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.
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 and |−4|=4.
Below is a more formal definition of absolute value.
For any real number x,
|x||x|=x for all x≥0=−x(read the opposite of x) for all x<0
The second part of this definition states that the absolute value of a negative number is its opposite (a positive number).
The absolute value of a negative number is its inverse, or opposite. Therefore, |−120|=−(−120)=120.
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 and y on a number line is found by:
distance=|x−y| or |y−x|
Find the distance between –5 and 8.
Use the definition of distance. Let x=−5 and y=8.
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:
Find the distance between -12 and 3.
Use the definition of distance. Let x=3 and y=−12.
The absolute value of 15 is 15, so 3 and -12 are 15 units apart.
Find the distance between 8 and -1.
We use the distance formula. Let x=8 and y=−1.
Notice that if we let x=−1 and y=8, we get:
So, it does not matter which number we pick for x and y. We will get the same answer.
Evaluate the absolute value.
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
- Solve: 6t−14<2t+7.
- The speed limit of a semi-truck 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.