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Solve for x: |x² - 4x + 3| = |x² - 5x + 4|.

1. x = 1 and x = 7/2
2. x = 1 only
3. x = 3 and x = 4
4. x = 7/2 only

Correct answer: x = 1 and x = 7/2

Solution

Factor: (x-1)(x-3) and (x-1)(x-4). |A|=|B| gives A=B or A=-B. Case A=B: (x-1)(x-3) = (x-1)(x-4) => (x-1)[(x-3)-(x-4)] = (x-1)(1) = 0 => x = 1. Case A=-B: (x-1)(x-3) = -(x-1)(x-4) => (x-1)[(x-3)+(x-4)] = (x-1)(2x-7) = 0 => x = 1 or x = 7/2. Combined solutions: x = 1 and x = 7/2.

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