Absolute value inequality

Definition

An absolute value inequality is an inequality that contains an absolute value expression. It can be solved by considering the positive and negative scenarios of the expression inside the absolute value.

5 Must Know Facts For Your Next Test

1. Absolute value inequalities can be written in two forms: $\left|x\right| \leq a$ and $\left|x\right| \geq a$.
2. For $\left|x\right| \leq a$, the solution is $-a \leq x \leq a$.
3. For $\left|x\right| \geq a$, the solution is $x \leq -a$ or $x \geq a$.
4. When solving absolute value inequalities, split them into two separate inequalities to handle both cases.
5. Graphing helps visualize solutions to absolute value inequalities, typically resulting in segments or rays on the number line.

Review Questions

• How do you solve an inequality of the form $\left|x - c\right| \leq d$?
• What is the solution set for $\left|2x - 3\right| > 5$?
• How do you graph the solution to the inequality $\left| x + 4 \right| < 7$?

Related terms

Absolute Value: The distance of a number from zero on the number line, always non-negative.

Inequality: A mathematical statement that compares two expressions using inequality symbols like <, >, ≤, or ≥.

Compound Inequality: An equation with two or more inequalities joined together by 'and' or 'or'.