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Absolute Value

Absolute value is simply the distance from zero of any given number or integer. Check out our modules and lessons below.

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Absolute Value

Suppose 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? 

Absolute Value

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, \begin{align*}|4|=4\end{align*} and \begin{align*}|-4|=4\end{align*}.

Below is a more formal definition of absolute value.

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

\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).

Let's evaluate the following absolute value:

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

The absolute value of a negative number is its inverse, or opposite. Therefore, \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 \begin{align*}x\end{align*} and \begin{align*}y\end{align*} on a number line is found by:

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

Let's use absolute value to find the following values:

  1. The distance between –5 and 8.

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

\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:

  1. The distance between -12 and 3.

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

\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.

   

 

 

Examples

Example 1

Earlier, you were told that two people stood back-to-back and walked in opposite directions. The first person walked 2 miles and the second person walked 5 miles. How far apart would they be?

This is a distance problem. Assume that the point that the people start walking from is 0 on the number line and that the person walking 2 miles was walking to the left or in the negative direction. Then the person who walked 5 miles is walking to the right or the positive direction. Therefore, you want to find the distance between -2 and 5. You can use the distance formula shown in this section. 

\begin{align*}distance=|5-(-2)|=|5+2|=|7|=7\end{align*}

The two people are 7 miles apart. 

Example 2

Find the distance between 8 and -1.

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

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

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

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

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

Review

Evaluate the absolute value.

  1. \begin{align*}|250|\end{align*}
  2. \begin{align*}|-12|\end{align*}
  3. \begin{align*}|-\frac{2}{5}|\end{align*}
  4. \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. \begin{align*}\frac{2}{3}\end{align*} and –11
  6. –10.5 and –9.75
  7. 36 and 14

Mixed Review

  1. Solve: \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.
  3. 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?
  4. Simplify \begin{align*}\sqrt{12} \times \sqrt{3}\end{align*}.
  5. 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?
  6. What is the additive inverse of 124?
  7. What is the multiplicative inverse of 14?
  8. Define the Addition Property of Equality.

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.7. 

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Vocabulary

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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