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Solve the inequality over the real numbers: ((x - 1)² * (x + 1)³) / (x⁴ * (x - 2)) <= 0.

1. x in [-1, 0) U (0, 1] U (1, 2), i.e. [-1, 2) excluding 0 (with x = 1 allowed); equivalently -1 <= x < 2, x != 0
2. x in (-infinity, -1] U (2, infinity)
3. x in (-1, 2)
4. x in [-1, 2]

Correct answer: x in [-1, 0) U (0, 1] U (1, 2), i.e. [-1, 2) excluding 0 (with x = 1 allowed); equivalently -1 <= x < 2, x != 0

Solution

Domain excludes x = 0 and x = 2 (denominator zero). The factor (x - 1)² >= 0 and x⁴ > 0 (for x != 0), so the sign of the whole expression matches the sign of (x + 1)³/(x - 2), i.e. of (x + 1)/(x - 2). We need (x+1)/(x-2) <= 0, which holds for -1 <= x < 2. Also the expression equals 0 at x = 1 (from (x-1)²) and at x = -1, both allowed. Removing x = 0 (not in domain), the solution is -1 <= x < 2 with x != 0.

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