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# 6.7: Absolute Value

Difficulty Level: Basic Created by: CK-12
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What if two people stood back-to-back and then walked in opposite directions? The first person walked 2 miles, while the second person walked 5 miles. How far apart would they be? Could you use a number line to represent this scenario? In this Concept, you'll review the meaning of absolute value and learn to find the distance between two numbers on a number line so that you can solve problems such as this one.

### Guidance

The absolute value of a number is the distance from zero on a number line. The numbers 4 and –4 are each four units away from zero on a number line. So, |4|=4\begin{align*}|4|=4\end{align*} and |4|=4\begin{align*}|-4|=4\end{align*}.

Below is a more formal definition of absolute value.

For any real number x\begin{align*}x\end{align*},

|x||x|=x for all x0=x(read the opposite of x) for all x<0\begin{align*}|x|& =x \ for \ all \ x \ge 0\\ |x|& =-x (read \ \text{the opposite of} \ x) \ for \ all \ x<0\end{align*}

The second part of this definition states that the absolute value of a negative number is its opposite (a positive number).

#### Example A

Evaluate |120|\begin{align*}|-120|\end{align*}.

Solution:

The absolute value of a negative number is its inverse, or opposite. Therefore, |120|=(120)=120\begin{align*}|-120|=-(-120)=120\end{align*}.

Distance on the Number Line

Because the absolute value is always positive, it can be used to find the distance between two values on a number line.

The distance between two values x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} on a number line is found by:

distance=|xy| or |yx|\begin{align*}distance=|x-y| \ or \ |y-x|\end{align*}

#### Example B

Find the distance between –5 and 8.

Solution:

Use the definition of distance. Let x=5\begin{align*}x=-5\end{align*} and y=8\begin{align*}y=8\end{align*}.

distance=|58|=|13|\begin{align*}distance=|-5-8|=|-13|\end{align*}

The absolute value of –13 is 13, so –5 and 8 are 13 units apart.

Check on the graph below that the length of the line between the points -5 and 8 is 13 units long:

#### Example C

Find the distance between -12 and 3.

Solution:

Use the definition of distance. Let x=3\begin{align*}x=3\end{align*} and y=12\begin{align*}y=-12\end{align*}.

distance=|3(12)|=|3+12|=|15|=15\begin{align*}distance=|3-(-12)|=|3+12|=|15|=15\end{align*}

The absolute value of 15 is 15, so 3 and -12 are 15 units apart.

### Guided Practice

Find the distance between 8 and -1.

Solution:

We use the distance formula. Let x=8\begin{align*}x=8\end{align*} and y=1\begin{align*}y=-1\end{align*}.

distance=|8(1)|=|8+1|=|9|=9\begin{align*}distance=|8-(-1)|=|8+1|=|9|=9\end{align*}

Notice that if we let x=1\begin{align*}x=-1\end{align*} and y=8\begin{align*}y=8\end{align*}, we get:

distance=|18|=|9|=9\begin{align*}distance=|-1-8|=|-9|=9\end{align*}.

So, it does not matter which number we pick for x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}. We will get the same answer.

### Practice

Evaluate the absolute value.

1. |250|\begin{align*}|250|\end{align*}
2. |12|\begin{align*}|-12|\end{align*}
3. |25|\begin{align*}|-\frac{2}{5}|\end{align*}
4. |110|\begin{align*}|\frac{1}{10}|\end{align*}

Find the distance between the points.

1. 12 and –11
2. 5 and 22
3. –9 and –18
4. –2 and 3
5. 23\begin{align*}\frac{2}{3}\end{align*} and –11
6. –10.5 and –9.75
7. 36 and 14

Mixed Review

1. Solve: 6t14<2t+7\begin{align*}6t-14<2t+7\end{align*}.
2. The speed limit of a semi-truck on the highway is between 45 mph and 65 mph.
1. Write this situation as a compound inequality
2. Graph the solutions on a number line.
1. Lloyd can only afford transportation costs of less than $276 per month. His monthly car payment is$181 and he sets aside \$25 per month for oil changes and other maintenance costs. How much can he afford for gas?
2. Simplify 12×3\begin{align*}\sqrt{12} \times \sqrt{3}\end{align*}.
3. A hush puppy recipe calls for 3.4 ounces of flour for one batch of 8 hush puppies. You need to make 56 hush puppies. How much flour do you need?
4. What is the additive inverse of 124?
5. What is the multiplicative inverse of 14?
6. Define the Addition Property of Equality.

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Color Highlighted Text Notes

### Vocabulary Language: English Spanish

distance on a line

The distance between two values $x$ and $y$ on a number line is found by: $distance=|x-y| \ or \ |y-x|$

Absolute Value

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

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Difficulty Level:
Basic
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