Jamie earned ten dollars cutting grass for his aunt, but he owes his brother twelve dollars from a movie that they went to last week. Does Jamie still owe his brother money if he gives him the whole ten dollars he made? If so, how much does he still owe? If not, how much does he have left over?

In this concept, you will learn to use a number line to find the difference between integers.

### Subtracting Integers Using a Number Line

An **integer **is any positive whole number or its opposite. Here, opposite means sign. So a positive integer has a negative opposite and vice versa.

A **positive number **is a number greater than zero. It can be written with or without a + symbol in front of it. A gain in something is written with a positive number.

A **negative number **is a number that is less than zero. It is always written with a - symbol in front of it. A loss is written with a negative number.

A **number line **is a line on which numbers are marked at intervals, used to illustrate simple numerical operations. Using a number line allows you to see where a number is in relationship to other numbers and from zero.

Number lines are useful tools for visualizing simple arithmetic operations, especially addition and subtraction. Here is an example of using a number line for subtraction.

-6 – 2 = ____

First, mark the location of the first integer.

Next, subtract two from this value. Subtraction means to move to the left.

The final mark is at -8.

The answer is -6 – 2 = -8.

Here is another example.

2 – 9 = ____

First, make a mark at the initial integer.

Next, subtract 9. Subtract means to move to the left.

The final mark is at -7.

The answer is 2 – 9 = -7.

### Examples

#### Example 1

Earlier, you were given a problem about Jamie.

He made ten dollars mowing his aunt's grass, but he owes his brother twelve. If he gives his brother his ten, does he still owe him money?

First, write a number sentence to represent the problem.

Jamie earned $10.00. He owes his brother $12.00.

10 - 12 = ____

Next, draw the problem on a number line.

Start at positive ten and then subtract 12. This means you move twelve units toward the negative side of the number line.

10 – 12 = -2

Then, notice whether the answer is positive or negative.

In this case, it is negative, so the answer is yes, Jaime still owes more money. He owes his brother $2.00.

#### Example 2

Write a number sentence for this situation and then use a number line to solve it.

Marcy lost 12 gems from a small jewel bag. Then she lost another four gems. How many did she lose?

To write this expression, use negative numbers for losses.

First, write the original number of gems lost as an integer.

\begin{align*}-12\end{align*}

Next write the additional number of gems lost as an integer.

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

Then, put them together in an expression.

Then, draw a number line.

Finally, start at -12 and move 4 to the left.

You are now at -16.

The answer is -16.

**Use a number line to evaluate the following problems.**

#### Example 3

4 – 10 = ____

First, draw the number line.

Next, start at 4 and move 10 to the left.

You are now at -6.

The answer is -6.

#### Example 4

-9 – 3 = ____

First, draw the number line.

Next, start at -9 and move 3 to the left.

You are now at -12.

The answer is -12.

#### Example 5

-12 – 4 = ____

First, draw the number line.

Next, start at -12 and move 4 to the left.

You are now at -16.

The answer is -16.

### Review

Find each difference by using a number line.

- \begin{align*}-5 - 2\end{align*}
- \begin{align*}7 - 12\end{align*}
- \begin{align*}-5 - 12\end{align*}
- \begin{align*}-25 - 42\end{align*}
- \begin{align*}5 - 32\end{align*}
- \begin{align*}-15 - 12\end{align*}
- \begin{align*}18 - 2\end{align*}
- \begin{align*}25 - 22\end{align*}
- \begin{align*}-45 - 32\end{align*}
- \begin{align*}-35 - 37\end{align*}
- \begin{align*}14 - 32\end{align*}
- \begin{align*}15 - 44\end{align*}
- \begin{align*}-41 - 12\end{align*}
- \begin{align*}-18 - 13\end{align*}
- \begin{align*}7 - 92\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.9.