To solve absolute value inequalities, the approach depends on whether the inequality is a "less than" or "greater than" type:
1. Absolute Value Inequalities of the Form |X| ≤ p (Less Than or Equal
To)
- Rewrite the inequality as a compound inequality without absolute value:
−p≤X≤p-p\leq X\leq p−p≤X≤p
- Solve this compound inequality as usual.
- The solution represents all values of XXX whose distance from zero is less than or equal to ppp.
- For example, ∣x∣≤3|x|\leq 3∣x∣≤3 becomes −3≤x≤3-3\leq x\leq 3−3≤x≤3 and the solution set is all xxx between −3-3−3 and 333 inclusive
2. Absolute Value Inequalities of the Form |X| ≥ p (Greater Than or Equal
To)
- Rewrite the inequality as two separate inequalities connected by "or":
X≤−porX≥pX\leq -p\quad \text{or}\quad X\geq pX≤−porX≥p
- Solve each inequality separately.
- The solution represents all values of XXX whose distance from zero is greater than or equal to ppp.
- For example, ∣x∣≥3|x|\geq 3∣x∣≥3 becomes x≤−3x\leq -3x≤−3 or x≥3x\geq 3x≥3
Step-by-Step Summary
- Isolate the absolute value expression on one side.
- For less than ( < or ≤) inequalities:
- Write a compound inequality: −p<X<p-p<X<p−p<X<p or −p≤X≤p-p\leq X\leq p−p≤X≤p.
- Solve for XXX.
- Combine solutions with "and".
- For greater than ( > or ≥) inequalities:
- Write two inequalities: X>pX>pX>p or X<−pX<-pX<−p (or with ≥ and ≤).
- Solve each inequality.
- Combine solutions with "or".
- Graph the solution on a number line to visualize the intervals
Important Notes
- The absolute value represents distance, so solutions describe intervals around zero.
- For "less than" inequalities, the solution is a bounded interval.
- For "greater than" inequalities, the solution is two unbounded intervals.
- Do not combine "greater than" solutions into a single inequality (e.g., don't write −3>X>3-3>X>3−3>X>3, which is incorrect)
This method applies to any absolute value inequality, including those with expressions inside the absolute value (e.g., ∣2x+5∣>11|2x+5|>11∣2x+5∣>11) by first isolating the absolute value expression and then applying the rules above
