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