same so the product is positive. |
$\begin{array}{c}\mathrm{-2}\left(\mathrm{-5}\right)\\ 10\end{array}$ |
ⓒ Multiply, with different signs. |
$\begin{array}{c}4\left(\mathrm{-8}\right)\\ \mathrm{-32}\end{array}$ |
ⓓ Multiply, with same signs. |
$\begin{array}{c}7\xb76\\ 42\end{array}$ |
Multiply: ⓐ $\mathrm{-6}\xb78$ ⓑ $\mathrm{-4}\left(\mathrm{-7}\right)$ ⓒ $9\left(\mathrm{-7}\right)$ ⓓ $5\xb712.$
Multiply: ⓐ $\mathrm{-8}\xb77$ ⓑ $\mathrm{-6}\left(\mathrm{-9}\right)$ ⓒ $7\left(\mathrm{-4}\right)$ ⓓ $3\xb713.$
When we multiply a number by 1, the result is the same number. What happens when we multiply a number by $\mathrm{-1}?$ Let’s multiply a positive number and then a negative number by $\mathrm{-1}$ to see what we get.
Each time we multiply a number by $\mathrm{-1},$ we get its opposite!
Multiplication by $\mathrm{-1}$
Multiplying a number by $\mathrm{-1}$ gives its opposite.
Example 1.47
Multiply: ⓐ $\mathrm{-1}\xb77$ ⓑ $\mathrm{-1}\left(\mathrm{-11}\right).$
ⓐ Multiply, noting that the signs are different so the product is negative. |
$\begin{array}{c}\mathrm{-1}\xb77\\ \mathrm{-7}\\ \mathrm{-7}\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}7.\end{array}$ |
ⓑ Multiply, noting that the signs are the same so the product is positive. |
$\begin{array}{c}\mathrm{-1}\left(\mathrm{-11}\right)\\ 11\\ 11\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}\mathrm{-11}.\end{array}$ |
Multiply: ⓐ $\mathrm{-1}\xb79$ ⓑ $\mathrm{-1}\xb7\left(\mathrm{-17}\right).$
Multiply: ⓐ $\mathrm{-1}\xb78$ ⓑ $\mathrm{-1}\xb7\left(\mathrm{-16}\right).$
Divide Integers
What about division? Division is the inverse operation of multiplication. So, $15\xf73=5$ because $5\xb73=15.$ 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.
Multiplication and Division of Signed Numbers
For multiplication and division of two signed numbers:
- If the signs are the same, the result is positive.
- If the signs are different, the result is negative.
Same signs | Result |
---|---|
Two positives Two negatives |
Positive Positive |
If the signs are the same, the result is positive. |
Different signs | Result |
---|---|
Positive and negative Negative and positive |
Negative Negative |
If the signs are different, the result is negative. |
Example 1.48
Divide: ⓐ $\mathrm{-27}\xf73$ ⓑ $\mathrm{-100}\xf7\left(\mathrm{-4}\right).$
ⓐ Divide. With different signs, the quotient is negative. |
$\begin{array}{c}-27\xf73\\ \mathrm{-9}\end{array}$ |
ⓑ Divide. With signs that are the same, the quotient is positive. |
$\begin{array}{c}-100\xf7\left(\mathrm{-4}\right)\\ 25\end{array}$ |
Divide: ⓐ $\mathrm{-42}\xf76$ ⓑ $\mathrm{-117}\xf7\left(\mathrm{-3}\right).$
Divide: ⓐ $\mathrm{-63}\xf77$ ⓑ $\mathrm{-115}\xf7\left(\mathrm{-5}\right).$
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
Simplify: $7\left(\mathrm{-2}\right)+4\left(\mathrm{-7}\right)-6.$
$7\left(\mathrm{-2}\right)+4\left(\mathrm{-7}\right)-6$ | |
Multiply first. | $\mathrm{-14}+\left(\mathrm{-28}\right)-6$ |
Add. | $\mathrm{-42}-6$ |
Subtract. | $\mathrm{-48}$ |
Simplify: $8\left(\mathrm{-3}\right)+5\left(\mathrm{-7}\right)-4.$
Simplify: $9\left(\mathrm{-3}\right)+7\left(\mathrm{-8}\right)-1.$
Example 1.50
Simplify: ⓐ ${\left(\mathrm{-2}\right)}^{4}$ ⓑ $\text{\u2212}{2}^{4}.$
ⓐ
Write in expanded form. Multiply. Multiply. Multiply. |
$\begin{array}{c}{\left(\mathrm{-2}\right)}^{4}\\ \left(\mathrm{-2}\right)\left(\mathrm{-2}\right)\left(\mathrm{-2}\right)\left(\mathrm{-2}\right)\\ 4\left(\mathrm{-2}\right)\left(\mathrm{-2}\right)\\ \mathrm{-8}\left(\mathrm{-2}\right)\\ 16\end{array}$ |
ⓑ
Write in expanded form. We are asked to find the opposite of$\phantom{\rule{0.2em}{0ex}}{2}^{4}.$ Multiply. Multiply. Multiply. |
$\begin{array}{c}\text{\u2212}{2}^{4}\\ \text{\u2212}\left(2\xb72\xb72\xb72\right)\\ \text{\u2212}\left(4\xb72\xb72\right)\\ \text{\u2212}\left(8\xb72\right)\\ -16\end{array}$ |
Notice the difference in parts ⓐ and ⓑ. In part ⓐ, the exponent means to raise what is in the parentheses, the $\left(\mathrm{-2}\right)$ to the ${4}^{\text{th}}$ power. In part ⓑ, the exponent means to raise just the 2 to the ${4}^{\text{th}}$ power and then take the opposite.
Simplify: ⓐ ${\left(\mathrm{-3}\right)}^{4}$ ⓑ $\text{\u2212}{3}^{4}.$
Simplify: ⓐ ${\left(\mathrm{-7}\right)}^{2}$ ⓑ $\text{\u2212}{7}^{2}.$
The next example reminds us to simplify inside parentheses first.
Example 1.51
Simplify: $12-3\left(9-12\right).$
$12-3\left(9-12\right)$ | |
Subtract in parentheses first. | $12-3\left(\mathrm{-3}\right)$ |
Multiply. | $12-\left(\mathrm{-9}\right)$ |
Subtract. | $21$ |
Simplify: $17-4\left(8-11\right).$
Simplify: $16-6\left(7-13\right).$
Example 1.52
Simplify: $8\left(\mathrm{-9}\right)\xf7{\left(\mathrm{-2}\right)}^{3}.$
$8\left(\mathrm{-9}\right)\xf7{\left(\mathrm{-2}\right)}^{3}$ | |
Exponents first. | $8\left(\mathrm{-9}\right)\xf7\left(\mathrm{-8}\right)$ |
Multiply. | $\mathrm{-72}\xf7\left(\mathrm{-8}\right)$ |
Divide. | $9$ |
Simplify: $12\left(\mathrm{-9}\right)\xf7{\left(\mathrm{-3}\right)}^{3}.$
Simplify: $18\left(\mathrm{-4}\right)\xf7{\left(\mathrm{-2}\right)}^{3}.$
Example 1.53
Simplify: $\mathrm{-30}\xf72+\left(\mathrm{-3}\right)\left(\mathrm{-7}\right).$
$\mathrm{-30}\xf72+\left(\mathrm{-3}\right)\left(\mathrm{-7}\right)$ | |
Multiply and divide left to right, so divide first. | $\mathrm{-15}+\left(\mathrm{-3}\right)\left(\mathrm{-7}\right)$ |
Multiply. | $\mathrm{-15}+21$ |
Add. | $6$ |
Simplify: $\mathrm{-27}\xf73+\left(\mathrm{-5}\right)\left(\mathrm{-6}\right).$
Simplify: $\mathrm{-32}\xf74+\left(\mathrm{-2}\right)\left(\mathrm{-7}\right).$
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
When $n=\mathrm{-5},$ evaluate: ⓐ $n+1$ ⓑ $\text{\u2212}n+1.$
ⓐ
Simplify. | −4 |
ⓑ
Simplify. | |
Add. | 6 |
When $n=\mathrm{-8},$ evaluate ⓐ $n+2$ ⓑ $\text{\u2212}n+2.$
When $y=\mathrm{-9},$ evaluate ⓐ $y+8$ ⓑ $\text{\u2212}y+8.$
Example 1.55
Evaluate ${\left(x+y\right)}^{2}$ when $x=\mathrm{-18}$ and $y=24.$
Add inside parenthesis. | (6)^{2} |
Simplify. | 36 |
Evaluate ${\left(x+y\right)}^{2}$ when $x=\mathrm{-15}$ and $y=29.$
Evaluate ${\left(x+y\right)}^{3}$ when $x=\mathrm{-8}$ and $y=10.$
Example 1.56
Evaluate $20-z$ when ⓐ $z=12$ and ⓑ $z=\mathrm{-12}.$
ⓐ
Subtract. | 8 |
ⓑ
Subtract. | 32 |
Evaluate: $17-k$ when ⓐ $k=19$ and ⓑ $k=\mathrm{-19}.$
Evaluate: $\mathrm{-5}-b$ when ⓐ $b=14$ and ⓑ $b=\mathrm{-14}.$
Example 1.57
Evaluate: $2{x}^{2}+3x+8$ when $x=4.$
Substitute $4\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x.$ Use parentheses to show multiplication.
Substitute. | |
Evaluate exponents. | |
Multiply. | |
Add. | 52 |
Evaluate: $3{x}^{2}-2x+6$ when $x=\mathrm{-3}.$
Evaluate: $4{x}^{2}-x-5$ when $x=\mathrm{-2}.$
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
Translate and simplify: the sum of 8 and $\mathrm{-12},$ increased by 3.
the sum of 8 and $\mathrm{-12},$ increased by 3. | |
Translate. | $\left[8+\left(\mathrm{-12}\right)\right]+3$ |
Simplify. Be careful not to confuse the brackets with an absolute value sign. | $\left(\mathrm{-4}\right)+3$ |
Add. | $\mathrm{-1}$ |
Translate and simplify the sum of 9 and $\mathrm{-16},$ increased by 4.
Translate and simplify the sum of $\mathrm{-8}$ and $\mathrm{-12},$ increased by 7.
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.
$a-b$ |
---|
$a$ minus $b$ the difference of $a$ and $b$ $b$ subtracted from $a$ $b$ less than $a$ |
Be careful to get a and b in the right order!
Example 1.59
Translate and then simplify ⓐ the difference of 13 and $\mathrm{-21}$ ⓑ subtract 24 from $\mathrm{-19}.$
ⓐ
Translate. Simplify. |
$\begin{array}{c}\text{the}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{difference}}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}13\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}-21\\ 13-\left(\mathrm{-21}\right)\\ 34\end{array}$ |
ⓑ Translate. Remember, "subtract $b$ from $a$ means $a-b$. Simplify. |
$\begin{array}{c}\mathbf{\text{subtract}}\phantom{\rule{0.2em}{0ex}}24\phantom{\rule{0.2em}{0ex}}\mathbf{\text{from}}\phantom{\rule{0.2em}{0ex}}-19\\ -19-24\\ 43\end{array}$ |
Translate and simplify ⓐ the difference of 14 and $\mathrm{-23}$ ⓑ subtract 21 from $\mathrm{-17}.$
Translate and simplify ⓐ the difference of 11 and $\mathrm{-19}$ ⓑ subtract 18 from $\mathrm{-11}.$
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
Translate to an algebraic expression and simplify if possible: the product of $\mathrm{-2}$ and 14.
$\text{the product}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}\mathrm{-2}\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}14$ | |
Translate. | $\left(\mathrm{-2}\right)\left(14\right)$ |
Simplify. | $\mathrm{-28}$ |
Translate to an algebraic expression and simplify if possible: the product of $\mathrm{-5}$ and 12.
Translate to an algebraic expression and simplify if possible: the product of 8 and $\mathrm{-13}.$
Example 1.61
Translate to an algebraic expression and simplify if possible: the quotient of $\mathrm{-56}$ and $\mathrm{-7}.$
$\text{the quotient}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}\mathrm{-56}\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}\mathrm{-7}$ | |
Translate. | $\mathrm{-56}\xf7\left(\mathrm{-7}\right)$ |
Simplify. | $8$ |
Translate to an algebraic expression and simplify if possible: the quotient of $\mathrm{-63}$ and $\mathrm{-9}.$
Translate to an algebraic expression and simplify if possible: the quotient of $\mathrm{-72}$ and $\mathrm{-9}.$
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 How to Apply a Strategy to Solve Applications with Integers
In the morning, the temperature in Urbana, Illinois was 11 degrees. By mid-afternoon, the temperature had dropped to $\mathrm{-9}$ degrees. What was the difference of the morning and afternoon temperatures?
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?
The temperature in Denver was $\mathrm{-6}$ degrees at lunchtime. By sunset the temperature had dropped to $\mathrm{-15}$ degrees. What was the difference in the lunchtime and sunset temperatures?
How To
Apply a Strategy to Solve Applications with Integers.
- Step 1. Read the problem. Make sure all the words and ideas are understood
- Step 2. Identify what we are asked to find.
- Step 3. Write a phrase that gives the information to find it.
- Step 4. Translate the phrase to an expression.
- Step 5. Simplify the expression.
- Step 6. Answer the question with a complete sentence.
Example 1.63
The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?
Step 1. Read the problem. Make sure all the words and ideas are understood. | |
Step 2. Identify what we are asked to find. | the number of yards lost |
Step 3. Write a phrase that gives the information to find it. | three times a 15-yard penalty |
Step 4. Translate the phrase to an expression. | $3\left(\mathrm{-15}\right)$ |
Step 5. Simplify the expression. | $\mathrm{-45}$ |
Step 6. Answer the question with a complete sentence. | The team lost 45 yards. |
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?
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?
Section 1.4 Exercises
Practice Makes Perfect
Multiply Integers
In the following exercises, multiply.
$\mathrm{-3}\xb79$
$13\left(\mathrm{-5}\right)$
$\mathrm{-1.3}$
$\mathrm{-1}\left(\mathrm{-19}\right)$
Divide Integers
In the following exercises, divide.
$35\xf7\left(\mathrm{-7}\right)$
$\mathrm{-84}\xf7\left(\mathrm{-6}\right)$
$\mathrm{-192}\xf712$
Simplify Expressions with Integers
In the following exercises, simplify each expression.
$8\left(\mathrm{-4}\right)+5\left(\mathrm{-4}\right)-6$
${\left(\mathrm{-3}\right)}^{5}$
$\text{\u2212}{6}^{2}$
$\mathrm{-4}\left(\mathrm{-6}\right)\left(3\right)$
$\left(6-11\right)\left(8-13\right)$
$23-2\left(4-6\right)$
$52\xf7\left(\mathrm{-4}\right)+\left(\mathrm{-32}\right)\xf7\left(\mathrm{-8}\right)$
$11-3\left[7-4\left(\mathrm{-2}\right)\right]$
${\left(\mathrm{-4}\right)}^{2}-32\xf7\left(12-4\right)$
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
$x+\left(\mathrm{-21}\right)$ whenⓐ $x=\mathrm{-27}$ⓑ $x=44$
- ⓐ $d+\left(\mathrm{-9}\right)$ when $d=\mathrm{-8}$
- ⓑ $\text{\u2212}d+\left(\mathrm{-9}\right)$ when $d=\mathrm{-8}$
$p+q$ when
$p=\mathrm{-9},q=17$
$t+u$ when $t=\mathrm{-6},u=\mathrm{-5}$
${\left(y+z\right)}^{2}$ when
$y=\mathrm{-3},z=15$
$\mathrm{-5}y+14$ when
ⓐ $y=9$
ⓑ $y=\mathrm{-9}$
$18-4n$ when
ⓐ $n=3$
ⓑ $n=\mathrm{-3}$
$3{u}^{2}-4u+5$ when $u=\mathrm{-3}$
$9a-2b-8$ when
$a=\mathrm{-6}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=\mathrm{-3}$
$7m-4n-2$ when
$m=\mathrm{-4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=\mathrm{-9}$
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
the sum of $\mathrm{-8}$ and $\mathrm{-9},$ increased by 23
subtract 11 from $\mathrm{-25}$
subtract $\mathrm{-6}$ from $\mathrm{-13}$
the product of $\text{\u22124 and 16}\phantom{\rule{0.2em}{0ex}}$
the quotient of $\mathrm{-40}$ and $\mathrm{-20}$
the quotient of $\mathrm{-7}$ and the sum of m and n
the product of $\mathrm{-10}$ and the difference of $p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q$
the product of $\mathrm{-13}$ and the difference of $c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$
Use Integers in Applications
In the following exercises, solve.
Temperature On January $15,$ the high temperature in Anaheim, California, was $84\text{\xb0}.$ That same day, the high temperature in Embarrass, Minnesota was $\mathrm{-12}\text{\xb0}.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?
Temperature On January $21,$ the high temperature in Palm Springs, California, was $89\text{\xb0},$ and the high temperature in Whitefield, New Hampshire was $\mathrm{-31}\text{\xb0}.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?
Football On the first down, the Chargers had the ball on their 25-yard line. They lost 6 yards on the first-down play, gained 10 yards on the second-down play, and lost 8 yards on the third-down play. What was the yard line at the end of the third-down play?
Football On first down, the Steelers had the ball on their 30-yard line. They gained 9 yards on the first-down play, lost 14 yards on the second-down play, and lost 2 yards on the third-down play. What was the yard line at the end of the third-down play?
Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?
Checking Account Selina has $165 in her checking account. She writes a check for $207. What is the new balance in her checking account?
Checking Account Diontre has a balance of $\text{\u2212}\mathrm{\$38}$ in his checking account. He deposits $225 to the account. What is the new balance?
Checking Account Reymonte has a balance of $\text{\u2212}\mathrm{\$49}$ in his checking account. He deposits $281 to the account. What is the new balance?
Everyday Math
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?
Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?
Writing Exercises
In your own words, state the rules for dividing integers.
Why is $\text{\u2212}{4}^{3}={\left(\mathrm{-4}\right)}^{3}?$
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?