Section 3.4: Absolute Value Equations
An absolute value equation is an equation in the form \(|A| = B\) where \(B \ge 0\). The solutions are found by splitting into two cases:
- \(|A| = B \Rightarrow A = B\) or \(A = -B\)
Example 1
Solve \(|x - 3| = 7\)
Case 1: \( x - 3 = 7 \Rightarrow x = 10 \)
Case 2: \( x - 3 = -7 \Rightarrow x = -4 \)
Solution: \( x = -4, 10 \)
Example 2
Solve \(|2x + 1| = 5\)
Case 1: \( 2x + 1 = 5 \Rightarrow 2x = 4 \Rightarrow x = 2 \)
Case 2: \( 2x + 1 = -5 \Rightarrow 2x = -6 \Rightarrow x = -3 \)
Solution: \( x = -3, 2 \)
Practice Problems
- Solve \(|x + 4| = 9
- Solve \(|3x - 5| = 8
- Solve \(|2x + 3| = 7
- Solve \(|x - 6| = 0
- Solve \(|5 - x| = 12