# 4.5: Sums of Integers Using Absolute Value

**At Grade**Created by: CK-12

**License**: CC BY-NC 3.0

The weather has been very cold this winter in North Dakota where Lauren lives! This morning the meteorologist said that the temperature was \begin{align*}-12^\circ F\end{align*}

In this concept, you will learn how to find the sum of integers using absolute value.

### Adding Integers Using Absolute Value

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

There are many different strategies for adding integers. One strategy for adding involves absolute value. The **absolute value** of a number is its distance from zero on the number line. Remember that the symbol for absolute value is | |. The absolute value of an integer will always be positive (or zero).

To add two integers using the absolute value strategy first look at the signs of the two integers.

- If the two integers have the same sign:
- Add their absolute values.
- Give the answer the same sign as the two original integers.

- If the two integers have different signs:
- Subtract the lesser absolute value from the greater absolute value.
- Give the answer the same sign as the integer with the greater absolute value.

Let's look at an example.

Find the sum of \begin{align*}-13 + (-12)\end{align*}

First, look at the signs of the integers. Both integers have the same sign, a negative sign.

Step 1 is to add their absolute values. \begin{align*}|-13| = 13\end{align*}

\begin{align*}13 + 12 = 25\end{align*}

Step 2 is to give the answer the same sign as the original two integers. In this case, that is a negative sign. So the 25 becomes -25.

The answer is -25. So \begin{align*}-13 + (-12) = -25\end{align*}

Let's look at another example.

Find the sum of \begin{align*}13 + (-12)\end{align*}

First, look at the signs of the integers. This time the integers have different signs, one negative and one positive.

Step 1 is to subtract the lesser absolute value from the greater absolute value. \begin{align*}|13| = 13\end{align*}

\begin{align*}13 - 12 = 1\end{align*}

Step 2 is to give the answer the same sign as the integer with the greater absolute value. 13 had the greater absolute value and it is a positive integer. So your final answer is positive.

The answer is 1. So \begin{align*}13 + (-12) = 1\end{align*}

You can use the same strategy to find the sum of more than two integers. Start by adding the first pair of integers. Then, add the next integer to the sum of the first pair. Continue in this way until you have added all of the integers.

### Examples

#### Example 1

Earlier, you were given a problem about Lauren and her cold winter in North Dakota. It is currently \begin{align*}-12^\circ F\end{align*}

In order to figure out what the temperature will be on Saturday, Lauren would need to add \begin{align*}-12 + 23\end{align*}

First, look at the signs of the integers. The integers have different signs, one negative and one positive.

Step 1 is to subtract the lesser absolute value from the greater absolute value. \begin{align*}|-12| = 12\end{align*}

\begin{align*}23 - 12 = 11\end{align*}

Step 2 is to give the answer the same sign as the integer with the greater absolute value. 23 had the greater absolute value and it is a positive integer. So your answer is positive.

The answer is 11. So \begin{align*}-12 + 23 = 11\end{align*}

Lauren can expect that it will be \begin{align*}11^\circ F\end{align*}

#### Example 2

Find the sum of \begin{align*}-12 + (-18)\end{align*}

First, look at the signs of the integers. Both integers have the same sign, a negative sign.

Step 1 is to add their absolute values. \begin{align*}|-12| = 12\end{align*}

\begin{align*}12 + 18 = 30\end{align*}

Step 2 is to give the answer the same sign as the original two integers. In this case, that is a negative sign. So the 30 becomes -30.

The answer is -30. So \begin{align*}-12 + (-18) = -30\end{align*}

#### Example 3

Find the sum of \begin{align*}-12 + 8\end{align*}

First, look at the signs of the integers. The integers have different signs, one negative and one positive.

Step 1 is to subtract the lesser absolute value from the greater absolute value. \begin{align*}|-12| = 12\end{align*}

\begin{align*}12 - 8 = 4\end{align*}

Step 2 is to give the answer the same sign as the integer with the greater absolute value. 12 had the greater absolute value and it is a negative integer. So your answer is negative and the 4 becomes a -4.

The answer is -4. So \begin{align*}-12 + 8 = -4\end{align*}

#### Example 4

Find the sum of \begin{align*}-9 + (-12)\end{align*}

First, look at the signs of the integers. Both integers have the same sign, a negative sign.

Step 1 is to add their absolute values. \begin{align*}|-9| = 9\end{align*}

\begin{align*}9 + 12 = 21\end{align*}

Step 2 is to give the answer the same sign as the original two integers. In this case, that is a negative sign. So the 21 becomes -21.

The answer is -21. So \begin{align*}-9 + (-12) = -21\end{align*}

#### Example 5

Find the sum of \begin{align*}7 + (-2) + (-10)\end{align*}

First, add the first two integers, \begin{align*}7 + (-2)\end{align*}

Step 1 is to subtract the lesser absolute value from the greater absolute value. \begin{align*}|7| = 7\end{align*}

\begin{align*}7 - 2 = 5\end{align*}

Step 2 is to give the answer the same sign as the integer with the greater absolute value. 7 had the greater absolute value and it is a positive integer. So your result is positive.

\begin{align*}7 + (-2) = 5\end{align*}

Next, take your result of 5 and add the third integer from the original sum, (-10). These two integers also have different signs, one negative and one positive.

Again, Step 1 is to subtract the lesser absolute value from the greater absolute value. \begin{align*}|5| = 5\end{align*}

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

Again, Step 2 is to give the answer the same sign as the integer with the greater absolute value. -10 had the greater absolute value and it is a negative integer. So your result becomes negative.

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

The final answer is -5. So \begin{align*}7 + (-2) + (-10) = -5\end{align*}

### Review

Use absolute values to find each sum.

- \begin{align*}20 + (−9)\end{align*}
20+(−9) - \begin{align*}−6 + (−9)\end{align*}
−6+(−9) - \begin{align*}4 + (−9)\end{align*}
4+(−9) - \begin{align*}−12 + (−19)\end{align*}
- \begin{align*}−2 + (−5)\end{align*}
- \begin{align*}−11 + (−13)\end{align*}
- \begin{align*}−30 + (−40)\end{align*}
- \begin{align*}−8 + 3 + (−9)\end{align*}
- \begin{align*}6 + 1 + (−9)\end{align*}
- \begin{align*}(−8) + −20\end{align*}
- \begin{align*}(−6) + 8 + (−4)\end{align*}
- \begin{align*}(2) + 8 + (−12)\end{align*}
- \begin{align*}5 + 7 + (−15)\end{align*}
- \begin{align*}-5 + −7 + (−15)\end{align*}
- \begin{align*}−15 + −17 + (12)\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 4.5.

### Image Attributions

**[1]****^**License: CC BY-NC 3.0

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In this concept, you will learn how to find the sum of integers using absolute value.

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Dec 02, 2015## Last Modified:

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