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The temperature in Anchorage, Alaska one morning 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 difference in temperatures was 45 degrees.
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?
The difference in temperatures was 9 degrees.
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?
$\begin{array}{cccc}\begin{array}{c}\mathbf{\text{Step 1.}}\phantom{\rule{0.2em}{0ex}}\text{Read the problem. Make sure all the words and}\hfill \\ \text{ideas are understood.}\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 2.}}\phantom{\rule{0.2em}{0ex}}\text{Identify what we are asked to find.}\hfill & & & & & \text{the number of yards lost}\hfill \\ \mathbf{\text{Step 3.}}\phantom{\rule{0.2em}{0ex}}\text{Write a phrase that gives the information to find it.}\hfill & & & & & \text{three times a 15-yard penalty}\hfill \\ \mathbf{\text{Step 4.}}\phantom{\rule{0.2em}{0ex}}\text{Translate the phrase to an expression.}\hfill & & & & & 3\left(\mathrm{-15}\right)\hfill \\ \mathbf{\text{Step 5.}}\phantom{\rule{0.2em}{0ex}}\text{Simplify the expression.}\hfill & & & & & \mathrm{-45}\hfill \\ \mathbf{\text{Step 6.}}\phantom{\rule{0.2em}{0ex}}\text{Answer the question with a complete sentence.}\hfill & & & & & \text{The team lost}\phantom{\rule{0.2em}{0ex}}45\phantom{\rule{0.2em}{0ex}}\text{yards.}\hfill \end{array}$
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?
The Bears lost 105 yards.
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?
A $16 fee was deducted from his checking account.
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.
$5\left(\mathrm{-6}\right)+7\left(\mathrm{-2}\right)-3$
$\mathrm{-47}$
$8\left(\mathrm{-4}\right)+5\left(\mathrm{-4}\right)-6$
${\left(\mathrm{-3}\right)}^{5}$
$\text{\u2212}{6}^{2}$
$\mathrm{-4}\left(\mathrm{-6}\right)\left(3\right)$
$\left(6-11\right)\left(8-13\right)$
$23-2\left(4-6\right)$
$65\xf7\left(\mathrm{-5}\right)+\left(\mathrm{-28}\right)\xf7\left(\mathrm{-7}\right)$
$\mathrm{-9}$
$52\xf7\left(\mathrm{-4}\right)+\left(\mathrm{-32}\right)\xf7\left(\mathrm{-8}\right)$
$9-2\left[3-8\left(\mathrm{-2}\right)\right]$
$\mathrm{-29}$
$11-3\left[7-4\left(\mathrm{-2}\right)\right]$
${\left(\mathrm{-4}\right)}^{2}-32\xf7\left(12-4\right)$
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
$y+\left(\mathrm{-14}\right)$ when
ⓐ
$y=\mathrm{-33}$
ⓑ
$y=30$
ⓐ $\mathrm{-47}$ ⓑ 16
$x+\left(\mathrm{-21}\right)$ when
ⓐ
$x=\mathrm{-27}$
ⓑ
$x=44$
ⓐ
$a+3$ when
$a=\mathrm{-7}$
ⓑ
$\text{\u2212}a+3$ when
$a=\mathrm{-7}$
ⓐ $\mathrm{-4}$ ⓑ 10
ⓐ
$d+\left(\mathrm{-9}\right)$ 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
$a=\mathrm{-6}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=\mathrm{-3}$
$\mathrm{-56}$
$7m-4n-2$ 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 3 and $\mathrm{-15},$ increased by 7
$\left(3+\left(\mathrm{-15}\right)\right)+7;-5$
the sum of $\mathrm{-8}$ and $\mathrm{-9},$ increased by 23
the difference of 10 and $\mathrm{-18}$
$10-\left(\mathrm{-18}\right);28$
subtract 11 from $\mathrm{-25}$
the difference of $\mathrm{-5}$ and $\mathrm{-30}$
$\mathrm{-5}-\left(\mathrm{-30}\right);25$
subtract $\mathrm{-6}$ from $\mathrm{-13}$
the product of $\text{\u22123 and 15}$
$\mathrm{-3}\xb715;-45$
the product of $\text{\u22124 and 16}\phantom{\rule{0.2em}{0ex}}$
the quotient of $\mathrm{-60}$ and $\mathrm{-20}$
$\mathrm{-60}\xf7\left(\mathrm{-20}\right);3$
the quotient of $\mathrm{-40}$ and $\mathrm{-20}$
the quotient of $\mathrm{-6}$ and the sum of a and b
$\frac{\mathrm{-6}}{a+b}$
the quotient of $\mathrm{-7}$ and the sum of m and n
the product of $\mathrm{-10}$ and the difference of $p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q$
$\mathrm{-10}\left(p-q\right)$
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?
$96\text{\xb0}$
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 At the first down, the Chargers had the ball on their 25 yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down?
21
Football At the first down, the Steelers had the ball on their 30 yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down?
Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?
$\text{\u2212}\mathrm{\$28}$
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?
$187
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?
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?
$\text{\u2212}\mathrm{\$3600}$
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?
In your own words, state the rules for multiplying integers.
Answers may vary
In your own words, state the rules for dividing integers.
Why is $\text{\u2212}{2}^{4}\ne {\left(\mathrm{-2}\right)}^{4}?$
Answers may vary
Why is $\text{\u2212}{4}^{3}={\left(\mathrm{-4}\right)}^{3}?$
ⓐ 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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