BackSolving Absolute Value Inequalities in Precalculus
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Absolute Value Inequalities
Understanding Absolute Value Inequalities
Absolute value inequalities are inequalities that involve the absolute value function. The absolute value of a number represents its distance from zero on the number line, regardless of direction. Solving these inequalities requires understanding how to rewrite and interpret the absolute value expression.
Absolute Value Definition: The absolute value of a real number is defined as if , and if .
General Form: An absolute value inequality can be written as , , , or , where is an algebraic expression and is a positive number.
Solving
To solve the inequality , we must rewrite it without the absolute value bars and solve the resulting compound inequality.
Step 1: Rewrite the Inequality For (where ), the equivalent compound inequality is .
Apply to the Given Problem: becomes:
Step 2: Solve for Add 5 to all parts: Divide all parts by 2:
Step 3: Write the Solution in Interval Notation The solution set is .
Graphing the Solution
The solution is represented on the number line as all points between and $7$, not including the endpoints.
Open Circles: Use open circles at and to indicate that these values are not included.
Shaded Region: Shade the region between and $7$.
Summary Table: Absolute Value Inequality Types
Form | Equivalent Compound Inequality | Interval Notation |
|---|---|---|
or | ||
or |
Example
Problem: Solve .
Solution: Rewrite as or .
Solve each:
Interval Notation:
