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Solve the inequality over the reals: ((2 - x²)*(x - 3)³) / ((x + 1)*(x² - 3x - 4)) >= 0.

1. [-sqrt(2), -1) U (-1, sqrt(2)] U [3, 4)
2. [-sqrt(2), sqrt(2)] U [3, 4]
3. (-1, sqrt(2)] U [3, 4)
4. [-sqrt(2), -1) U (3, 4)

Correct answer: [-sqrt(2), -1) U (-1, sqrt(2)] U [3, 4)

Solution

Denominator = (x+1)(x-4)(x+1) = (x+1)²*(x-4). Since (x+1)² > 0 for x != -1, the inequality reduces to (2 - x²)*(x - 3)³/(x - 4) >= 0 with x != -1, 4. Critical points: x = -sqrt(2), sqrt(2), 3, 4. Sign analysis shows the expression is >= 0 on [-sqrt(2), sqrt(2)] and [3, 4), with x = -1 excluded (it lies in (-sqrt(2), sqrt(2))) and x = 4 excluded. Zeros at +-sqrt(2) and 3 are included.

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