Step-by-Step Solution
Key Concept: Solve absolute value equations by breaking them into cases: first remove the outer absolute value to get two equations, then remove the inner absolute value in each case to create four linear equations total.
<p><strong>Step 1:</strong> Remove outer absolute value: |4 - |x - 1|| = 3 gives two cases:</p><p>Case 1: 4 - |x - 1| = 3 → |x - 1| = 1</p><p>Case 2: 4 - |x - 1| = -3 → |x - 1| = 7</p><p><strong>Step 2:</strong> Solve Case 1: |x - 1| = 1</p><p>• x - 1 = 1 → x = 2</p><p>• x - 1 = -1 → x = 0</p><p><strong>Step 3:</strong> Solve Case 2: |x - 1| = 7</p><p>• x - 1 = 7 → x = 8</p><p>• x - 1 = -7 → x = -6</p><p><strong>Step 4:</strong> Verify all solutions in original equation:</p><p>• x = 0: |4 - |0 - 1|| = |4 - 1| = 3 ✓</p><p>• x = 2: |4 - |2 - 1|| = |4 - 1| = 3 ✓</p><p>• x = 8: |4 - |8 - 1|| = |4 - 7| = 3 ✓</p><p>• x = -6: |4 - |-6 - 1|| = |4 - 7| = 3 ✓</p><p>∴ Answer: <strong>x = -6, 0, 2, 8</strong></p>
Correct Answer: x = -6, 0, 2, 8