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Subtract Integers

Model subtraction of integers

Model Subtraction of Integers

Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.

We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read $5-3$ as five take away three. When we use counters, we can think of subtraction the same way.

We will model four subtraction facts using the numbers $5$ and $3.$

\[5-3\ -5-(-3)\ -5-3\ 5-(-3)\]

Example

Try it.

Model: $5-3.$

Solution

Interpret the expression.	$5-3$ means $5$ take away $3$.
Model the first number. Start with 5 positives.
Take away the second number. So take away 3 positives.
Find the counters that are left.
	$5-3=2$. The difference between $5$ and $3$ is $2$.

Example

Try it.

Model: $-5-(-3)\text{.}$

Solution

Interpret the expression.	$-5-(-3)$ means $-5$ take away $-3$.
Model the first number. Start with 5 negatives.
Take away the second number. So take away 3 negatives.
Find the number of counters that are left.
	$-5-(-3)=-2$. The difference between $-5$ and $-3$ is $-2$.

Notice that and are very much alike.

First, we subtracted $3$ positives from $5$ positives to get $2$ positives.
Then we subtracted $3$ negatives from $5$ negatives to get $2$ negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

Example

Try it.

Model: $-5-3.$

Solution

Interpret the expression.	$-5-3$ means $-5$ take away $3$.
Model the first number. Start with 5 negatives.
Take away the second number. So we need to take away 3 positives.
But there are no positives to take away. Add neutral pairs until you have 3 positives.
Now take away 3 positives.
Count the number of counters that are left.
	$-5-3=-8$. The difference of $-5$ and $3$ is $-8$.

Condensed — the full section is in OpenStax Prealgebra 2e.

Simplify Expressions with Integers

Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.

Subtract $-23-7.$
Think: We start with $23$ negative counters.
We have to subtract $7$ positives, but there are no positives to take away.
So we add $7$ neutral pairs to get the $7$ positives. Now we take away the $7$ positives.
So what’s left? We have the original $23$ negatives plus $7$ more negatives from the neutral pair. The result is $30$ negatives.
\[-23-7=-30\]
Notice, that to subtract $\text{7,}$ we added $7$ negatives.
Subtract $30-(-12).$
Think: We start with $30$ positives.
We have to subtract $12$ negatives, but there are no negatives to take away.
So we add $12$ neutral pairs to the $30$ positives. Now we take away the $12$ negatives.
What’s left? We have the original $30$ positives plus $12$ more positives from the neutral pairs. The result is $42$ positives.
\[30-(-12)=42\]
Notice that to subtract $-12,$ we added $12.$

While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:

Look at these two examples.

We see that $6-4$ gives the same answer as $6+(-4).$

Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract $6-4$ long ago. But knowing that $6-4$ gives the same answer as $6+(-4)$ helps when we are subtracting negative numbers.

Example

Try it.

Simplify:

ⓐ $\ 13-8\ \text{and}\ 13+(-8)\$
ⓑ $\ -17-9\ \text{and}\ -17+(-9)$

Solution

ⓐ
	$13-8$ and $13+(-8)$
Subtract to simplify.	$13-8=5$
Add to simplify.	$13+(-8)=5$
Subtracting 8 from 13 is the same as adding −8 to 13.

ⓑ
	$-17-9$ and $-17+(-9)$
Subtract to simplify.	$-17-9=-26$
Add to simplify.	$-17+(-9)=-26$
Subtracting 9 from −17 is the same as adding −9 to −17.

Now look what happens when we subtract a negative.

Condensed — the full section is in OpenStax Prealgebra 2e.

Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

Example

Try it.

Evaluate $x-4\ \text{when}\$

ⓐ $\ x=3\$
ⓑ $\ x=-6.$

Solution

ⓐ To evaluate $x-4$ when $x=3$, substitute $3$ for $x$ in the expression.



Subtract.

ⓑ To evaluate $x-4$ when $x=-6,$ substitute $-6$ for $x$ in the expression.



Subtract.

Example

Try it.

Evaluate $20-z\ \text{when}\$

ⓐ $\ z=12\$
ⓑ $\ z=-12$

Solution

ⓐ To evaluate $20-z\ \text{when}\ z=12,$ substitute $12$ for $z$ in the expression.



Subtract.

ⓑ To evaluate $20-z\ \text{when}\ z=-12,\ \text{substitute}\ -12\ \text{for}\ z\ \text{in the expression.}$



Subtract.

Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression $a-b$ may be read in several ways as shown below.

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

Example

Try it.

Translate and then simplify:

ⓐ the difference of $13$ and $-21$
ⓑ subtract $24$ from $-19$

Solution

ⓐ A difference means subtraction. Subtract the numbers in the order they are given.


Translate.
Simplify.

ⓑ Subtract means to take $24$ away from $-19.$


Translate.
Simplify.

Subtract Integers in 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 problem, we first need to determine what we are asked to find. Then we can 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

Try it.

In the morning, the temperature in Urbana, Illinois was $11$ degrees Fahrenheit. By mid-afternoon, the temperature had dropped to $-9$ degrees Fahrenheit. What was the difference between the morning and afternoon temperatures?

Solution

Step 1. Identify what we are asked to find.	the difference between the morning and afternoon temperatures
Step 2. Write a phrase that gives the information to find it.	the difference of $11$ and $-9$
Step 3. Translate the phrase to an expression. The word difference indicates subtraction.	$11-(-9)$
Step 4. Simplify the expression.	$20$
Step 5. Write a complete sentence that answers the question.	The difference in temperature was $20$ degrees Fahrenheit.

Geography provides another application of negative numbers with the elevations of places below sea level.

Example

Try it.

Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is $14,497$ feet above sea level and the elevation of Death Valley is $282$ feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?

Solution

Step 1. What are we asked to find?	The difference in elevation between Mt. Whitney and Death Valley
Step 2. Write a phrase.	elevation of Mt. Whitney−elevation of Death Valley
Step 3. Translate.	$14,497-(-282)$
Step 4. Simplify.	$14,779$
Step 5. Write a complete sentence that answers the question.	The difference in elevation is $14,779$ feet.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Subtraction of Integers

$5-3$	$-5-(-3)$
$2$	$-2$
2 positives	2 negatives
When there would be enough counters of the color to take away, subtract.
$-5-3$	$5-(-3)$
$-8$	$8$
5 negatives, want to subtract 3 positives	5 positives, want to subtract 3 negatives
need neutral pairs	need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.

Subtraction Property
- $a-b=a+(\text{-b})$
- $a-(-b)=a+b$
Solve Application Problems
- Step 1. Identify what you are asked to find.
- Step 2. Write a phrase that gives the information to find it.
- Step 3. Translate the phrase to an expression.
- Step 4. Simplify the expression.
- Step 5. Answer the question with a complete sentence.

Subtract Integers

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

Try it.

$8-2$

Solution

Try it.

$9-3$

Try it.

$-5-(-1)$

Solution

−4

Try it.

$-6-(-4)$

Try it.

$-5-4$

Solution

−9

Try it.

$-7-2$

Try it.

$8-(-4)$

Solution

Try it.

$7-(-3)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Try it.

ⓐ $\ 15-6\$
ⓑ $\ 15+(-6)$

Solution

ⓐ 9
ⓑ 9

Try it.

ⓐ $\ 12-9\$
ⓑ $\ 12+(-9)$

Try it.

ⓐ $\ 44-28\$
ⓑ $\ 44+(-28)$

Solution

ⓐ 16
ⓑ 16

Try it.

ⓐ $\ 35-16\$
ⓑ $35+(-16)$

Try it.

ⓐ $\ 8-(-9)\$
ⓑ $8+9$

Solution

ⓐ 17
ⓑ 17

Try it.

ⓐ $\ 4-(-4)\$
ⓑ $\ 4+4$

Try it.

ⓐ $\ 27-(-18)\$
ⓑ $\ 27+18$

Solution

ⓐ 45
ⓑ 45

Try it.

ⓐ $\ 46-(-37)\$
ⓑ $\ 46+37$

In the following exercises, simplify each expression.

Try it.

$15-(-12)$

Solution

Try it.

$14-(-11)$

Try it.

$10-(-19)$

Solution

Try it.

$11-(-18)$

Try it.

$48-87$

Solution

−39

Try it.

$45-69$

Try it.

$31-79$

Solution

−48

Try it.

$39-81$

Try it.

$-31-11$

Solution

−42

Try it.

$-32-18$

Try it.

$-17-42$

Solution

−59

Try it.

$-19-46$

Try it.

$-103-(-52)$

Solution

−51

Try it.

$-105-(-68)$

Try it.

$-45-(-54)$

Solution

Try it.

$-58-(-67)$

Try it.

$8-3-7$

Solution

−2

Try it.

$9-6-5$

Try it.

$-5-4+7$

Solution

−2

Try it.

$-3-8+4$

Try it.

$-14-(-27)+9$

Solution

Try it.

$-15-(-28)+5$

Try it.

$71+(-10)-8$

Solution

Try it.

$64+(-17)-9$

Try it.

$-16-(-4+1)-7$

Solution

−20

Try it.

$-15-(-6+4)-3$

Try it.

$(2-7)-(3-8)$

Solution

Try it.

$(1-8)-(2-9)$

Try it.

$-(6-8)-(2-4)$

Solution

Try it.

$-(4-5)-(7-8)$

Try it.

$25-[10-(3-12)]$

Solution

Try it.

$32-[5-(15-20)]$

Try it.

$6⋅3-4⋅3-7⋅2$

Solution

–8

Try it.

$5⋅7-8⋅2-4⋅9$

Try it.

${5}^{2}-{6}^{2}$

Solution

−11

Try it.

${6}^{2}-{7}^{2}$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression for the given values.

Try it.

$x-6\ \text{when}\$

ⓐ $\ x=3\$
ⓑ $\ x=-3$

Solution

ⓐ −3
ⓑ −9

Try it.

$x-4\ \text{when}\$

ⓐ $\ x=5\$
ⓑ $\ x=-5$

Try it.

$5-y\ \text{when}\$

ⓐ $\ y=2\$
ⓑ $\ y=-2$

Solution

ⓐ 3
ⓑ 7

Try it.

$8-y\ \text{when}\$

ⓐ $\ y=3\$
ⓑ $\ y=-3$

Try it.

$4{x}^{2}-15x+1\ \text{when}\ x=3$

Solution

−8

Try it.

$5{x}^{2}-14x+7\ \text{when}\ x=2$

Try it.

$-12-5{x}^{2}\ \text{when}\ x=6$

Solution

−192

Try it.

$-19-4{x}^{2}\ \text{when}\ x=5$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

Try it.

ⓐ The difference of $3$ and $-10$
ⓑ Subtract $-20$ from $45$

Solution

ⓐ 3 − (−10) = 13
ⓑ 45 − (−20) = 65

Try it.

ⓐ The difference of $8$ and $-12$
ⓑ Subtract $-13$ from $50$

Try it.

ⓐ The difference of $-6$ and $9$
ⓑ Subtract $-12$ from $-16$

Solution

ⓐ −6 − 9 = −15
ⓑ −16 − (−12) = −4

Try it.

ⓐ The difference of $-8$ and $9$
ⓑ Subtract $-15$ from $-19$

Try it.

ⓐ $\ 8$ less than $-17$
ⓑ $\ -24$ minus $37$

Solution

ⓐ −17 − 8 = −25
ⓑ −24 − 37 = −61

Try it.

ⓐ $\ 5$ less than $-14$
ⓑ $\ -13$ minus $42$

Try it.

ⓐ $\ 21$ less than$\ 6$
ⓑ $\ 31$ subtracted from $-19$

Solution

ⓐ 6 − 21 = −15
ⓑ −19 − 31 = −50

Try it.

ⓐ $\ 34$ less than$\ 7$
ⓑ $\ 29$ subtracted from $-50$

Subtract Integers in Applications

In the following exercises, solve the following applications.

Try it.

Temperature One morning, the temperature in Urbana, Illinois, was $\text{28^{\circ} Fahrenheit.}$ By evening, the temperature had dropped $\text{38^{\circ} Fahrenheit.}$ What was the temperature that evening?

Solution

−10°

Try it.

Temperature On Thursday, the temperature in Spincich Lake, Michigan, was $\text{22^{\circ} Fahrenheit.}$ By Friday, the temperature had dropped $\text{35^{\circ} Fahrenheit.}$ What was the temperature on Friday?

Try it.

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

Solution

96°

Try it.

Temperature On January 21, the high temperature in Palm Springs, California, was $\text{89^{\circ},}$ and the high temperature in Whitefield, New Hampshire was $\text{-31^{\circ}}.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Try it.

Football At the first down, the Warriors football team had the ball on their $\text{30-yard line.}$ On the next three downs, they gained $\text{2 yards,}$ lost $\text{7 yards,}$ and lost $\text{4 yards.}$ What was the yard line at the end of the third down?

Solution

21-yard line

Try it.

Football At the first down, the Barons football team had the ball on their $\text{20-yard line.}$ On the next three downs, they lost $\text{8 yards,}$ gained $\text{5 yards,}$ and lost $\text{6 yards.}$ What was the yard line at the end of the third down?

Try it.

Checking Account John has $\text{\$148}$ in his checking account. He writes a check for $\text{\$83.}$ What is the new balance in his checking account?

Solution

$65

Try it.

Checking Account Ellie has $\text{\$426}$ in her checking account. She writes a check for $\text{\$152.}$ What is the new balance in her checking account?

Try it.

Checking Account Gina has $\text{\$210}$ in her checking account. She writes a check for $\text{\$250.}$ What is the new balance in her checking account?

Solution

−$40

Try it.

Checking Account Frank has $\text{\$94}$ in his checking account. He writes a check for $\text{\$110.}$ What is the new balance in his checking account?

Try it.

Checking Account Bill has a balance of $\text{-\$14}$ in his checking account. He deposits $\text{\$40}$ to the account. What is the new balance?

Solution

$26

Try it.

Checking Account Patty has a balance of $\text{-\$23}$ in her checking account. She deposits $\text{\$80}$ to the account. What is the new balance?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: $12-(8-1).$
If you missed this problem, review .

@ action

$5$
Translate the difference of $\text{20}$ and $\text{-15}$ into an algebraic expression.
If you missed this problem, review .

@ action

$20-(-15)$
Add: $-18+7.$
If you missed this problem, review .

@ action

$-11$

Model: $5-3.$

@ action

Interpret the expression.	$5-3$ means $5$ take away $3$.
Model the first number. Start with 5 positives.
Take away the second number. So take away 3 positives.
Find the counters that are left.
	$5-3=2$. The difference between $5$ and $3$ is $2$.

Model the expression:
$6-4$

@ action

2
Model the expression:
$7-4$

@ action

3

Model: $-5-(-3)\text{.}$

@ action

Interpret the expression.	$-5-(-3)$ means $-5$ take away $-3$.
Model the first number. Start with 5 negatives.
Take away the second number. So take away 3 negatives.
Find the number of counters that are left.
	$-5-(-3)=-2$. The difference between $-5$ and $-3$ is $-2$.

Model the expression:
$-6-(-4)$

@ action

−2
Model the expression:
$-7-(-4)$

@ action

−3

Model: $-5-3.$

@ action

Interpret the expression.	$-5-3$ means $-5$ take away $3$.
Model the first number. Start with 5 negatives.
Take away the second number. So we need to take away 3 positives.
But there are no positives to take away. Add neutral pairs until you have 3 positives.
Now take away 3 positives.
Count the number of counters that are left.
	$-5-3=-8$. The difference of $-5$ and $3$ is $-8$.

Model the expression:
$-6-4$

@ action

−10
Model the expression:
$-7-4$

@ action

−11

Model: $5-(-3).$

@ action

Interpret the expression.	$5-(-3)$ means $5$ take away $-3$.
Model the first number. Start with 5 positives.
Take away the second number, so take away 3 negatives.
But there are no negatives to take away. Add neutral pairs until you have 3 negatives.
Then take away 3 negatives.
Count the number of counters that are left.
	The difference of $5$ and $-3$ is $8$. $5-(-3)=8$

Model the expression:
$6-(-4)$

@ action

10
Model the expression:
$7-(-4)$

@ action

11
Model each subtraction.
1. ⓐ 8 − 2
2. ⓑ −5 − 4
3. ⓒ 6 − (−6)
4. ⓓ −8 − (−3)
@ action

ⓐ
$8-2$
This means $8$ take away $2$.
Start with 8 positives.
Take away 2 positives.
How many are left? $6$
$8-2=6$
ⓑ
$-5-4$
This means $-5$ take away $4$.
Start with 5 negatives.
You need to take away 4 positives.
Add 4 neutral pairs to get 4 positives.
Take away 4 positives.
How many are left?
$-9$
$-5-4=-9$
ⓒ
$6-(-6)$
This means $6$ take away $-6$.
Start with 6 positives.
Add 6 neutrals to get 6 negatives to take away.
Remove 6 negatives.
How many are left?
$12$
$6-(-6)=12$
ⓓ
$-8-(-3)$
This means $-8$ take away $-3$.
Start with 8 negatives.
Take away 3 negatives.
How many are left?
$-5$
$-8-(-3)=-5$

Model each subtraction expression:

ⓐ $\ 2-8$
ⓑ $-3-(-8)$

@ action

ⓐ We start with 2 positives.
We need to take away 8 positives, but we have only 2.
Add neutral pairs until there are 8 positives to take away.
Then take away eight positives.
Find the number of counters that are left. There are 6 negatives.
	$2-8=-6$

ⓑ We start with 3 negatives.
We need to take away 8 negatives, but we have only 3.
Add neutral pairs until there are 8 negatives to take away.
Then take away the 8 negatives.
Find the number of counters that are left. There are 5 positives.
	$-3-(-8)=5$

Model each subtraction expression.
1. ⓐ $\ 7-9$
2. ⓑ $\ -5-(-9)$
@ action

ⓐ

−2
ⓑ

4
Model each subtraction expression.
1. ⓐ $\ 4-7$
2. ⓑ $\ -7-(-10)$
@ action

ⓐ

−3
ⓑ

3
Simplify:
1. ⓐ $\ 13-8\ \text{and}\ 13+(-8)\$
2. ⓑ $\ -17-9\ \text{and}\ -17+(-9)$
@ action

ⓐ
$13-8$ and $13+(-8)$
Subtract to simplify. $13-8=5$
Add to simplify. $13+(-8)=5$
Subtracting 8 from 13 is the same as adding −8 to 13.
ⓑ
$-17-9$ and $-17+(-9)$
Subtract to simplify. $-17-9=-26$
Add to simplify. $-17+(-9)=-26$
Subtracting 9 from −17 is the same as adding −9 to −17.
Simplify each expression:
1. ⓐ $\ 21-13\ \text{and}\ 21+(-13)\$
2. ⓑ $\ -11-7\ \text{and}\ -11+(-7)$
@ action
1. ⓐ 8, 8
2. ⓑ −18, −18
Simplify each expression:
1. ⓐ $\ 15-7\ \text{and}\ 15+(-7)\$
2. ⓑ $\ -14-8\ \text{and}\ -14+(-8)$
@ action
1. ⓐ 8, 8
2. ⓑ −22, −22
Simplify:
1. ⓐ $\ 9-(-15)\ \text{and}\ 9+15\$
2. ⓑ $\ -7-(-4)\ \text{and}\ -7+4$
@ action

ⓐ
$9-(-15)$ and $9+15$
Subtract to simplify. $9-(-15)=24$
Add to simplify. $9+15=24$
Subtracting −15 from 9 is the same as adding 15 to 9.
ⓑ
$-7-(-4)$ and $-7+4$
Subtract to simplify. $-7-(-4)=-3$
Add to simplify. $-7+4=-3$
Subtracting −4 from −7 is the same as adding 4 to −7
Simplify each expression:
1. ⓐ $\ 6-(-13)\ \text{and}\ 6+13\$
2. ⓑ $\ -5-(-1)\ \text{and}\ -5+1$
@ action
1. ⓐ 19, 19
2. ⓑ −4, −4
Simplify each expression:
1. ⓐ $\ 4-(-19)\ \text{and}\ 4+19\$
2. ⓐ $-4-(-7)\ \text{and}\ -4+7$
@ action
1. ⓐ 23, 23
2. ⓑ 3, 3
Simplify: $-74-(-58).$

@ action

We are taking 58 negatives away from 74 negatives. $-74-(-58)$
Subtract. $-16$
Simplify the expression:
$-67-(-38)$

@ action

−29
Simplify the expression:
$-83-(-57)$

@ action

−26
Simplify: $7-(-4-3)-9.$

@ action

We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.
Simplify inside the parentheses first.
Subtract from left to right.
Subtract.
Simplify the expression:
$8-(-3-1)-9$

@ action

3
Simplify the expression:
$12-(-9-6)-14$

@ action

13
Simplify: $3\cdot 7-4\cdot 7-5\cdot 8.$

@ action

We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.
Multiply first.
Subtract from left to right.
Subtract.
Simplify the expression:
$6\cdot 2-9\cdot 1-8\cdot 9.$

@ action

−69
Simplify the expression:
$2\cdot 5-3\cdot 7-4\cdot 9$

@ action

−47
Evaluate $x-4\ \text{when}\$
1. ⓐ $\ x=3\$
2. ⓑ $\ x=-6.$
@ action

ⓐ To evaluate $x-4$ when $x=3$, substitute $3$ for $x$ in the expression.

Subtract.
ⓑ To evaluate $x-4$ when $x=-6,$ substitute $-6$ for $x$ in the expression.
Subtract.
Evaluate each expression:
$y-7\ \text{when}\$
1. ⓐ $\ y=5\ \$
2. ⓑ $\ y=-8$
@ action
1. ⓐ −2
2. ⓑ −15
Evaluate each expression:
$m-3\ \text{when}\$
1. ⓐ $\ m=1\$
2. ⓑ $\ m=-4$
@ action
1. ⓐ −2
2. ⓑ −7
Evaluate $20-z\ \text{when}\$
1. ⓐ $\ z=12\$
2. ⓑ $\ z=-12$
@ action

ⓐ To evaluate $20-z\ \text{when}\ z=12,$ substitute $12$ for $z$ in the expression.

Subtract.
ⓑ To evaluate $20-z\ \text{when}\ z=-12,\ \text{substitute}\ -12\ \text{for}\ z\ \text{in the expression.}$

Subtract.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Subtract Integers

Model subtraction of integers
Simplify expressions with integers
Evaluate variable expressions with integers
Translate words phrases to algebraic expressions
Subtract integers in applications
First, we subtracted
Then we subtracted
Subtract

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

@ action Arithmetic

Order of operations Fractions Percentages Powers and roots

Interpret the expression.	\(5-3\) means \(5\) take away \(3\).
Model the first number. Start with 5 positives.
Take away the second number. So take away 3 positives.
Find the counters that are left.
	\(5-3=2\). The difference between \(5\) and \(3\) is \(2\).

Interpret the expression.	\(-5-(-3)\) means \(-5\) take away \(-3\).
Model the first number. Start with 5 negatives.
Take away the second number. So take away 3 negatives.
Find the number of counters that are left.
	\(-5-(-3)=-2\). The difference between \(-5\) and \(-3\) is \(-2\).

Interpret the expression.	\(-5-3\) means \(-5\) take away \(3\).
Model the first number. Start with 5 negatives.
Take away the second number. So we need to take away 3 positives.
But there are no positives to take away. Add neutral pairs until you have 3 positives.
Now take away 3 positives.
Count the number of counters that are left.
	\(-5-3=-8\). The difference of \(-5\) and \(3\) is \(-8\).

ⓐ
	\(13-8\) and \(13+(-8)\)
Subtract to simplify.	\(13-8=5\)
Add to simplify.	\(13+(-8)=5\)
Subtracting 8 from 13 is the same as adding −8 to 13.

ⓑ
	\(-17-9\) and \(-17+(-9)\)
Subtract to simplify.	\(-17-9=-26\)
Add to simplify.	\(-17+(-9)=-26\)
Subtracting 9 from −17 is the same as adding −9 to −17.

\(5-3\)	\(-5-(-3)\)
\(2\)	\(-2\)
2 positives	2 negatives
When there would be enough counters of the color to take away, subtract.
\(-5-3\)	\(5-(-3)\)
\(-8\)	\(8\)
5 negatives, want to subtract 3 positives	5 positives, want to subtract 3 negatives
need neutral pairs	need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.

Interpret the expression.	\(5-(-3)\) means \(5\) take away \(-3\).
Model the first number. Start with 5 positives.
Take away the second number, so take away 3 negatives.
But there are no negatives to take away. Add neutral pairs until you have 3 negatives.
Then take away 3 negatives.
Count the number of counters that are left.
	The difference of \(5\) and \(-3\) is \(8\). \(5-(-3)=8\)

ⓐ
	\(8-2\) This means \(8\) take away \(2\).
Start with 8 positives.
Take away 2 positives.
How many are left?	\(6\)
	\(8-2=6\)

ⓑ
	\(-5-4\) This means \(-5\) take away \(4\).
Start with 5 negatives.
You need to take away 4 positives. Add 4 neutral pairs to get 4 positives.
Take away 4 positives.
How many are left?
	\(-9\)
	\(-5-4=-9\)

ⓒ
	\(6-(-6)\) This means \(6\) take away \(-6\).
Start with 6 positives.
Add 6 neutrals to get 6 negatives to take away.
Remove 6 negatives.
How many are left?
	\(12\)
	\(6-(-6)=12\)

ⓓ
	\(-8-(-3)\) This means \(-8\) take away \(-3\).
Start with 8 negatives.
Take away 3 negatives.
How many are left?
	\(-5\)
	\(-8-(-3)=-5\)

ⓐ
	\(9-(-15)\) and \(9+15\)
Subtract to simplify.	\(9-(-15)=24\)
Add to simplify.	\(9+15=24\)
Subtracting −15 from 9 is the same as adding 15 to 9.

ⓑ
	\(-7-(-4)\) and \(-7+4\)
Subtract to simplify.	\(-7-(-4)=-3\)
Add to simplify.	\(-7+4=-3\)
Subtracting −4 from −7 is the same as adding 4 to −7

We are taking 58 negatives away from 74 negatives.	\(-74-(-58)\)
Subtract.	\(-16\)

Simplify inside the parentheses first.
Subtract from left to right.
Subtract.