Section 3.5: Absolute Value Inequalities

Absolute value inequalities are solved using two different approaches depending on the inequality:

  • |A| < B → AND: \( -B < A < B \)
  • |A| > B → OR: \( A < -B \) or \( A > B \)
  • Ensure \( B > 0 \). No solution if \( B \le 0 \) for < inequalities, all real numbers for > inequalities.

Example 1: Less Than

Solve \(|x - 3| < 5\)

Step 1: Write as AND inequality: \(-5 < x - 3 < 5\)

Step 2: Add 3 to all parts: \(-2 < x < 8\)

Solution: \(-2 < x < 8\)

Example 2: Greater Than

Solve \(|2x + 1| > 7\)

Step 1: Write as OR inequality: \( 2x + 1 < -7 \) or \( 2x + 1 > 7 \)

Step 2: Solve each part: \( 2x < -8 \Rightarrow x < -4 \) and \( 2x > 6 \Rightarrow x > 3 \)

Solution: \( x < -4 \) or \( x > 3 \)

Practice Problems

  1. Solve \(|x + 2| < 6
  2. Solve \(|3x - 5| > 10
  3. Solve \(|2x + 1| < 7
  4. Solve \(|x - 4| > 3
  5. Solve \(|5 - 2x| < 8
← Previous Next →