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# Sums of Integers on a Number Line

## Add positive and negative numbers using a number line.

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Sums of Integers on a Number Line

Leslie is saving up for concert tickets and wants to keep track of the amount of money she has. She started with $20 that her parents gave her. Then, she spent$6 on a sandwich. Next, she made $15 babysitting. Last, she made$10 mowing her neighbor's lawn. How could Leslie represent her earning/spending of money with integers and then add those integers in order to figure out how much money she now has?

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

### Adding Integers Using a Number Line

Integers are the set of whole numbers and their opposites.

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

1. First, draw a number line.
2. Then, find the location of the first integer on the number line.
3. Next, if the second integer is positive, move that many units to the right from the location of the first integer. If the second integer is negative, move that many units to the left from the location of the first integer.

Let's look at an example.

Use a number line to find the sum of \begin{align*}\text{-}4 + (\text{-}6)\end{align*}.

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

Next, notice that the second integer, -6, is negative. This means you will be moving to the LEFT. Starting at -4, move to the LEFT 6 units.

You end up on -10. The answer is -10.

Let's look at another example.

Use a number line to find the sum of \begin{align*}4 + (\text{-}6)\end{align*}.

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

Next, notice that the second integer, -6, is negative. This means you will be moving to the left. Starting at 4, move to the left 6 units.

You end up on -2. The answer is -2.

Let's look at one more example.

Use a number line to find the sum of \begin{align*}\text{-}4 + 6\end{align*}.

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

Next, notice that the second integer, 6, is positive. This means you will be moving to the right. Starting at -4, move to the right 6 units.

You end up on 2. The answer is 2.

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 Leslie, who was saving up for concert tickets.

She started with $20, then spent$6, then earned $15, and finally earned another$10. Leslie wanted to know how much money she now has.

First, change the money amounts to integers.

• She started with $20 so that is +20. • She then spent$6 so that is -6.
• Next she earned $15 so that is +15. • Finally she earned another$10 so that is +10.

In order to figure out how much money Leslie has, you need to find the sum of the four amounts.

\begin{align*}20 + (-6) + 15 + 10\end{align*}

You can use a number line to help her to find the sum.



Start by finding the location of 20 on the number line.

Next, notice that the second integer, -6, is negative. This means you will be moving to the left. Starting at 20, move to the left 6 units.

You are now at 14.

Next, notice that the third integer, 15, is positive. This means you will be moving to the right. Starting at 14, move to the right 15 units.

You are now at 29.

Finally, notice that the fourth integer, 10, is also positive. This means you will again be moving to the right. Starting at 29, move to the right 10 units.

Leslie now has \$39.

#### Example 2

Use a number line to find the sum of \begin{align*}\text{-}8 + 3 + (\text{-}2)\end{align*}.

In this problem you are finding the sum of three integers. You can use the same process as before, you will just need to repeat step 3.

Then, find the location of -8 (the first integer in your sum) on the number line.

Next, notice that the second integer, 3, is positive. This means you will be moving to the right. Starting at -8, move to the right 3 units.

You are now at -5.

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

You end up on -7.

#### Example 3

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

Then, find the location of -4 on the number line.

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

You end up on -7.

#### Example 4

Use a number line to find the sum of \begin{align*}\text{-}8 + 7\end{align*}.

Then, find the location of -8 on the number line.

Next, notice that the second integer, 7, is positive. This means you will be moving to the right. Starting at -8, move to the right 7 units.

You end up on -1.

#### Example 5

Use a number line to find the sum of \begin{align*}8 + (\text{-}12)\end{align*}.

Then, find the location of 8 on the number line.

Next, notice that the second integer, -12, is negative. This means you will be moving to the left. Starting at 8, move to the left 12 units.

You end up on -4.

### Review

1. \begin{align*}2 + 5\end{align*}
2. \begin{align*}(\text{-}4) + 4\end{align*}
3. \begin{align*}(\text{-}3) + (\text{-}3)\end{align*}
4.  \begin{align*}8 + (\text{-}6)\end{align*}
5. \begin{align*}\text{-}8 + (\text{-}6)\end{align*}
6. \begin{align*}\text{-}2 + (\text{-}6)\end{align*}
7.  \begin{align*}8 + 5 + (\text{-}2)\end{align*}
8.  \begin{align*}5 + (\text{-}6) + 2\end{align*}
9. \begin{align*}2 + (\text{-}2) + 7\end{align*}
10. \begin{align*}\text{-}2 + (\text{-}6) + 5\end{align*}
11. \begin{align*}\text{-}8 + 3 + (\text{-}4)\end{align*}
12. \begin{align*}\text{-}6 + 6 + (\text{-}5)\end{align*}
13. \begin{align*}\text{-}8 + (\text{-}8) + 3\end{align*}
14. \begin{align*}\text{-}7 + (\text{-}6) + 3\end{align*}
15. \begin{align*}\text{-}9 + (\text{-}6) + 11\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Sum

The sum is the result after two or more amounts have been added together.

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