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Solve the inequality over the real numbers: [ (x - 2)² * (1 - x) * (x - 3)³ * (x - 4)² ] / (x + 1) <= 0.

1. x in (-1, 1] union {2} union [3, infinity)
2. x in (-infinity, -1) union [1, 3]
3. x in (-1, 1] union [3, 4]
4. x in [1, 3] union {4}

Correct answer: x in (-1, 1] union {2} union [3, infinity)

Solution

Even-power factors only create zeros (which satisfy <= 0) and never change sign. The odd factors (x-3)³ and (1-x) plus the linear denominator (x+1) determine the sign intervals. x = -1 is excluded (denominator zero). x = 2 is an isolated solution (a squared factor vanishes there).

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