Over the past few months Mark has borrowed a lot of money from his sister. At this point he is $25 in debt to his sister! His sister makes him a deal and says that she will take away $8 of the debt if Mark cleans her room for her. How could Mark use a number line to find the difference of \begin{align*}-25-(-8)\end{align*} in order to determine how much he will still be in debt to his sister after cleaning his sister's room?

In this concept, you will learn how to subtract integers with the help of a number line.

### Subtracting Integers Using a Number Line

**Integers** are the set of whole numbers and their opposites.

There are many different strategies for subtracting integers. One strategy for subtracting integers is to use a number line. To subtract two integers using a number line:

- First, draw a number line.
- Then, find the location of the first integer on the number line.
- Next, if the second integer is positive, move that many units to the left from the location of the first integer. If the second integer is negative, move the absolute value of that many units to the right from the location of the first integer. *
*Note that these are the opposite movements from when you ADD integers using a number line! This is because addition and subtraction are opposite operations*. - Your answer will be the point you end on.

Let's look at an example.

Use a number line to find the difference of \begin{align*}4-3\end{align*}.

First, draw your number line.

Then, find the location of 4 (the first integer in your difference) on the number line.

Next, notice that the second integer, 3, is positive. This means you will be moving to the left. Starting at 4, move to the left 3 units.

You end up on 1. The answer is 1.

So \begin{align*}4-3=1\end{align*}.

Let's look at another example.

Use a number line to find the difference of \begin{align*}4-(-3)\end{align*}.

First, draw your number line.

Then, find the location of 4 (the first integer in your difference) on the number line.

Next, notice that the second integer, -3, is negative. This means you will be moving to the right. Starting at 4, move to the right 3 units.

You end up on 7. The answer is 7.

So \begin{align*}4-(-3)=7\end{align*}.

Let's look at one more example.

Use a number line to find the difference of \begin{align*}-4-(-3)\end{align*}.

First, draw your number line.

Then, find the location of -4 (the first integer in your difference) on the number line.

Next, notice that the second integer, -3, is negative. This means you will be moving to the right. Starting at -4, move to the right 3 units.

You end up on -1. The answer is -1.

So \begin{align*}-4-(-3)=-1\end{align*}.

Remember that whether you move to the right or to the left from your starting point only depends on the sign of the second integer. The sign of the first integer just helps you to find the correct starting position on the number line.

### Examples

#### Example 1

Earlier, you were given a problem about Mark, who was in debt to his sister.

Mark is $25 in debt, but has the opportunity of getting rid of $8 of that debt if he cleans his sister's room. Mark wants to subtract \begin{align*}-25-(-8)\end{align*} in order to determine what his new debt would be if he cleaned his sister's room.

First, draw your number line.

Then, find the location of -25 on the number line.

Next, notice that the second integer, -8, is negative. This means you will be moving to the right. Starting at -25, move to the right 8 units.

You end up on -17. The answer is -17.

So \begin{align*}-25-(-8)=-17\end{align*}.

If Mark cleans his sister's room, he will be left with -$17 (he will be $17 in debt).

#### Example 2

Use a number line to find the difference of \begin{align*}-4-3\end{align*}.

First, draw your number line.

Then, find the location of -4 (the first integer in your difference) on the number line.

Next, notice that the second integer, 3, is positive. This means you will be moving to the left. Starting at -4, move to the left 3 units.

You end up on -7. The answer is -7. \begin{align*}-4-3=-7\end{align*}.

#### Example 3

Use a number line to find the difference of \begin{align*}-5-2\end{align*}.

First, draw your number line.

Then, find the location of -5 on the number line.

Next, notice that the second integer, 2, is positive. This means you will be moving to the left. Starting at -5, move to the left 2 units.

You end up on -7. The answer is -7. \begin{align*}-5-2=-7\end{align*}.

#### Example 4

Use a number line to find the difference of \begin{align*}6-(-2)\end{align*}.

First, draw your number line.

Then, find the location of 6 on the number line.

Next, notice that the second integer, -2, is negative. This means you will be moving to the right. Starting at 6, move to the right 2 units.

You end up on 8. The answer is 8. \begin{align*}6-(-2)=8\end{align*}.

#### Example 5

Use a number line to find the difference of \begin{align*}-7-(-5)\end{align*}.

First, draw your number line.

Then, find the location of -7 on the number line.

Next, notice that the second integer, -5, is negative. This means you will be moving to the right. Starting at -7, move to the right 5 units.

You end up on -2. The answer is -2. \begin{align*}-7-(-5)=-2\end{align*}.

### Review

Subtract the following integers using a number line.

- \begin{align*}8-6\end{align*}

- \begin{align*}-5-3\end{align*}

- \begin{align*}-3-(-6)\end{align*}

- \begin{align*}2-(-5)\end{align*}

- \begin{align*}-6-3\end{align*}

- \begin{align*}8-(-3)\end{align*}

- \begin{align*}-7-(-7)\end{align*}

- \begin{align*}-5-4\end{align*}

- \begin{align*}1-(-8)\end{align*}

- \begin{align*}-4-7\end{align*}

Subtract these integers without a number line.

- \begin{align*}-24-37\end{align*}
- \begin{align*}-34-(-7)\end{align*}
- \begin{align*}-44-(-37)\end{align*}
- \begin{align*}-82-(-7)\end{align*}
- \begin{align*}-64-97\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.7.