<p>\(1 \leq x \leq 3/2\)</p>
<p>\(x \in [1, 3/2]\)</p>
<p>\(x \in (1, 3/2)\)</p>
<p>\(x \in [0, 1]\)</p>
Step-by-Step Solution
Key Concept: The equation involves absolute values that change sign at x = 1, 3/2, and 2. Partition the real line into intervals and solve separately in each region where the expressions inside absolute values maintain constant signs.
<p><strong>Step 1:</strong> Identify critical points where expressions change sign: x = 1, x = 3/2, x = 2. These divide the real line into four intervals.</p><p><strong>Step 2: Case 1 (x < 1):</strong> All expressions negative: -(2x-3) - (x-1) = -(x-2) → -2x+3-x+1 = -x+2 → -3x+4 = -x+2 → x = 1. But x < 1, so no solution here.</p><p><strong>Step 3: Case 2 (1 ≤ x < 3/2):</strong> (2x-3) negative, others: -(2x-3) - (x-1) = -(x-2) → -2x+3-x+1 = -x+2 → -2x = -2 → x = 1. Check: |2(1)-3| + |1-1| = 1 + 0 = 1 and |1-2| = 1 ✓</p><p><strong>Step 4: Case 3 (3/2 ≤ x < 2):</strong> (2x-3) positive, (x-1) positive, (x-2) negative: (2x-3) + (x-1) = -(x-2) → 3x-4 = -x+2 → 4x = 6 → x = 3/2. Check: |2(3/2)-3| + |3/2-1| = 0 + 1/2 = 1/2 and |3/2-2| = 1/2 ✓</p><p><strong>Step 5: Case 4 (x ≥ 2):</strong> All positive: (2x-3) + (x-1) = (x-2) → 3x-4 = x-2 → 2x = 2 → x = 1. But x ≥ 2, so no solution here.</p><p><strong>∴ Answer: x = 1 or x = 3/2</strong></p>
Correct Answer: A