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# 4.8: Differences of Integers Using Absolute Value

Difficulty Level: At Grade Created by: CK-12
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Practice Differences of Integers Using Absolute Value

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This summer Logan is spending all of July driving across the United States. At the start of the trip Logan had $500. Over the course of the trip, Logan spent$792 traveling and charged it all to his credit card. How can Logan figure out how much money he has now?

In this concept, you will learn how to find the difference of integers using opposites.

### Guidance

The opposite of any integer, b\begin{align*}b\end{align*}, is b\begin{align*}-b\end{align*}. To find the opposite of an integer, change its sign.

Here are some examples of integers and their opposites.

• The opposite of 2 is -2.
• The opposite of -4 is 4.
• The opposite of -100 is 100.
• The opposite of 17 is -17.

There are many different strategies for subtracting integers. One strategy for subtracting involves opposites. To use this strategy you will be changing your subtraction problem into an addition problem.

To find ab\begin{align*}a-b\end{align*}, add a+(b)\begin{align*}a+(-b)\end{align*}.

In other words, to subtract two integers:

1. Take the opposite of the integer being subtracted (the second integer).

Let's look at an example.

Find the difference of 5(8)\begin{align*}5-(-8)\end{align*}.

First, you need to take the opposite of your second integer, -8. The opposite of -8 is 8.

5(8)=5+8\begin{align*}5-(-8)=5+8\end{align*}

5+8=13\begin{align*}5+8=13\end{align*}

The answer is 13. 5(8)=13\begin{align*}5-(-8)=13\end{align*}.

Let's look at another example.

Find the difference of 122\begin{align*}-12-2\end{align*}.

First, you need to take the opposite of your second integer, 2. The opposite of 2 is -2.

122=12+(2)\begin{align*}-12-2=-12+(-2)\end{align*}

Next, you can solve your addition problem. Here you have two integers with the same sign. To add, find the absolute value of each integer and add. |12|=12\begin{align*}|-12| = 12\end{align*} and |2|=2\begin{align*}|-2| = 2\end{align*} so you have

12+2=14\begin{align*}12+2=14\end{align*}.

Finally, determine the sign of your final answer by looking at the signs of the original integers. -12 and -2 were both negative so your answer will be negative.

12+(2)=14\begin{align*}-12+(-2)=-14\end{align*}

The answer is 14.122=14\begin{align*}-14.-12-2=-14\end{align*}.

### Guided Practice

The temperature inside a laboratory freezer was 10 C\begin{align*}-10^\circ \ C\end{align*}. A scientist at the lab then lowered the temperature inside the freezer so it was 5 C\begin{align*}5^\circ \ C\end{align*} less. What was the new temperature inside the freezer?

First, turn the word problem into a subtract problem. The room started out at -10 and then the temperature went down by 5 degrees. The subtraction problem is

105\begin{align*}-10-5\end{align*}.

To solve the subtraction problem, you first need to take the opposite of your second integer, 5. The opposite of 5 is -5.

105=10+(5)\begin{align*}-10-5=-10+(-5)\end{align*}

Next, you can solve your addition problem. Here we have two integers with the same sign. To add, find the absolute value of each integer and add. |10|=10\begin{align*}|-10| = 10\end{align*} and |5|=5\begin{align*}|-5| = 5\end{align*} so you have

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

Finally, determine the sign of your final answer by looking at the signs of the original integers. -10 and -5 were both negative so your answer will be negative.

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

The answer is -15. 105=15\begin{align*}-10-5=-15\end{align*}

The new temperature inside the freezer is 15 C\begin{align*}-15^\circ \ C\end{align*}.

### Examples

#### Example 1

Find the difference of 57\begin{align*}-5-7\end{align*}.

First, you need to take the opposite of your second integer, 7. The opposite of 7 is -7.

57=5+(7)\begin{align*}-5-7=-5+(-7)\end{align*}

Next, you can solve your addition problem. Here you have two integers with the same sign. To add, find the absolute value of each integer and add. |5|=5\begin{align*}|-5| = 5\end{align*}  and |7|=7\begin{align*}|-7| = 7\end{align*}  so you have

5+7=12\begin{align*}5+7=12\end{align*}

Finally, determine the sign of your final answer by looking at the signs of the original integers. -5 and -7 were both negative so your answer will be negative.

5+(7)=12\begin{align*}-5+(-7)=-12\end{align*}

The answer is -12. 57=12\begin{align*}-5-7=-12\end{align*}.

#### Example 2

Find the difference of 8(4)\begin{align*}8-(-4)\end{align*}.

First, you need to take the opposite of your second integer, -4. The opposite of -4 is 4.

8(4)=8+4\begin{align*}8-(-4)=8+4\end{align*}

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

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

#### Example 3

Find the difference of 12(8)\begin{align*}-12-(-8)\end{align*}.

First, you need to take the opposite of your second integer, -8. The opposite of -8 is 8.

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

Next, you can solve your addition problem. Here you have two integers with different signs. To add, find the absolute value of each integer and subtract. |12|=12\begin{align*}|-12| = 12\end{align*}  and |8|=8\begin{align*}|8| = 8\end{align*} so you have

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

Finally, determine the sign of your final answer by looking at the sign of the integer with the greater absolute value. -12 had the greater absolute value and it is negative, so your answer will be negative.

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

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

Remember Logan and his trip across the United States? He started out with $500 and spent$792 on his credit card over the course of his trip. He wants to know how much money he has now.

Since Logan spent more money than he has, he is now in debt and essentially has negative money. To figure out how much money he has, you need to subtract:

500792\begin{align*}500-792\end{align*}

First, you need to take the opposite of your second integer, 792. The opposite of 792 is -792.

500792=500+(792)\begin{align*}500-792=500+(-792)\end{align*}

Next, you can solve your addition problem. Here we have two integers with different signs. To add, find the absolute value of each integer and subtract. |500|=500\begin{align*}|500| = 500\end{align*}  and |792|=792\begin{align*}|-792| = 792\end{align*}  so you have

792500=292\begin{align*}792-500=292\end{align*}.

Finally, determine the sign of your final answer by looking at the sign of the integer with the greater absolute value. -792 had the greater absolute value and it is negative, so your answer will be negative.

500+(792)=292\begin{align*}500+(-792)=-292\end{align*}

The answer is -292. 500792=292\begin{align*}500-792=-292\end{align*}.

Logan now has -\$292.

### Video Review

https://youtu.be/-BzR1DwJznY

### Explore More

Find each difference using opposites.

1. \begin{align*}15 − 7\end{align*}

2. \begin{align*}−7 − 12\end{align*}

3. \begin{align*}0 − 4\end{align*}

4. \begin{align*}13 − (−9)\end{align*}

5. \begin{align*}−21 − 4\end{align*}

6. \begin{align*}33 − (−4)\end{align*}

7. \begin{align*}−11 − (−8)\end{align*}

8. \begin{align*}18 − 28\end{align*}

9. \begin{align*}28 − (−8)\end{align*}

10. \begin{align*}13 − 18\end{align*}

11. \begin{align*}−21 − (−8)\end{align*}

12. \begin{align*}−9 − (−38)\end{align*}

13. \begin{align*}22 − 8\end{align*}

14. \begin{align*}25 − 38\end{align*}

15. \begin{align*}19 − (−19)\end{align*}

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