Absolute value inequality

Definition

An absolute value inequality is an inequality that contains an absolute value expression. It takes the form $|ax + b| < c$, $|ax + b| \leq c$, $|ax + b| > c$, or $|ax + b| \geq c$.

5 Must Know Facts For Your Next Test

1. Absolute value inequalities split into two separate inequalities: one for the positive and one for the negative case.
2. For $|ax + b| < c$ where $c > 0$, it splits into $-c < ax + b < c$.
3. For $|ax + b| > c$ where $c > 0$, it splits into $ax + b > c$ or $ax + b < -c$.
4. If the inequality symbol is $\leq$ or $\geq$, include equal signs in the split inequalities.
5. When solving, isolate the absolute value expression first before splitting into two separate inequalities.

Review Questions

• What are the two inequalities formed from the absolute value inequality $|3x - 2| < 5$?
• How do you solve an absolute value inequality of the form $|x - 4| \geq 7$?
• What steps must you follow to isolate and solve an absolute value inequality?

Related terms

Absolute Value: The distance of a number from zero on a number line, denoted by vertical bars around the number (e.g., |x|).

Linear Inequality: An inequality that involves a linear function, typically taking forms like ax + b < c or ax + b > c.

Compound Inequality: Two or more inequalities joined by 'and' or 'or' (e.g., a < x < b).