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Solve the inequality over the real numbers: ((2x - 1)(x - 1)² (x - 2)³) / (x - 4)⁴ > 0.

1. x in (-infinity, 1/2) U (2, infinity), excluding x = 1 and x = 4
2. x in (1/2, 2)
3. x in (1/2, 4) only
4. x in (-infinity, 1/2) U (1, 2)

Correct answer: x in (-infinity, 1/2) U (2, infinity), excluding x = 1 and x = 4

Solution

Factors raised to even powers never flip the sign, but create excluded points (zeros or undefined). Only the odd-power factors determine where the whole expression is positive.

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