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# 4.5: Sums of Integers Using Absolute Value

Difficulty Level: At Grade Created by: CK-12
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The weather has been very cold this winter in North Dakota where Lauren lives! This morning the meteorologist said that the temperature was 12F\begin{align*}-12^\circ F\end{align*}. Lauren was hoping to go for a ride on her horse on Saturday and luckily for Lauren, it is supposed to warm up a bit by the weekend. The meteorologist said the temperature will be 23F\begin{align*}23^\circ F\end{align*} higher on Saturday. How could Lauren use absolute value to help her to add the two integers in order to figure out what the temperature will be on Saturday?

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

Let's look at an example.

Find the sum of 13+(12)\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. |13|=13\begin{align*}|-13| = 13\end{align*} and |12|=12\begin{align*}|-12| = 12\end{align*}, so you have

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

Let's look at another example.

Find the sum of 13+(12)\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. |13|=13\begin{align*}|13| = 13\end{align*} and |12|=12\begin{align*}|-12| = 12\end{align*}, so the lesser absolute value is 12 and the greater absolute value is 13. So you have

1312=1\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 13+(12)=1\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 12F\begin{align*}-12^\circ F\end{align*}, but it is supposed to warm up by 23F\begin{align*}23^\circ F\end{align*} on Saturday when Lauren is hoping to go for a ride on her horse. Lauren wanted to figure out what the temperature will be on Saturday.

In order to figure out what the temperature will be on Saturday, Lauren would need to add 12+23\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. |12|=12\begin{align*}|-12| = 12\end{align*} and |23|=23\begin{align*}|23| = 23\end{align*}, so the lesser absolute value is 12 and the greater absolute value is 23. So you have

2312=11\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 12+23=11\begin{align*}-12 + 23 = 11\end{align*}.

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

#### Example 2

Find the sum of 12+(18)\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. |12|=12\begin{align*}|-12| = 12\end{align*} and |18|=18\begin{align*}|-18| = 18\end{align*}, so you have

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

#### Example 3

Find the sum of 12+8\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. |12|=12\begin{align*}|-12| = 12\end{align*} and |8|=8\begin{align*}|8| = 8\end{align*}, so the lesser absolute value is 8 and the greater absolute value is 12. So you have

128=4\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 12+8=4\begin{align*}-12 + 8 = -4\end{align*}.

#### Example 4

Find the sum of 9+(12)\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. |9|=9\begin{align*}|-9| = 9\end{align*} and |12|=12\begin{align*}|-12| = 12\end{align*}, so you have

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

#### Example 5

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

First, add the first two integers, 7+(2)\begin{align*}7 + (-2)\end{align*}. These two integers have different signs, one negative and one positive.

Step 1 is to subtract the lesser absolute value from the greater absolute value. |7|=7\begin{align*}|7| = 7\end{align*} and |2|=2\begin{align*}|-2| = 2\end{align*}, so the lesser absolute value is 2 and the greater absolute value is 7. So you have

72=5\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.

7+(2)=5\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. |5|=5\begin{align*}|5| = 5\end{align*} and |10|=10\begin{align*}|-10| = 10\end{align*}, so the lesser absolute value is 5 and the greater absolute value is 10. So you have

105=5\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.

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

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

### Review

Use absolute values to find each sum.

1.  20+(9)\begin{align*}20 + (−9)\end{align*}
2.  6+(9)\begin{align*}−6 + (−9)\end{align*}
3. 4+(9)\begin{align*}4 + (−9)\end{align*}
4.  12+(19)\begin{align*}−12 + (−19)\end{align*}
5. 2+(5)\begin{align*}−2 + (−5)\end{align*}
6.  11+(13)\begin{align*}−11 + (−13)\end{align*}
7. 30+(40)\begin{align*}−30 + (−40)\end{align*}
8. 8+3+(9)\begin{align*}−8 + 3 + (−9)\end{align*}
9. 6+1+(9)\begin{align*}6 + 1 + (−9)\end{align*}
10. (8)+20\begin{align*}(−8) + −20\end{align*}
11. (6)+8+(4)\begin{align*}(−6) + 8 + (−4)\end{align*}
12. (2)+8+(12)\begin{align*}(2) + 8 + (−12)\end{align*}
13. 5+7+(15)\begin{align*}5 + 7 + (−15)\end{align*}
14. 5+7+(15)\begin{align*}-5 + −7 + (−15)\end{align*}
15. 15+17+(12)\begin{align*}−15 + −17 + (12)\end{align*}

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