Step-by-Step Solution
Key Concept: Divide the number line into intervals based on critical points (x = 1 and x = 2.5) where absolute value expressions change sign, then solve the resulting linear equations in each interval separately.
<p><strong>Step 1: Identify critical points</strong></p><p>Critical points where expressions inside absolute values equal zero: x = 1 and x = 2.5</p><p><strong>Step 2: Case 1 (x < 1)</strong></p><p>Both expressions negative: |x - 1| = -(x - 1) = 1 - x and |2x - 5| = -(2x - 5) = 5 - 2x</p><p>(1 - x) - (5 - 2x) = 2x</p><p>1 - x - 5 + 2x = 2x</p><p>x - 4 = 2x</p><p>x = -4 ✓ (valid since -4 < 1)</p><p><strong>Step 3: Case 2 (1 ≤ x < 2.5)</strong></p><p>|x - 1| = x - 1 and |2x - 5| = -(2x - 5) = 5 - 2x</p><p>(x - 1) - (5 - 2x) = 2x</p><p>x - 1 - 5 + 2x = 2x</p><p>3x - 6 = 2x</p><p>x = 6 ✗ (invalid since 6 ≮ 2.5)</p><p><strong>Step 4: Case 3 (x ≥ 2.5)</strong></p><p>Both positive: |x - 1| = x - 1 and |2x - 5| = 2x - 5</p><p>(x - 1) - (2x - 5) = 2x</p><p>x - 1 - 2x + 5 = 2x</p><p>-x + 4 = 2x</p><p>x = 4/3 ✗ (invalid since 4/3 < 2.5)</p><p><strong>Verification at x = -4:</strong> |-4 - 1| - |2(-4) - 5| = |-5| - |-13| = 5 - 13 = -8 = 2(-4) ✓</p><p>∴ Answer: x = -4</p>
Correct Answer: x = -4