Step-by-Step Solution
Key Concept: An absolute value equation |x| = 2x - 1 requires solving two cases separately: when x ≥ 0 (where |x| = x) and when x < 0 (where |x| = -x). Critical step: verify each solution satisfies the original equation AND its case condition.
<p><strong>Step 1: Case 1 (x ≥ 0):</strong> When x ≥ 0, we have |x| = x</p><p>x = 2x - 1</p><p>-x = -1</p><p>x = 1</p><p><em>Check:</em> x = 1 ≥ 0 ✓ and |1| = 2(1) - 1 → 1 = 1 ✓</p><p><strong>Step 2: Case 2 (x < 0):</strong> When x < 0, we have |x| = -x</p><p>-x = 2x - 1</p><p>-3x = -1</p><p>x = 1/3</p><p><em>Check condition:</em> x = 1/3 is NOT less than 0 ✗ (violates x < 0 requirement)</p><p><em>Check in original:</em> |1/3| = 2(1/3) - 1 → 1/3 ≠ -1/3 ✗</p><p><strong>Step 3: Verification</strong> Only x = 1 satisfies both its case condition AND the original equation.</p><p>∴ <strong>Answer: x = 1</strong> (Note: The provided answer x = 1/3 does not satisfy the original equation)</p>
Correct Answer: x = 1/3