Robin learned how to build arrows and a bow with a 50 lb draw out of PVC pipe and other cheap materials at her summer camp. She made 3 for her best friends, and had orders for another 8. Then, she discovered that business was more complicated than she'd hoped. In the same day, her best friends' mothers all made them return the bows, and she got an order for 32 from a PE teacher. How many more will she need to make to fill the demand?

In this concept, you will learn how to take the difference between two integers with different signs.

### Subtracting Integers with Different Signs

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.

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.

There are two possibilities when subtracting numbers with different signs. Either the negative number is first or the positive number is. If the negative number is first, then add the two integers and put a negative sign in front of the answer, like this:

If the positive number is first, then you are subtracting a negative number. Whenever you subtract a negative, you can re-write it as an addition problem, and then solve, like this:

\begin{align*}5--10=5+10=15\end{align*}

Notice that if the first number was negative in a difference between numbers with different signs, the answer will be negative. If the first number was positive, the answer will be positive.

### Examples

#### Example 1

Earlier, you were given a problem about Robin and her entrepreneurial struggles.

She started a bow-making business. She had 8 orders to fill when her first 3 were returned and she got an order for 32 more. How many does she need to make total?

To solve this problem, first she assigns signs to each quantity and writes them as integers.

The ones that are returned, she doesn't have to make, so she calls them positive. She has +3 bows. The ones she still has to make, she assigns a negative. Those are the 8 she already owes: -8.

Next, she figures out what she'll have left once these orders are filled by adding these two numbers together:

\begin{align*}3-8=-5\end{align*}

Then, she sets up a new math problem as a difference of two integers with different signs:

\begin{align*}-5-32=\end{align*}

\begin{align*}5+32=37\end{align*}

Then, she checks the sign of the first number. It was negative. She decides, with a sigh, that the answer must be negative too.

The answer is she must make 37 bows in order to stay in business.

#### Example 2

Write the following scenario as a mathematical expression and solve.

Here are a few to practice with. Choose a method and find the difference of each pair of integers. Javier owes Louisa $20 for dinner. Then she gives him $10 for the movies. How much does Javier owe Louisa total?

First, write the expression.

Javier starts out owing 20 which is -20. Then he subtracts 10 from that.\begin{align*}-20-10\end{align*}Next, add the two together.\begin{align*}20+10=30\end{align*}Then, check the sign of the first number.

Since it was negative, put a negative in front of the answer. The answer is -30. This means that Javier owes Louisa $30 total.

#### Example 3

Find the difference.

-5 – 7 = ____

First, add the two numbers together.

5+7=12

Next, check the sign of the first number.

In this case, it is negative, so the answer is negative.

The answer is -12.

#### Example 4

Find the difference.

2 – -8 = ____

First, re-write the double negative as a positive.

2+8

Next, add the two together.

2+8=10

Then, check the sign of the first number in the original problem.

In this case, it is positive so the answer is positive.

The answer is 10.

#### Example 5

Find the difference.

-13 – 5 = ____

First, add the two integers together.

13+5=18

Next, check the sign of the first integer in the original problem.

In this case, it is negative so the answer is negative.

The answer is -18.

### Review

Find the differences of the following integer pairs.

- -2 – 4 = ____
- -8 – 9 = ____
- -6 – 7 = ____
- -11 – 12 = ____
- -13 – 22 = ____
- -89 – 11 = ____
- 2 – 7 = ____
- 4 – 9 = ____
- 5 – 8 = ____
- 13 – 20 = ____
- 12 – 23 = ____
- 25 – 30 = ____
- 45 – 90 = ____
- 34 – 67 = ____
- -2 – -3 = ____
- -8 – -3 = ____
- -9 – -7 = ____
- -5 – -10 = ____
- -9 – -12 = ____
- -10 – -10 = ____
- -14 – -15 = ____
- 5 – -8 = ____
- 6 – -7 = ____
- 10 – -9 = ____
- 11 – -7 = ____
- 18 – -9 = ____
- 22 – -5 = ____
- 34 – -3 = ____
- 35 – -35 = ____
- 45 – -10 = ____

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.11.