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$\text{Simplify:}\phantom{\rule{0.2em}{0ex}}\mathrm{-30}\xf72+(\mathrm{-3})(\mathrm{-7}).$
First we will multiply and divide from left to right. Then we will add.
$\mathrm{-30}\xf72+(\mathrm{-3})(\mathrm{-7})$ | |
Divide. | $\mathrm{-15}+(\mathrm{-3})(\mathrm{-7})$ |
Multiply. | $\mathrm{-15}+21$ |
Add. | $\text{6}$ |
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.
$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}-3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=\mathrm{-4}.$
Simplify exponents. | |
Multiply. | |
Subtract. | |
Add. |
Keep in mind that when we substitute $\mathrm{-4}$ for $x,$ we use parentheses to show the multiplication. Without parentheses, it would look like $2\xb7{\mathrm{-4}}^{2}-3\xb7\mathrm{-4}+8.$
Evaluate:
$3{x}^{2}-2x+6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=\mathrm{-3}$
39
Evaluate:
$4{x}^{2}-x-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=\mathrm{-2}$
13
$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=\mathrm{-1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2.$
Substitute $x=\mathrm{-1}$ and $y=2$ . | |
Multiply. | |
Simplify. |
Evaluate:
$7x+6y-12\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=\mathrm{-2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$
−8
Evaluate:
$8x-6y+13\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=\mathrm{-3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\mathrm{-5}$
19
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 .
Translate to an algebraic expression and simplify if possible: the product of $\mathrm{-2}$ and $14.$
The word product tells us to multiply.
the product of $\mathrm{-2}$ and $14$ | |
Translate. | $\left(\mathrm{-2}\right)\left(14\right)$ |
Simplify. | $\mathrm{-28}$ |
Translate to an algebraic expression and simplify if possible:
$\text{the product of \u22125 and 12}$
−5 (12) = −60
Translate to an algebraic expression and simplify if possible:
$\text{the product of 8 and \u221213}$
8 (−13) = −104
Translate to an algebraic expression and simplify if possible: the quotient of $\mathrm{-56}$ and $\mathrm{-7}.$
The word quotient tells us to divide.
the quotient of −56 and −7 | |
Translate. | $\mathrm{-56}\xf7\left(\mathrm{-7}\right)$ |
Simplify. | $8$ |
Translate to an algebraic expression and simplify if possible:
$\text{the quotient of \u221263 and \u22129}$
−63 ÷ −9 = 7
Translate to an algebraic expression and simplify if possible:
$\text{the quotient of \u221272 and \u22129}$
−72 ÷ −9 = 8
Same signs | Product |
---|---|
•Two positives
•Two negatives |
Positive
Positive |
Different signs | Product |
---|---|
•Positive • negative
•Negative • positive |
Negative
Negative |
Multiply Integers
In the following exercises, multiply each pair of integers.
$\mathrm{-3}\xb79$
$\mathrm{-8}(6)$
$\mathrm{-10}(\mathrm{-6})$
$13(\mathrm{-5})$
$\mathrm{-1}\xb73$
$\mathrm{-1}(\mathrm{-19})$
Divide Integers
In the following exercises, divide.
$\mathrm{-28}\xf77$
$35\xf7(\mathrm{-7})$
$\mathrm{-84}\xf7(\mathrm{-6})$
$\mathrm{-192}\xf712$
$62\xf7(\mathrm{-1})$
Simplify Expressions with Integers
In the following exercises, simplify each expression.
$8(\mathrm{-4})+5(\mathrm{-4})\mathrm{-6}$
$\mathrm{-7}(\mathrm{-4})\mathrm{-5}(\mathrm{-3})$
${(\mathrm{-4})}^{3}$
${(\mathrm{-3})}^{5}$
$-{6}^{2}$
$\mathrm{-4}(\mathrm{-6})(3)$
$\mathrm{-5}\xb73\xb710$
$(6-11)(8-13)$
$23-2(4-6)$
$\mathrm{-8}(\mathrm{-6})\xf7(\mathrm{-4})$
$52\xf7(\mathrm{-4})+(\mathrm{-32})\xf7(\mathrm{-8})$
$11-3[7-4(\mathrm{-2})]$
${(\mathrm{-4})}^{2}-32\xf7(12-4)$
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
$\mathrm{-2}x+17\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$
$\mathrm{-5}y+14\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$
$10-3m\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$
$18-4n\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$
${p}^{2}-5p+5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=\mathrm{-1}$
8
${q}^{2}-2q+9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=\mathrm{-2}$
$2{w}^{2}-3w+7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}w=\mathrm{-2}$
21
$3{u}^{2}-4u+5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=\mathrm{-3}$
$6x-5y+15\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\mathrm{-1}$
38
$3p-2q+9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q=\mathrm{-2}$
$9a-2b-8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=\mathrm{-6}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=\mathrm{-3}$
−56
$7m-4n-2\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=\mathrm{-4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=\mathrm{-9}$
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
The product of $\mathrm{-4}$ and $16$
The quotient of $\mathrm{-60}$ and $\mathrm{-20}$
−60 ÷ (−20) = 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 $p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q$
−10 ( p − q )
The product of $\mathrm{-13}$ and the difference of $c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$
Stock market Javier owns $300$ shares of stock in one company. On Tuesday, the stock price dropped $\text{\$12}$ per share. What was the total effect on Javier’s portfolio?
−$3,600
Weight loss In the first week of a diet program, eight women lost an average of $\text{3 pounds}$ each. What was the total weight change for the eight women?
In your own words, state the rules for multiplying two integers.
Sample answer: Multiplying two integers with the same sign results in a positive product. Multiplying two integers with different signs results in a negative product.
In your own words, state the rules for dividing two integers.
Why is ${\mathrm{-2}}^{4}\ne {(\mathrm{-2})}^{4}?$
Sample answer: In one expression the base is positive and then we take the opposite, but in the other the base is negative.
Why is ${\mathrm{-4}}^{2}\ne {\left(\mathrm{-4}\right)}^{2}?$
ⓐ 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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