<p>Number of solutions of the equation \(|2 - |x|| = x + 4\) is</p>
Step-by-Step Solution
Key Concept: Rewrite |2 - |x|| = x + 4 by removing absolute values from outside-in: first handle |x|, then |2 - |x||. Graph both sides or analyze cases systematically to avoid missing solutions or accepting extraneous ones.
<p><strong>Step 1:</strong> Since |2 - |x|| ≥ 0 always, we need x + 4 ≥ 0, so x ≥ -4.</p><p><strong>Step 2:</strong> Consider cases based on the sign of x.</p><p><strong>Case 1 (x ≥ 0):</strong> Then |x| = x, so |2 - x| = x + 4</p><p>• Subcase 1a (0 ≤ x ≤ 2): 2 - x = x + 4 → -2x = 2 → x = -1 (invalid, not in [0,2])</p><p>• Subcase 1b (x > 2): x - 2 = x + 4 → -2 = 4 (no solution)</p><p><strong>Step 3:</strong> <strong>Case 2 (-4 ≤ x < 0):</strong> Then |x| = -x, so |2 - (-x)| = |2 + x| = x + 4</p><p>• Subcase 2a (-4 ≤ x ≤ -2): 2 + x = -(x + 4) → 2 + x = -x - 4 → 2x = -6 → x = -3 ✓ (satisfies -4 ≤ -3 ≤ -2)</p><p>• Subcase 2b (-2 < x < 0): 2 + x = x + 4 → 2 = 4 (no solution)</p><p><strong>Step 4:</strong> Verify x = -3: |2 - |-3|| = |2 - 3| = |-1| = 1 and -3 + 4 = 1 ✓</p><p>∴ Answer: <strong>1 solution</strong> (B)</p>
Correct Answer: B