Module 3: The Absolute Value - Portland Community College
Section II: Functions, Inequalities, and the Absolute Value
[pic]
Module 3: The Absolute Value
The absolute value function is defined as follows:
[pic]
The graph of [pic] is shown in Figure 1 below.
[pic]
Figure 1: [pic]
[pic]
[pic] example: Solve [pic].
SOLUTION: Since both [pic] and [pic], both [pic] and [pic] are solutions, so the solution set is [pic].
[pic]
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|The Absolute-Value Principle for Equations |
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|For any positive number a and any algebraic expression X: |
| |
|▪ if [pic] then the equation [pic] has two solutions. |
| |
|▪ if [pic] then the equation [pic] has one solution. |
| |
|▪ if [pic] then the equation [pic] has no solutions. |
|[pic] |CLICK HERE FOR AN ABSOLUTE VALUE EQUATION EXAMPLE |
[pic]
|[pic] |CLICK HERE FOR ANOTHER ABSOLUTE VALUE EQUATION EXAMPLE |
[pic]
[pic] example: Solve [pic].
SOLUTION: Removing the absolute value symbols yields
[pic] or [pic]
which implies that
[pic] or [pic].
Thus, the solution set is [pic].
[pic]
[pic] example: Solve [pic].
SOLUTION: In order that [pic] we need [pic]. Thus, [pic] and the solution set is {–10}.
[pic]
[pic] example: Solve [pic] graphically.
SOLUTION: Below, we've graphed [pic] and [pic] (i.e., the left and right sides of the inequality). Since the absolute value function is below the horizontal line [pic] when [pic], the solution set for [pic] is [pic] which can be written in interval notation as [pic]. Another way to say this is that the y-values (i.e., the output values) of [pic] are less than [pic] when [pic], so the solution set for [pic] is the interval [pic].
[pic]
Figure 2: [pic] and [pic]
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|Principles for Solving Absolute Value Problems |
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|For any positive number a and any algebraic expression X : |
| |
|▪ the solutions of [pic] are those numbers that satisfy [pic]. |
| |
|▪ the solutions of [pic] are those numbers that satisfy [pic]. |
| |
|▪ the solutions of [pic] are those numbers that satisfy [pic]. |
|[pic] |CLICK HERE FOR AN ABSOLUTE VALUE INEQUALITY EXAMPLE |
[pic]
|[pic] |CLICK HERE FOR ANOTHER ABSOLUTE VALUE INEQUALITY EXAMPLE |
[pic]
[pic] example: Solve [pic].
SOLUTION: Removing the absolute value symbols yields
[pic] or [pic],
which can be simplified to
[pic] or [pic].
Thus, the solution set for [pic] is [pic]. We’ve graphed this set on the number line below.
[pic]
[pic]
[pic] Try these yourself and check your answers.
a. Solve [pic].
b. Solve [pic].
c. Solve [pic].
SOLUTIONS:
a.
[pic]
Thus, the solution set is {4, –1}.
b. The equation [pic] has no solutions since the absolute value of a number is never negative.
c.
[pic]
Thus, the solution set is [pic].
[pic]
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