Foundations

4 Add and Subtract Integers

Learning Objectives

By the end of this section, you will be able to:

  • Use negatives and opposites
  • Simplify: expressions with absolute value
  • Add integers
  • Subtract integers

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

Use Negatives and Opposites

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than 0. The negative numbers are to the left of zero on the number line. See (Figure).

The number line shows the location of positive and negative numbers.

A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See (Figure).

The numbers on a number line increase in value going from left to right and decrease in value going from right to left.

A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.

Doing the Manipulative Mathematics activity “Number Line-part 2” will help you develop a better understanding of integers.

Remember that we use the notation:

a < b (read “a is less than b”) when a is to the left of b on the number line.

a > b (read “a is greater than b”) when a is to the right of b on the number line.

Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in (Figure) are called the integers. The integers are the numbers \text{…}-3,-2,-1,0,1,2,3\text{…}

All the marked numbers are called integers.

A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.

Order each of the following pairs of numbers, using < or >: ⓐ 14___6 ⓑ -1___9 ⓒ -1___-4 ⓓ 2___-20.

Solution

It may be helpful to refer to the number line shown.

A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.

ⓐ
14 is to the right of 6 on the number line.		 \begin{array}{c}14___6\\ 14>6\end{array}
ⓑ
−1 is to the left of 9 on the number line.		 \begin{array}{c}-1___9\\ -1<9\end{array}
ⓒ
−1 is to the right of −4 on the number line.		 \begin{array}{c}-1___-4\\ -1>-4\end{array}
ⓓ
2 is to the right of −20 on the number line.		 \begin{array}{c}2___-20\\ 2>-20\end{array}

Order each of the following pairs of numbers, using < or >\text{:} ⓐ 15___7 ⓑ -2___5 ⓒ -3___-7
ⓓ 5___-17.

ⓐ > ⓑ < ⓒ > ⓓ >

Order each of the following pairs of numbers, using < or >\text{:} ⓐ 8___13 ⓑ 3___-4 ⓒ -5___-2
ⓓ 9___-21.

ⓐ < ⓑ > ⓒ < ⓓ >

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and -2 are the same distance from zero, they are called opposites. The opposite of 2 is -2, and the opposite of -2 is 2.

Opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

(Figure) illustrates the definition.

The opposite of 3 is -3.

A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.

Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.

\begin{array}{cccc}10-4\hfill & & & \text{Between two numbers, it indicates the operation of}\phantom{\rule{0.2em}{0ex}}\mathit{\text{subtraction}}.\hfill \\ & & & \text{We read}\phantom{\rule{0.2em}{0ex}}10-4\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}\text{``}10\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}4.\text{''}\hfill \\ -8\hfill & & & \text{In front of a number, it indicates a}\phantom{\rule{0.2em}{0ex}}\mathit{\text{negative}}\phantom{\rule{0.2em}{0ex}}\text{number.}\hfill \\ & & & \text{We read}\phantom{\rule{0.2em}{0ex}}-8\phantom{\rule{0.2em}{0ex}}\text{as ``negative eight.''}\hfill \\ \text{−}x\hfill & & & \text{In front of a variable, it indicates the}\phantom{\rule{0.2em}{0ex}}\mathit{\text{opposite}}.\phantom{\rule{0.2em}{0ex}}\text{We read}\phantom{\rule{0.2em}{0ex}}\text{−}x\phantom{\rule{0.2em}{0ex}}\text{as ``the opposite of}\phantom{\rule{0.2em}{0ex}}x.\text{''}\hfill \\ \text{−}\left(-2\right)\hfill & & & \text{Here there are two}\phantom{\rule{0.2em}{0ex}}\text{``}-\text{''}\phantom{\rule{0.2em}{0ex}}\text{signs. The one in the parentheses tells us the number is}\hfill \\ & & & \text{negative}\phantom{\rule{0.2em}{0ex}}2.\phantom{\rule{0.2em}{0ex}}\text{The one outside the parentheses tells us to take the}\phantom{\rule{0.2em}{0ex}}\mathit{\text{opposite}}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}-2.\hfill \\ & & & \text{We read}\phantom{\rule{0.2em}{0ex}}\text{−}\left(-2\right)\phantom{\rule{0.2em}{0ex}}\text{as ``the opposite of negative two.''}\hfill \end{array}
10-4 Between two numbers, it indicates the operation of subtraction.
We read 10-4 as “10 minus 4.”
-8 In front of a number, it indicates a negative number.
We read −8 as “negative eight.”
\text{−}x In front of a variable, it indicates the opposite. We read \text{−}x as “the opposite of x.”
\text{−}\left(-2\right) Here there are two “−” signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the opposite of −2.
We read \text{−}\left(-2\right) as “the opposite of negative two.”
Opposite Notation

\text{−}a means the opposite of the number a.

The notation \text{−}a is read as “the opposite of a.”

Find: ⓐ the opposite of 7 ⓑ the opposite of -10 ⓒ \text{−}\left(-6\right).

Solution
ⓐ −7 is the same distance from 0 as 7, but on the opposite side of 0. .
The opposite of 7 is −7.
ⓑ 10 is the same distance from 0 as −10, but on the opposite side of 0. .
The opposite of −10 is 10.
ⓒ −(−6) .
The opposite of −(−6) is −6.

Find: ⓐ the opposite of 4 ⓑ the opposite of -3 ⓒ \text{−}\left(-1\right).

ⓐ-4ⓑ 3 ⓒ 1

Find: ⓐ the opposite of 8 ⓑ the opposite of -5 ⓒ \text{−}\left(-5\right).

ⓐ-8ⓑ 5 ⓒ 5

Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the integers. The integers are the numbers \text{…}-3,-2,-1,0,1,2,3\text{…}

Integers

The whole numbers and their opposites are called the integers.

The integers are the numbers

\text{…}-3,-2,-1,0,1,2,3\text{…}

When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether \text{−}x is positive or negative. We can see this in (Figure).

Evaluate ⓐ \text{−}x, when x=8 ⓑ \text{−}x, when x=-8.

Solution
  1. ⓐ

    .
    −x
    . .
    Write the opposite of 8. .



  2. ⓑ

    .
    −x
    . .
    Write the opposite of −8. 8

Evaluate \text{−}n, when ⓐ n=4 ⓑ n=-4.

ⓐ-4ⓑ 4

Evaluate \text{−}m, when ⓐ m=11 ⓑ m=-11.

ⓐ-11ⓑ 11

Simplify: Expressions with Absolute Value

We saw that numbers such as 2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-2 are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

Absolute Value

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number n is written as |n|.

For example,

  • -5\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}5 units away from 0, so |-5|=5.
  • \phantom{\rule{0.65em}{0ex}}5\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}5 units away from 0, so |5|=5.

(Figure) illustrates this idea.

The integers 5\phantom{\rule{0.2em}{0ex}}\text{and are}\phantom{\rule{0.2em}{0ex}}5 units away from 0.

A number line is shown ranging from negative 5 to 5. A bracket labeled “5 units” lies above the points negative 5 to 0. An arrow labeled “negative 5 is 5 units from 0, so absolute value of negative 5 equals 5.” is written above the labeled bracket. A bracket labeled “5 units” lies above the points “0” to “5”. An arrow labeled “5 is 5 units from 0, so absolute value of 5 equals 5.” and is written above the labeled bracket.

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from 0\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0 on the number line is zero units.

Property of Absolute Value

|n|\ge 0 for all numbers

Absolute values are always greater than or equal to zero!

Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero.

Simplify: ⓐ |3| ⓑ |-44| ⓒ |0|.

Solution

The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative.

ⓐ|3|
\phantom{\rule{1.2em}{0ex}}3

ⓑ|-44|
\phantom{\rule{1.5em}{0ex}}44

ⓒ|0|
\phantom{\rule{1.2em}{0ex}}0

Simplify: ⓐ |4| ⓑ |-28| ⓒ |0|.

ⓐ 4 ⓑ 28 ⓒ 0

Simplify: ⓐ |-13| ⓑ |47| ⓒ |0|.

ⓐ 13 ⓑ 47 ⓒ 0

In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!

Fill in <,>,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}= for each of the following pairs of numbers:

ⓐ|-5|___-|-5|ⓑ8___-|-8|ⓒ-9___-|-9|ⓓ\text{−}\left(-16\right)___-|-16|

Solution
ⓐ
Simplify.
Order.
\begin{array}{ccc}|-5|& ___\hfill & -|-5|\hfill \\ 5& ___\hfill & -5\hfill \\ 5& >\hfill & -5\hfill \\ |-5|& >\hfill & -|-5|\hfill \end{array}
ⓑ
Simplify.
Order.
\begin{array}{ccc}8& ___\hfill & -|-8|\hfill \\ 8& ___\hfill & -8\hfill \\ 8& >\hfill & -8\hfill \\ 8& >\hfill & -|-8|\hfill \end{array}
ⓒ
Simplify.
Order.
\begin{array}{ccc}9& ___\hfill & -|-9|\hfill \\ -9& ___\hfill & -9\hfill \\ -9& =\hfill & -9\hfill \\ -9& =\hfill & -|-9|\hfill \end{array}
ⓓ
Simplify.
Order.
\begin{array}{ccc}\text{−}\left(-16\right)& ___\hfill & -|-16|\hfill \\ 16& ___\hfill & -16\hfill \\ 16& >\hfill & -16\hfill \\ \text{−}\left(-16\right)& >\hfill & -|-16|\hfill \end{array}

Fill in <, >, or = for each of the following pairs of numbers: ⓐ |-9|___-|-9| ⓑ 2___-|-2| ⓒ -8___|-8|
ⓓ \text{−}\left(-9\right)___-|-9|.

ⓐ > ⓑ > ⓒ < ⓓ >

Fill in <, >, or = for each of the following pairs of numbers: ⓐ 7___-|-7| ⓑ \text{−}\left(-10\right)___-|-10|
ⓒ |-4|___-|-4| ⓓ -1___|-1|.

ⓐ > ⓑ > ⓒ > ⓓ <

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

Grouping Symbols
\begin{array}{cccccccc}\text{Parentheses}\hfill & \left(\right)\hfill & & & & & \text{Braces}\hfill & \left\{\right\}\hfill \\ \text{Brackets}\hfill & \left[\right]\hfill & & & & & \text{Absolute value}\hfill & \phantom{\rule{0.15em}{0ex}}|\phantom{\rule{0.2em}{0ex}}|\hfill \end{array}

In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.

Simplify: 24-|19-3\left(6-2\right)|.

Solution
24-|19-3\left(6-2\right)|
Work inside parentheses first: subtract 2 from 6. 24-|19-3\left(4\right)|
Multiply 3(4). 24-|19-12|
Subtract inside the absolute value bars. 24-|7|
Take the absolute value. 24-7
Subtract. 17

Simplify: 19-|11-4\left(3-1\right)|.

16

Simplify: 9-|8-4\left(7-5\right)|.

9

Evaluate: ⓐ |x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-35 ⓑ |\text{−}y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-20 ⓒ \text{−}|u|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=12 ⓓ \text{−}|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-14.

Solution

ⓐ|x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-35

|x|
. .
Take the absolute value. 35



ⓑ|\text{−}y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-20

|-y|
. .
Simplify. |20|
Take the absolute value. 20



ⓒ\text{−}|u|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=12

-|u|
. .
Take the absolute value. -12



ⓓ\text{−}|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-14

-|p|
. .
Take the absolute value. -14

Evaluate: ⓐ |x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-17 ⓑ |\text{−}y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-39 ⓒ \text{−}|m|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=22 ⓓ \text{−}|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-11.

ⓐ17ⓑ39ⓒ-22ⓓ-11

Evaluate: ⓐ |y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-23 ⓑ |\text{−}y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-21 ⓒ \text{−}|n|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}n=37 ⓓ \text{−}|q|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-49.

ⓐ23ⓑ21ⓒ-37ⓓ-49

Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

Doing the Manipulative Mathematics activity “Addition of Signed Numbers” will help you develop a better understanding of adding integers.”

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.

We will use the counters to show how to add the four addition facts using the numbers 5,-5 and 3,-3.

\begin{array}{cccccccccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill & & & \hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \end{array}

To add 5+3, we realize that 5+3 means the sum of 5 and 3.

We start with 5 positives. .
And then we add 3 positives. .
We now have 8 positives. The sum of 5 and 3 is 8. .

Now we will add -5+\left(-3\right). Watch for similarities to the last example 5+3=8.

To add -5+\left(-3\right), we realize this means the sum of -5\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-3.

We start with 5 negatives. .
And then we add 3 negatives. .
We now have 8 negatives. The sum of −5 and −3 is −8. .

In what ways were these first two examples similar?

  • The first example adds 5 positives and 3 positives—both positives.
  • The second example adds 5 negatives and 3 negatives—both negatives.

In each case we got 8—either 8 positives or 8 negatives.

When the signs were the same, the counters were all the same color, and so we added them.

This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text “8 positives.” Centered under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase “8 negatives”. Centered under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.

Add: ⓐ 1+4 ⓑ -1+\left(-4\right).

Solution

ⓐ

.
1 positive plus 4 positives is 5 positives.

ⓑ

.
1 negative plus 4 negatives is 5 negatives.

Add: ⓐ 2+4 ⓑ -2+\left(-4\right).

ⓐ 6 ⓑ -6

Add: ⓐ 2+5 ⓑ -2+\left(-5\right).

ⓐ 7 ⓑ -7

So what happens when the signs are different? Let’s add -5+3. We realize this means the sum of -5 and 3. When the counters were the same color, we put them in a row. When the counters are a different color, we line them up under each other.

−5 + 3 means the sum of −5 and 3.
We start with 5 negatives. .
And then we add 3 positives. .
We remove any neutral pairs. .
We have 2 negatives left. .
The sum of −5 and 3 is −2. −5 + 3 = −2

Notice that there were more negatives than positives, so the result was negative.

Let’s now add the last combination, 5+\left(\text{−}3\right).

5 + (−3) means the sum of 5 and −3.
We start with 5 positives. .
And then we add 3 negatives. .
We remove any neutral pairs. .
We have 2 positives left. .
The sum of 5 and −3 is 2. 5 + (−3) = 2

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “negative 5 plus 3” and below is written “More negatives – the sum is negative.” The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “5 plus negative 3” and below is written “More positives – the sum is positive.”

Add: ⓐ -1+5 ⓑ 1+\left(-5\right).

Solution

ⓐ

−1 + 5
.
There are more positives, so the sum is positive. 4



ⓑ

1 + (−5)
.
There are more negatives, so the sum is negative. −4

Add: ⓐ -2+4 ⓑ 2+\left(-4\right).

ⓐ 2 ⓑ -2

Add: ⓐ -2+5 ⓑ 2+\left(-5\right).

ⓐ 3 ⓑ -3

Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.

When you need to add numbers such as 37+\left(\text{−}53\right), you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?

Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because 53-37=16, there are 16 more red counters.

Therefore, the sum of 37+\left(\text{−}53\right) is -16.

37+\left(\text{−}53\right)=-16

Let’s try another one. We’ll add -74+\left(\text{−}27\right). Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is -101.

-74+\left(\text{−}27\right)=-101

Let’s look again at the results of adding the different combinations of 5,-5 and 3,-3.

Addition of Positive and Negative Integers
\begin{array}{cccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill \\ \hfill 8\hfill & & & \hfill -8\hfill \\ \hfill \text{both positive, sum positive}\hfill & & & \hfill \text{both negative, sum negative}\hfill \end{array}

When the signs are the same, the counters would be all the same color, so add them.

\begin{array}{cccc}\hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \\ \hfill -2\hfill & & & \hfill 2\hfill \\ \hfill \text{different signs, more negatives, sum negative}\hfill & & & \hfill \text{different signs, more positives, sum positive}\hfill \end{array}

When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left.

Visualize the model as you simplify the expressions in the following examples.

Simplify: ⓐ 19+\left(-47\right) ⓑ -14+\left(-36\right).

Solution
  1. ⓐ Since the signs are different, we subtract \text{19 from 47}\text{.} The answer will be negative because there are more negatives than positives.
     \begin{array}{cccc}& & & \hfill \phantom{\rule{0.3em}{0ex}}19+\left(-47\right)\hfill \\ \text{Add.}\hfill & & & \hfill \phantom{\rule{0.3em}{0ex}}-28\hfill \end{array}
  2. ⓑ Since the signs are the same, we add. The answer will be negative because there are only negatives.
     \begin{array}{cccc}& & & \hfill -14+\left(-36\right)\hfill \\ \text{Add.}\hfill & & & \hfill -50\hfill \end{array}

Simplify: ⓐ -31+\left(-19\right) ⓑ 15+\left(-32\right).

ⓐ-50ⓑ-17

Simplify: ⓐ -42+\left(-28\right) ⓑ 25+\left(-61\right).

ⓐ-70ⓑ-36

The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations!

Simplify: -5+3\left(-2+7\right).

Solution
-5+3\left(-2+7\right)
Simplify inside the parentheses. -5+3\left(5\right)
Multiply. -5+15
Add left to right. 10

Simplify: -2+5\left(-4+7\right).

13

Simplify: -4+2\left(-3+5\right).

0

Subtract Integers

Doing the Manipulative Mathematics activity “Subtraction of Signed Numbers” will help you develop a better understanding of subtracting integers.

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

Perhaps when you were younger, you read \text{``}5-3\text{''} as \text{``}5 take away 3.\text{''} When you use counters, you can think of subtraction the same way!

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

\begin{array}{cccccccccc}\hfill 5-3\hfill & & & \hfill -5-\left(-3\right)\hfill & & & \hfill -5-3\hfill & & & \hfill 5-\left(-3\right)\hfill \end{array}

To subtract 5-3, we restate the problem as \text{``}5 take away 3.\text{''}

We start with 5 positives. .
We ‘take away’ 3 positives. .
We have 2 positives left.
The difference of 5 and 3 is 2. 2

Now we will subtract -5-\left(-3\right). Watch for similarities to the last example 5-3=2.

To subtract -5-\left(-3\right), we restate this as \text{``}-5 take away -3\text{''}

We start with 5 negatives. .
We ‘take away’ 3 negatives. .
We have 2 negatives left.
The difference of −5 and −3 is −2. −2

Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.

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

Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation “5 minus 3 equals 2.” The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation “negative 5, minus, negative 3, equals negative 2.”

Subtract: ⓐ 7-5 ⓑ -7-\left(-5\right).

Solution
ⓐ
Take 5 positive from 7 positives and get 2 positives.		 \begin{array}{c}7-5\\ 2\end{array}
ⓑ
Take 5 negatives from 7 negatives and get 2 negatives. 		\begin{array}{c}-7-\left(-5\right)\\ -2\end{array}

Subtract: ⓐ 6-4 ⓑ -6-\left(-4\right).

ⓐ 2 ⓑ -2

Subtract: ⓐ 7-4 ⓑ -7-\left(-4\right).

ⓐ 3 ⓑ -3

What happens when we have to subtract one positive and one negative number? We’ll need to use both white and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

  • To subtract -5-3, we restate it as -5 take away 3.

We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.

Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.

−5 − 3 means −5 take away 3.
We start with 5 negatives. .
We now add the neutrals needed to get 3 positives. .
We remove the 3 positives. .
We are left with 8 negatives. .
The difference of −5 and 3 is −8. −5 − 3 = −8

And now, the fourth case, 5-\left(-3\right). We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.

5 − (−3) means 5 take away −3.
We start with 5 positives. .
We now add the needed neutrals pairs. .
We remove the 3 negatives. .
We are left with 8 positives. .
The difference of 5 and −3 is 8. 5 − (−3) = 8

Subtract: ⓐ -3-1 ⓑ 3-\left(-1\right).

Solution

ⓐ

Take 1 positive from the one added neutral pair. .
.		 −3 − 1

−4

ⓑ

Take 1 negative from the one added neutral pair. .
.		 3 − (−1)

4

Subtract: ⓐ -6-4 ⓑ 6-\left(-4\right).

ⓐ-10ⓑ 10

Subtract: ⓐ -7-4 ⓑ 7-\left(-4\right).

ⓐ-11ⓑ 11

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In (Figure), -3-1 is the same as -3+\left(-1\right) and 3-\left(-1\right) is the same as 3+1. You will often see this idea, the subtraction property, written as follows:

Subtraction Property
a-b=a+\left(\text{−}b\right)

Subtracting a number is the same as adding its opposite.

Look at these two examples.

Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as “6 minus 4” and below as “2.” The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as “6 plus, open parenthesis, negative 4, close parenthesis” and below as “2.”

6-4\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}6+\left(-4\right).

Of course, when you have a subtraction problem that has only positive numbers, like 6-4, you just do the subtraction. You already knew how to subtract 6-4 long ago. But knowing that 6-4 gives the same answer as 6+\left(-4\right) helps when you are subtracting negative numbers. Make sure that you understand how 6-4 and 6+\left(-4\right) give the same results!

Simplify: ⓐ 13-8 and 13+\left(-8\right) ⓑ -17-9 and -17+\left(-9\right).

Solution
ⓐ
Subtract.		 \begin{array}{c}13-8\\ 5\end{array} \begin{array}{c}13+\left(-8\right)\\ 5\end{array}
ⓑ
Subtract.		 \begin{array}{c}-17-9\\ -29\end{array} \begin{array}{c}-17+\left(-9\right)\\ -26\end{array}

Simplify: ⓐ 21-13 and 21+\left(-13\right) ⓑ -11-7 and -11+\left(-7\right).

ⓐ 8 ⓑ −18

Simplify: ⓐ 15-7 and 15+\left(-7\right) ⓑ -14-8 and -14+\left(-8\right).

ⓐ8ⓑ-22

Look at what happens when we subtract a negative.

This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.

8-\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}8+5

Subtracting a negative number is like adding a positive!

You will often see this written as a-\left(\text{−}b\right)=a+b.

Does that work for other numbers, too? Let’s do the following example and see.

Simplify: ⓐ 9-\left(-15\right) and 9+15 ⓑ -7-\left(-4\right) and -7+4.

Solution

ⓐ
\begin{array}{ccccccccc}& & & & & \hfill 9-\left(-15\right)\hfill & & & \hfill 9+15\hfill \\ \text{Subtract.}\hfill & & & & & \hfill 24\hfill & & & \hfill 24\hfill \end{array}

ⓑ
\begin{array}{ccccccccc}& & & & & \hfill -7-\left(-4\right)\hfill & & & \hfill -7+4\hfill \\ \text{Subtract.}\hfill & & & & & \hfill -3\hfill & & & \hfill -3\hfill \end{array}

ⓐ
Subtract.		 \begin{array}{c}9-\left(-15\right)\\ 24\end{array} \begin{array}{c}9+15\\ 24\end{array}
ⓑ
Subtract.		 \begin{array}{c}-7-\left(-4\right)\\ -3\end{array} \begin{array}{c}-7+4\\ -3\end{array}

Simplify: ⓐ 6-\left(-13\right) and 6+13 ⓑ -5-\left(-1\right) and -5+1.

ⓐ19ⓑ-4

Simplify: ⓐ 4-\left(-19\right) and 4+19 ⓑ -4-\left(-7\right) and -4+7.

ⓐ 23 ⓑ 3

Let’s look again at the results of subtracting the different combinations of 5,-5 and 3,-3.

Subtraction of Integers
\begin{array}{cccc}\hfill 5-3\hfill & & & \hfill -5-\left(-3\right)\hfill \\ \hfill 2\hfill & & & \hfill -2\hfill \\ \hfill 5\phantom{\rule{0.2em}{0ex}}\text{positives take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \hfill 5\phantom{\rule{0.2em}{0ex}}\text{negatives take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \hfill 2\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \hfill 2\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \end{array}

When there would be enough counters of the color to take away, subtract.

\begin{array}{cccc}\hfill -5-3\hfill & & & \hfill 5-\left(-3\right)\hfill \\ \hfill -8\hfill & & & \hfill 8\hfill \\ \hfill 5\phantom{\rule{0.2em}{0ex}}\text{negatives, want to take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \hfill 5\phantom{\rule{0.2em}{0ex}}\text{positives, want to take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \hfill \text{need neutral pairs}\hfill & & & \hfill \text{need neutral pairs}\hfill \end{array}

When there would be not enough counters of the color to take away, add.

What happens when there are more than three integers? We just use the order of operations as usual.

Simplify: 7-\left(-4-3\right)-9.

Solution
7-\left(-4-3\right)-9
Simplify inside the parentheses first. 7-\left(-7\right)-9
Subtract left to right. 14-9
Subtract. 5

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

3

Simplify: 12-\left(-9-6\right)-14.

13

Access these online resources for additional instruction and practice with adding and subtracting integers. You will need to enable Java in your web browser to use the applications.

Key Concepts

  • Addition of Positive and Negative Integers
    \begin{array}{cccc}5+3\hfill & & & \phantom{\rule{2.5em}{0ex}}-5+\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}8\hfill & & & \phantom{\rule{3.45em}{0ex}}-8\hfill \\ \text{both positive,}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{both negative,}\hfill \\ \text{sum positive}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{sum negative}\hfill \\ \\ \\ -5+3\hfill & & & \phantom{\rule{2.5em}{0ex}}5+\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}-2\hfill & & & \phantom{\rule{3.45em}{0ex}}2\hfill \\ \text{different signs,}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{different signs,}\hfill \\ \text{more negatives}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{more positives}\hfill \\ \text{sum negative}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{sum positive}\hfill \end{array}
  • Property of Absolute Value: |n|\ge 0 for all numbers. Absolute values are always greater than or equal to zero!
  • Subtraction of Integers
    \begin{array}{cccc}5-3\hfill & & & -5-\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}2\hfill & & & \phantom{\rule{0.95em}{0ex}}-2\hfill \\ 5\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & 5\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \text{take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{2 positives}\hfill & & & \text{2 negatives}\hfill \\ \\ \\ -5-3\hfill & & & 5-\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}-8\hfill & & & \phantom{\rule{0.95em}{0ex}}8\hfill \\ 5\phantom{\rule{0.2em}{0ex}}\text{negatives, want to}\hfill & & & 5\phantom{\rule{0.2em}{0ex}}\text{positives, want to}\hfill \\ \text{subtract}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \text{subtract}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{need neutral pairs}\hfill & & & \text{need neutral pairs}\hfill \end{array}
  • Subtraction Property: Subtracting a number is the same as adding its opposite.

Practice Makes Perfect

Use Negatives and Opposites of Integers

In the following exercises, order each of the following pairs of numbers, using < or >.


ⓐ9___4
ⓑ-3___6
ⓒ-8___-2
ⓓ1___-10

ⓐ > ⓑ < ⓒ < ⓓ >


ⓐ-7___3
ⓑ-10___-5
ⓒ2___-6
ⓓ8___9

In the following exercises, find the opposite of each number.


ⓐ 2
ⓑ -6

ⓐ-2ⓑ 6


ⓐ 9
ⓑ -4

In the following exercises, simplify.

\text{−}\left(-4\right)

4

\text{−}\left(-8\right)

\text{−}\left(-15\right)

15

\text{−}\left(-11\right)

In the following exercises, evaluate.

\text{−}c when
ⓐ c=12
ⓑ c=-12

ⓐ-12ⓑ 12

\text{−}d when
ⓐ d=21
ⓑ d=-21

Simplify Expressions with Absolute Value

In the following exercises, simplify.


ⓐ|-32|
ⓑ|0|
ⓒ|16|

ⓐ 32 ⓑ 0 ⓒ 16


ⓐ|0|
ⓑ|-40|
ⓒ|22|

In the following exercises, fill in <, >, or = for each of the following pairs of numbers.


ⓐ-6___|-6|
ⓑ\text{−}|-3|___-3

ⓐ < ⓑ =


ⓐ|-5|___\text{−}|-5|
ⓑ9___\text{−}|-9|

In the following exercises, simplify.

\text{−}\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{−}|-5|

5,-5

\text{−}|-9|\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{−}\left(-9\right)

8|-7|

56

5|-5|

|15-7|-|14-6|

0

|17-8|-|13-4|

18-|2\left(8-3\right)|

8

18-|3\left(8-5\right)|

In the following exercises, evaluate.


ⓐ\text{−}|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=19
ⓑ\text{−}|q|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-33

ⓐ-19ⓑ-33


ⓐ\text{−}|a|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=60
ⓑ\text{−}|b|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}b=-12

Add Integers

In the following exercises, simplify each expression.

-21+\left(-59\right)

-80

-35+\left(-47\right)

48+\left(-16\right)

32

34+\left(-19\right)

-14+\left(-12\right)+4

-22

-17+\left(-18\right)+6

135+\left(-110\right)+83

108

6 -38+27+\left(-8\right)+126

19+2\left(-3+8\right)

29

24+3\left(-5+9\right)

Subtract Integers

In the following exercises, simplify.

8-2

6

-6-\left(-4\right)

-5-4

-9

-7-2

8-\left(-4\right)

12

7-\left(-3\right)


ⓐ44-28
ⓑ44+\left(-28\right)

ⓐ 16 ⓑ 16


ⓐ35-16
ⓑ35+\left(-16\right)


ⓐ27-\left(-18\right)
ⓑ27+18

ⓐ 45 ⓑ 45


ⓐ46-\left(-37\right)
ⓑ46+37

In the following exercises, simplify each expression.

15-\left(-12\right)

27

14-\left(-11\right)

48-87

-39

45-69

-17-42

-59

-19-46

-103-\left(-52\right)

-51

-105-\left(-68\right)

-45-\left(54\right)

9

-58-\left(-67\right)

8-3-7

-2

9-6-5

-5-4+7

-2

-3-8+4

-14-\left(-27\right)+9

22

64+\left(-17\right)-9

\left(2-7\right)-\left(3-8\right)\left(2\right)

0

\left(1-8\right)-\left(2-9\right)

\text{−}\left(6-8\right)-\left(2-4\right)

4

\text{−}\left(4-5\right)-\left(7-8\right)

25-\left[10-\left(3-12\right)\right]

6

32-\left[5-\left(15-20\right)\right]

6.3-4.3-7.2

-8

5.7-8.2-4.9

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

-11

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

Everyday Math

Elevation The highest elevation in the United States is Mount McKinley, Alaska, at 20,320 feet above sea level. The lowest elevation is Death Valley, California, at 282 feet below sea level.

Use integers to write the elevation of:

ⓐ Mount McKinley.
ⓑ Death Valley.

ⓐ 20,329 ⓑ -282

Extreme temperatures The highest recorded temperature on Earth was 58\text{°} Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature was 90\text{°}\phantom{\rule{0.2em}{0ex}}\text{below}\phantom{\rule{0.2em}{0ex}}0\text{°} Celsius, recorded in Antarctica in 1983.

Use integers to write the:

  1. ⓐ highest recorded temperature.
  2. ⓑ lowest recorded temperature.

State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of ?540 million. That same month, Texas estimated it would have a budget deficit of ?27 billion.

Use integers to write the budget of:

ⓐ Pennsylvania.
ⓑ Texas.

ⓐ ?540 million ⓑ \text{−}?27 billion

College enrollments Across the United States, community college enrollment grew by 1,400,000 students from Fall 2007 to Fall 2010. In California, community college enrollment declined by 110,171 students from Fall 2009 to Fall 2010.

Use integers to write the change in enrollment:

  1. ⓐ in the U.S. from Fall 2007 to Fall 2010.
  2. ⓑ in California from Fall 2009 to Fall 2010.

Stock Market The week of September 15, 2008 was one of the most volatile weeks ever for the US stock market. The closing numbers of the Dow Jones Industrial Average each day were:

Monday -504
Tuesday +142
Wednesday -449
Thursday +410
Friday +369

What was the overall change for the week? Was it positive or negative?

-32

Stock Market During the week of June 22, 2009, the closing numbers of the Dow Jones Industrial Average each day were:

Monday -201
Tuesday -16
Wednesday -23
Thursday +172
Friday -34

What was the overall change for the week? Was it positive or negative?

Writing Exercises

Give an example of a negative number from your life experience.

Answers may vary

What are the three uses of the \text{``}-\text{''} sign in algebra? Explain how they differ.

Explain why the sum of -8 and 2 is negative, but the sum of 8 and -2 is positive.

Answers may vary

Give an example from your life experience of adding two negative numbers.

Self Check

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

A table is shown with four columns and five rows. The column titles, from left to right, are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!” The first column includes the phrases “use negatives and opposites of integers.”, “Simplify: expressions with absolute value.”, “add integers.” and “subtract integers.”

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Glossary

absolute value
The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as |n|.
integers
The whole numbers and their opposites are called the integers: …−3, −2, −1, 0, 1, 2, 3…
opposite
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: \text{−}a means the opposite of the number. The notation \text{−}a is read “the opposite of a.”

License

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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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