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# Differences of Integers Using Absolute Value

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

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Differences of Integers Using Absolute Value

Remember Cameron and the diving from the Find Differences of Integers on a Number Line Concept?

While on the plane, Cameron looked at photos from his Dad's deep sea dive. On a shark dive, Cameron's Dad had gone down to a depth of 80 feet with hopes of seeing a shark. After ten minutes or so, he had spotted a beautiful shark swimming above him. Cameron’s Dad went up about 20 feet to try to catch a picture of the shark.

He did get a few good shots before the shark swam away.

“What depth did you see the shark at?” Cameron asked his Dad showing him the picture.

Do you know? To figure this out, you will need to subtract integers. Subtracting integers is the focus of this Concept. By the end of it, you will know the depth of the shark.

### Guidance

Previously we worked on subtracting integers using a number line. Another strategy for subtracting integers involves using opposites. Remember, you can find the opposite of an integer by changing the sign of an integer. The opposite of any integer, \begin{align*}b\end{align*}, would be \begin{align*}-b\end{align*}.

For any two integers, \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, the difference of \begin{align*}a-b\end{align*} can be found by adding \begin{align*}a+(-b)\end{align*} So, to subtract two integers, take the opposite of the integer being subtracted and then add that opposite to the first integer.

Sure.

For any two integers, \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, the difference of \begin{align*}a-b\end{align*} can be found by adding \begin{align*}a+(-b)\end{align*}. So, to subtract two integers, take the opposite of the integer being subtracted and then add that opposite to the first integer.

Write this down in your notebook and then continue with the Concept.

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

The integer being subtracted is -8. The opposite of that integer is 8, so add 8 to 5.

\begin{align*}5-(-8)=5+8=13\end{align*}.

So, the difference of \begin{align*}5-(-8)\end{align*} is 13.

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

The integer being subtracted is -2. The opposite of that integer is 2, so add 2 to -12.

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

\begin{align*}|-12|=12\end{align*} and \begin{align*}|2|=2\end{align*}, so subtract the lesser absolute value from the greater absolute value.

\begin{align*}12-2=10\end{align*}

Give that answer the same sign as the integer with the greater absolute value. \begin{align*}12>2\end{align*}, so -12 has a greater absolute value than 2. Give the answer a negative sign.

So, the difference of \begin{align*}-12-(-2)\end{align*} is -10.

Find the difference of \begin{align*}-20-3\end{align*}.

The integer being subtracted is 3. The opposite of that integer is -3, so add -3 to -20.

\begin{align*}-20-3=-20+(-3)\end{align*}.

Add as you would add any integers with the same sign––a negative sign.

\begin{align*}|-20|=20\end{align*} and \begin{align*}|-3|=3\end{align*}, so add their absolute values:

\begin{align*}20+3=23\end{align*}

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

So, the difference of \begin{align*}-20-3\end{align*} is -23.

Now take a few minutes to practice what you have learned. Find the differences using opposites.

#### Example A

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

Solution: \begin{align*}-12\end{align*}

#### Example B

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

Solution: \begin{align*}12\end{align*}

#### Example C

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

Solution: \begin{align*}-4\end{align*}

Here is the original problem once again.

While on the plane, Cameron looked at photos from his Dad's deep sea dive. On a shark dive, Cameron's Dad had gone down to a depth of 80 feet with hopes of seeing a shark. After ten minutes or so, he had spotted a beautiful shark swimming above him. Cameron’s Dad went up about 20 feet to try to catch a picture of the shark.

He did get a few good shots before the shark swam away.

“What depth did you see the shark at?” Cameron asked his Dad showing him the picture.

To find the depth that Cameron’s Dad saw the shark, we need to write a subtraction problem and solve it. Remember that depth has to do with below the surface, so we use negative integers to represent different depths.

-80 was his starting depth, then he went up -20 so we take away 20 feet.

\begin{align*}-80 - (-20) = -60 \ feet\end{align*}

Cameron’s Dad saw the shark at 60 feet below the surface.

### Vocabulary

Difference
the answer in a subtraction problem.
Integer
the set of whole numbers and their opposites.

### Guided Practice

Here is one for you to try on your own.

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

The problem says that the temperature was lowered. This means that the temperature decreased, so you should subtract. To find the new temperature, you can subtract the amount by which the temperature was lowered from the original temperature, using one of these equations.

\begin{align*}-10^\circ C-5^\circ C &= ?\\ \text{or} \qquad -10-5&=?\end{align*}

The integer being subtracted is 5. The opposite of that integer is -5, so add -5 to -10.

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

Add as you would add any integers with the same sign––a negative sign.

\begin{align*}|-10|=10\end{align*} and \begin{align*}|-5|=5\end{align*}, so add their absolute values.

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

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

So, the difference of \begin{align*}-10-5\end{align*} is -15.

This means that the new temperature inside the freezer must be \begin{align*}-15^\circ\end{align*} Celsius.

### Practice

Directions: 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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### Vocabulary Language: English

TermDefinition
Difference The result of a subtraction operation is called a difference.
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...

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