Foundations

5 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 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·b means add a, b times. Here, we are using the model just to help us discover the pattern.

Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”

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.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

\begin{array}{cccccccc}\hfill 5·3& =\hfill & 15\hfill & & & \hfill -5\left(3\right)& =\hfill & -15\hfill \\ \hfill 5\left(-3\right)& =\hfill & -15\hfill & & & \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\hfill \end{array}

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 7·4& =\hfill & 28\hfill \\ \hfill -8\left(-6\right)& =\hfill & 48\hfill \end{array}
Different signs Product Example
Positive · negative
Negative · positive
Negative
Negative
\begin{array}{ccc}\hfill 7\left(-9\right)& =\hfill & -63\hfill \\ \hfill -5·10& =\hfill & -50\hfill \end{array}

Multiply: -9·3 -2\left(-5\right) 4\left(-8\right) 7·6.

Solution

Multiply, noting that the signs are different so the product is negative.
\begin{array}{c}-9·3\\ -27\end{array}

Multiply, noting that the signs are the same so the product is positive.
\begin{array}{c}-2\left(-5\right)\\ 10\end{array}

Multiply, with different signs.
\begin{array}{c}4\left(-8\right)\\ -32\end{array}

Multiply, with same signs.
\begin{array}{c}7·6\\ 42\end{array}

Multiply: -6·8 -4\left(-7\right) 9\left(-7\right) 5·12.

-48 28 -63 60

Multiply: -8·7 -6\left(-9\right) 7\left(-4\right) 3·13.

-56 54 -28 39

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.

\begin{array}{ccccccc}& & & \hfill -1·4\hfill & & & \hfill -1\left(-3\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill -4\hfill & & & \hfill 3\hfill \\ & & & \hfill -4\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}4.\hfill & & & \hfill 3\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-3.\hfill \end{array}

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

Multiplication by -1
-1a=\text{−}a

Multiplying a number by -1 gives its opposite.

Multiply: -1·7 -1\left(-11\right).

Solution

Multiply, noting that the signs are different so the product is negative.
\begin{array}{c}-1·7\\ -7\\ -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}-1\left(-11\right)\\ 11\\ 11\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-11.\end{array}

Multiply: -1·9 -1·\left(-17\right).

-9 17

Multiply: -1·8 -1·\left(-16\right).

-8 16

Divide Integers

What about division? Division is the inverse operation of multiplication. So, 15÷3=5 because 15·3=5. 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.

\begin{array}{cccccccccccccc}\hfill 5·3& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15÷3\hfill & =\hfill & 5\hfill & & & & & \hfill -5\left(3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15÷3\hfill & =\hfill & -5\hfill \\ \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15÷\left(-3\right)\hfill & =\hfill & -5\hfill & & & & & \hfill 5\left(-3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15÷\left(-3\right)\hfill & =\hfill & 5\hfill \end{array}

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.

Divide: -27÷3 -100÷\left(-4\right).

Solution

Divide. With different signs, the quotient is negative.
\begin{array}{c}-27÷3\\ -9\end{array}

Divide. With signs that are the same, the quotient is positive.
\begin{array}{c}-100÷\left(-4\right)\\ 25\end{array}

Divide: -42÷6 -117÷\left(-3\right).

-7 39

Divide: -63÷7 -115÷\left(-5\right).

-9 23

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.

Simplify: 7\left(-2\right)+4\left(-7\right)-6.

Solution
7\left(-2\right)+4\left(-7\right)-6
Multiply first. -14+\left(-28\right)-6
Add. -42-6
Subtract. -48

Simplify: 8\left(-3\right)+5\left(-7\right)-4.

-63

Simplify: 9\left(-3\right)+7\left(-8\right)-1.

-84

Simplify: {\left(-2\right)}^{4} \text{−}{2}^{4}.

Solution

Write in expanded form.
Multiply.
Multiply.
Multiply.
\begin{array}{c}{\left(-2\right)}^{4}\\ \left(-2\right)\left(-2\right)\left(-2\right)\left(-2\right)\\ 4\left(-2\right)\left(-2\right)\\ -8\left(-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{−}{2}^{4}\\ \text{−}\left(2·2·2·2\right)\\ \text{−}\left(4·2·2\right)\\ \text{−}\left(8·2\right)\\ 16\end{array}

Notice the difference in parts and . In part , the exponent means to raise what is in the parentheses, the \left(-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(-3\right)}^{4} \text{−}{3}^{4}.

81 -81

Simplify: {\left(-7\right)}^{2} \text{−}{7}^{2}.

49 -49

The next example reminds us to simplify inside parentheses first.

Simplify: 12-3\left(9-12\right).

Solution
12-3\left(9-12\right)
Subtract in parentheses first. 12-3\left(-3\right)
Multiply. 12-\left(-9\right)
Subtract. 21

Simplify: 17-4\left(8-11\right).

29

Simplify: 16-6\left(7-13\right).

52

Simplify: 8\left(-9\right)÷{\left(-2\right)}^{3}.

Solution
8\left(-9\right)÷{\left(-2\right)}^{3}
Exponents first. 8\left(-9\right)÷\left(-8\right)
Multiply. -72÷\left(-8\right)
Divide. 9

Simplify: 12\left(-9\right)÷{\left(-3\right)}^{3}.

4

Simplify: 18\left(-4\right)÷{\left(-2\right)}^{3}.

9

Simplify: -30÷2+\left(-3\right)\left(-7\right).

Solution
-30÷2+\left(-3\right)\left(-7\right)
Multiply and divide left to right, so divide first. -15+\left(-3\right)\left(-7\right)
Multiply. -15+21
Add. 6

Simplify: -27÷3+\left(-5\right)\left(-6\right).

21

Simplify: -32÷4+\left(-2\right)\left(-7\right).

6

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.

When n=-5, evaluate: n+1 \text{−}n+1.

Solution


.
. .
Simplify. −4


.
. .
Simplify. .
Add. 6

When n=-8, evaluate n+2 \text{−}n+2.

-6 10

When y=-9, evaluate y+8 \text{−}y+8.

-1 17

Evaluate {\left(x+y\right)}^{2} when x=-18 and y=24.

Solution
.
. .
Add inside parenthesis. (6)2
Simplify. 36

Evaluate {\left(x+y\right)}^{2} when x=-15 and y=29.

196

Evaluate {\left(x+y\right)}^{3} when x=-8 and y=10.

8

Evaluate 20-z when z=12 and z=-12.

Solution


.
. .
Subtract. 8




.
. .
Subtract. 32

Evaluate: 17-k when k=19 and k=-19.

-2 36

Evaluate: -5-b when b=14 and b=-14.

-19 9

Evaluate: 2{x}^{2}+3x+8 when x=4.

Solution

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=-3.

39

Evaluate: 4{x}^{2}-x-5 when x=-2.

13

Translate Phrases to Expressions with Integers

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

Translate and simplify: the sum of 8 and -12, increased by 3.

Solution
the sum of 8 and -12, increased by 3.
Translate. \left[8+\left(-12\right)\right]+3
Simplify. Be careful not to confuse the brackets with an absolute value sign. \left(-4\right)+3
Add. -1

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

\left(9+\left(-16\right)\right)+4-3

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

\left(-8+\left(-12\right)\right)+7-13

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!

Translate and then simplify the difference of 13 and -21 subtract 24 from -19.

Solution

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(-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 -23 subtract 21 from -17.

14-\left(-23\right);37-17-21;-38

Translate and simplify the difference of 11 and -19 subtract 18 from -11.

11-\left(-19\right);30-11-18;-29

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.”

Translate to an algebraic expression and simplify if possible: the product of -2 and 14.

Solution
\text{the product}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}-2\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}14
Translate. \left(-2\right)\left(14\right)
Simplify. -28

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

-5\left(12\right);-60

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

-8\left(13\right);-104

Translate to an algebraic expression and simplify if possible: the quotient of -56 and -7.

Solution
\text{the quotient}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}-56\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}-7
Translate. -56÷\left(-7\right)
Simplify. 8

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

-63÷\left(-9\right);7

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

-72÷\left(-9\right);8

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.

How to Apply a Strategy to Solve Applications with Integers

The temperature in Urbana, Illinois one morning was 11 degrees. By mid-afternoon, the temperature had dropped to -9 degrees. What was the difference of the morning and afternoon temperatures?

Solution

This is a table with two columns. The left column includes steps to solve the problem. The right column includes the math to solve the problem. In the first row, the left column says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The right column is blank.In the second row, the left column says “Step 2. Identify what we are asked to find”. The right column says, “the difference of the morning and afternoon temperatures.”In the third row, the left column says, “Step 3. Write a phrase that gives the information to find it.” Next to this in the right column, it says “the difference of 11 and negative 9.”In the fourth row, the left column says, “Step 4. Translate the phrase to an expression.” The right column contains 11 minus negative 9.In the fifth row, the left column says, “Step 5. Simplify the expression.” The right column contains 20.The final row says, “Step five. Write a complete sentence that answers the question.” Next to this in the right column, it says “the difference in temperatures was 20 degrees.”

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

The difference in temperatures was 9 degrees.

Apply a Strategy to Solve Applications with Integers.
  1. Read the problem. Make sure all the words and ideas are understood
  2. Identify what we are asked to find.
  3. Write a phrase that gives the information to find it.
  4. Translate the phrase to an expression.
  5. Simplify the expression.
  6. Answer the question with a complete sentence.

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?

Solution
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(-15\right)
Step 5. Simplify the expression. -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?

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.

Key Concepts

  • Multiplication and Division of Two Signed Numbers
    • Same signs—Product is positive
    • Different signs—Product is negative
  • Strategy for Applications
    1. Identify what you are asked to find.
    2. Write a phrase that gives the information to find it.
    3. Translate the phrase to an expression.
    4. Simplify the expression.
    5. Answer the question with a complete sentence.

Practice Makes Perfect

Multiply Integers

In the following exercises, multiply.

-4·8

-32

-3·9

9\left(-7\right)

-63

13\left(-5\right)

-1.6

-6

-1.3

-1\left(-14\right)

14

-1\left(-19\right)

Divide Integers

In the following exercises, divide.

-24÷6

-4

35÷\left(-7\right)

-52÷\left(-4\right)

13

-84÷\left(-6\right)

-180÷15

-12

-192÷12

Simplify Expressions with Integers

In the following exercises, simplify each expression.

5\left(-6\right)+7\left(-2\right)-3

-47

8\left(-4\right)+5\left(-4\right)-6

{\left(-2\right)}^{6}

64

{\left(-3\right)}^{5}

\text{−}{4}^{2}

-16

\text{−}{6}^{2}

-3\left(-5\right)\left(6\right)

90

-4\left(-6\right)\left(3\right)

\left(8-11\right)\left(9-12\right)

9

\left(6-11\right)\left(8-13\right)

26-3\left(2-7\right)

41

23-2\left(4-6\right)

65÷\left(-5\right)+\left(-28\right)÷\left(-7\right)

-9

52÷\left(-4\right)+\left(-32\right)÷\left(-8\right)

9-2\left[3-8\left(-2\right)\right]

-29

11-3\left[7-4\left(-2\right)\right]

{\left(-3\right)}^{2}-24÷\left(8-2\right)

5

{\left(-4\right)}^{2}-32÷\left(12-4\right)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

y+\left(-14\right) when
y=-33
y=30

-47 16

x+\left(-21\right) when
x=-27
x=44


a+3 when a=-7
\text{−}a+3 when a=-7

-4 10


d+\left(-9\right) when d=-8
\text{−}d+\left(-9\right) when d=-8

m+n when
m=-15,n=7

-8

p+q when
p=-9,q=17

r+s when r=-9,s=-7

-16

t+u when t=-6,u=-5

{\left(x+y\right)}^{2} when
x=-3,y=14

121

{\left(y+z\right)}^{2} when
y=-3,z=15

-2x+17 when
x=8
x=-8

1 33

-5y+14 when
y=9
y=-9

10-3m when
m=5
m=-5

-5 25

18-4n when
n=3
n=-3

2{w}^{2}-3w+7 when
w=-2

21

3{u}^{2}-4u+5 when u=-3

9a-2b-8 when
a=-6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=-3

-56

7m-4n-2 when
m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=-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 -15, increased by 7

\left(3+\left(-15\right)\right)+7;-5

the sum of -8 and -9, increased by 23

the difference of 10 and -18

10-\left(-18\right);28

subtract 11 from -25

the difference of -5 and -30

-5-\left(-30\right);25

subtract -6 from -13

the product of \text{−3 and 15}

-3·15;-45

the product of \text{−4 and 16}\phantom{\rule{0.2em}{0ex}}

the quotient of -60 and -20

-60÷\left(-20\right);3

the quotient of -40 and -20

the quotient of -6 and the sum of a and b

\frac{-6}{a+b}

the quotient of -7 and the sum of m and n

the product of -10 and the difference of p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q

-10\left(p-q\right)

the product of -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{°}. That same day, the high temperature in Embarrass, Minnesota was -12\text{°}. What was the difference between the temperature in Anaheim and the temperature in Embarrass?

96\text{°}

Temperature On January 21, the high temperature in Palm Springs, California, was 89\text{°}, and the high temperature in Whitefield, New Hampshire was -31\text{°}. 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{−}?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{−}?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{−}?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?

\text{−}?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?

Writing Exercises

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{−}{2}^{4}\ne {\left(-2\right)}^{4}?

Answers may vary

Why is \text{−}{4}^{3}={\left(-4\right)}^{3}?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A table is shown that is composed of four columns and seven rows. The titles of the columns are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The first column reads “multiple integers.”, “divide integers.”, “simplify expressions with integers.”, “evaluate variable expressions with integers.”, “translate English phrases to algebraic expressions.” and “use integers in applications.”

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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Elementary Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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