What if you were given two points like -8 and 12? How could you find the distance between them on a number line? After completing this Concept, you'll be able to use absolute value properties to solve problems like this one.
Timmy is trying out his new roller skates. He’s not allowed to cross the street yet, so he skates back and forth in front of his house. If he skates 20 yards east and then 10 yards west, how far is he from where he started? What if he skates 20 yards west and then 10 yards east?
The absolute value of a number is its distance from zero on a number line. There are always two numbers on the number line that are the same distance from zero. For instance, the numbers 4 and -4 are each a distance of 4 units away from zero.
In fact, for any real number
Absolute value has no effect on a positive number, but changes a negative number into its positive inverse.
Evaluate the following absolute values.
Absolute value is very useful in finding the distance between two points on the number line. The distance between any two points
For example, the distance from 3 to -1 on the number line is
We could have also found the distance by subtracting in the opposite order:
Find the distance between the following points on the number line.
a) 6 and 15
b) -5 and 8
c) -3 and -12
Distance is the absolute value of the difference between the two points.
Remember: When we computed the change in
Solve an Absolute Value Equation
We now want to solve equations involving absolute values. Consider the following equation:
This means that the distance from the number
When we solve absolute value equations we always consider two possibilities:
- The expression inside the absolute value sign is not negative.
- The expression inside the absolute value sign is negative.
Then we solve each equation separately.
Solve the following absolute value equations.
a) There are two possibilities:
b) There are two possibilities:
Watch this video for help with the Examples above.
- The absolute value of a number is its distance from zero on a number line.
|x|=xif xis not negative, and |x|=−xif xis negative.
- An equation or inequality with an absolute value in it splits into two equations, one where the expression inside the absolute value sign is positive and one where it is negative. When the expression within the absolute value is positive, then the absolute value signs do nothing and can be omitted. When the expression within the absolute value is negative, then the expression within the absolute value signs must be negated before removing the signs.
- Inequalities of the type
|x|<acan be rewritten as “ −a<x<a.”
- Inequalities of the type
|x|>bcan be rewritten as “ x<−bor x>b.”
Find the distance between the values
The distance is the absolute value of the difference:
The distance between the two points is
Evaluate the absolute values.
Find the distance between the points.
- 12 and -11
- 5 and 22
- -9 and -18
- -2 and 3
- -0.012 and 1.067