6.7: Absolute Value
What if two people stood backtoback 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,
Below is a more formal definition of absolute value.
For any real number
The second part of this definition states that the absolute value of a negative number is its opposite (a positive number).
Example A
Evaluate
Solution:
The absolute value of a negative number is its inverse, or opposite. Therefore,
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
Example B
Find the distance between –5 and 8.
Solution:
Use the definition of distance. Let
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
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
Notice that if we let
So, it does not matter which number we pick for
Practice
Evaluate the absolute value.

250 
−12 
−25 
110
Find the distance between the points.
 12 and –11
 5 and 22
 –9 and –18
 –2 and 3

23 and –11  –10.5 and –9.75
 36 and 14
Mixed Review
 Solve:
6t−14<2t+7 .  The speed limit of a semitruck on the highway is between 45 mph and 65 mph.
 Write this situation as a compound inequality
 Graph the solutions on a number line.
 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?
 Simplify
12−−√×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?
 What is the additive inverse of 124?
 What is the multiplicative inverse of 14?
 Define the Addition Property of Equality.
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Here you'll review the meaning of absolute value and learn how to find the distance between two numbers on a number line.