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# Sums of Integers Using Absolute Value

## Add positive and negative numbers by using the distance a value is from zero.

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Sums of Integers Using Absolute Value

Molly lives in Alaska. The temperature outside her home at 6:00 A.M. one day last February was 2F\begin{align*}-2^\circ F\end{align*}. Six hours later, the temperature had risen by 5F\begin{align*}5^\circ F\end{align*}. What was the temperature six hours later?

### Guidance

Previously we worked with number lines to add integers. Using a number line is one strategy for adding integers.

Now let’s look at another strategy.

Another strategy for adding integers involves using absolute values. An absolute value is the distance or the number of units that a number is from zero. Remember, with absolute value, the sign doesn’t matter. You will see the symbol | |\begin{align*}| \ |\end{align*} with an integer in the middle when absolute value is required.

Here are the steps to the absolute value strategy.

• To add two integers with the same sign, add their absolute values. Then give the answer the same sign as the two original integers.
• To add two integers with different signs, subtract the lesser absolute value from the greater absolute value. Then give the answer the same sign as the integer with the greater absolute value.

Take a few minutes to write these steps down in your notebook. Then continue.

How do we apply these steps?

Let’s see what this looks like in action.

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

Both integers being added have the same sign––a negative sign. So, add their absolute values.

Since |13|=13\begin{align*}|-13|=13\end{align*} and |12|=12\begin{align*}|-12|=12\end{align*}, add those values.

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

Give that answer, 25, the same sign as the original two integers––a negative sign.

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

Now you have seen this strategy with two negative numbers. Next, let’s see how it applies with a negative and a positive number.

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

The two integers being added have different signs. So, subtract their absolute values.

|13|=13\begin{align*}|13|=13\end{align*} and |12|=12\begin{align*}|-12|=12\end{align*}, so subtract the lesser absolute value from the greater absolute value :

1312=1\begin{align*}13-12=1\end{align*}.

Give that answer, 1, the same sign as the integer with the greater absolute value. 13>12, so 13 has a greater absolute value than -12. Give the answer a positive sign.

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

What happens if we are finding the sum of more than two integers?

You can use this same strategy to add three or more integers. When adding three or more integers, remember that the associative property of addition states that the grouping of numbers being added does not matter.

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

According to the associative property of addition, the integers being added can be grouped in any way. Here is one way to group numbers. Notice that we used brackets because parentheses are helpful when separating a negative sign and an addition sign. Brackets can mean the same thing as parentheses in thee examples.

[7+2]+(10)\begin{align*}[7+2]+(-10)\end{align*}

If you group the numbers this way, you will add 7+2\begin{align*}7+2\end{align*} first. Then you will add (-10) to that sum.

To add 7+2\begin{align*}7+2\end{align*}, first notice that both integers have the same sign––a positive sign. So, add their absolute values.

|7|+|2|=7+2=9\begin{align*}|7|+|2|=7+2=9\end{align*}

Since the two original integers both had positive signs, give the sum a positive sign.

[7+2]+(10)=9+(10)\begin{align*}[7+2]+(-10)=9+(-10)\end{align*}

Now, add 9+(10)\begin{align*}9+(-10)\end{align*}. Since both integers have different signs, find the absolute value of each integer.

|9|=9\begin{align*}|9|=9\end{align*} and |10|=10\begin{align*}|-10|=10\end{align*}, so subtract the lesser absolute value from the greater absolute value.

109=1\begin{align*}10-9=1\end{align*}

Give that answer, 1, the same sign as the integer with the greater absolute value. 10>9\begin{align*}10>9\end{align*}, so -10 has a greater absolute value than 9. Give the answer a negative sign.

So, 7+2+(10)=9+(10)=1\begin{align*}7+2+(-10)=9+(-10)=-1\end{align*}.

Could I use a number line too?

Sure. A number line would have worked too. It just would have involved the step of drawing the number line and then working through the math. Either way, you would still end up with the same answer.

Let’s practice using this method of adding integers.

#### Example A

4+7+5\begin{align*}-4 + 7 + -5\end{align*}

Solution:2\begin{align*}-2\end{align*}

#### Example B

9+12+8\begin{align*}-9 + -12 + 8\end{align*}

Solution:13\begin{align*}-13\end{align*}

#### Example C

12+29+18\begin{align*}-12 + 29 + -18\end{align*}

Solution:1\begin{align*}-1\end{align*}

Now let's go back to the problem from the beginning of the Concept.

Molly lives in Alaska. The temperature outside her home at 6:00 A.M. one day last February was 2F\begin{align*}-2^\circ F\end{align*}. Six hours later, the temperature had risen by 5F\begin{align*}5^\circ F\end{align*}. What was the temperature six hours later?

The problem says that the temperature had risen six hours later. This means that the temperature had increased, so you should add. To find the new temperature, you can add the amount of the increase to the previous temperatures.

You can find the temperature six hours later by using one of these equations.

2F+5For2+5=?=?

Both integers have different signs. So, find the absolute values of both integers. Then subtract the integer with the lesser absolute value from the absolute value of the integer with the greater absolute value.

|2|=2\begin{align*}|-2|=2\end{align*} and |5|=5\begin{align*}|5|=5\end{align*}, so subtract: 52=3\begin{align*}5-2=3\end{align*}.

Since 5>2\begin{align*}5>2\end{align*}, and 5F\begin{align*}5^\circ F\end{align*} has a positive sign, the temperature six hours later must be 3F\begin{align*}3^\circ F\end{align*}.

### Vocabulary

Integer
the set of whole numbers and their opposites.
Sum
Absolute Value
the distance or number of units that an integer is from zero.

### Guided Practice

Here is one for you to try on your own.

Find the sum of these integers.

12+18\begin{align*}-12+-18\end{align*}

To find this sum, we can add the absolute value of each integer since both integers are negative.

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

Now we add a negative sign.

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

### Practice

Directions: 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. \begin{align*}-8+3+(-9)\end{align*}

9. \begin{align*}6+1+(-9)\end{align*}

10. \begin{align*}(-8)+-20\end{align*}

11. \begin{align*}(-6)+8+(-4)\end{align*}

12. \begin{align*}(2)+8+(-12)\end{align*}

13. \begin{align*}5+7+(-15)\end{align*}

14. \begin{align*}-5+-7+(-15)\end{align*}

15. \begin{align*}-15+-17+(12)\end{align*}

### Vocabulary Language: English

Absolute Value

Absolute Value

The absolute value of a number is the distance the number is from zero. Absolute values are never negative.
Integer

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

Sum

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