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Solve over the real numbers: (x - 1)² (x + 1)³ (x - 4) < 0. Give the solution set.

1. (-1, 1) U (1, 4)
2. (-infinity, -1) U (4, infinity)
3. (-1, 4)
4. (1, 4) only

Correct answer: (-1, 1) U (1, 4)

Solution

Zeros: x = -1 (odd mult 3), x = 1 (even mult 2), x = 4 (odd mult 1). (x-1)² >= 0, zero only at x = 1, so x = 1 is excluded. Sign of (x+1)³ (x-4) is negative for -1 < x < 4. Multiplied by the positive even factor, the product is < 0 on (-1, 4) except at x = 1 where it is 0. Endpoints -1, 4 give 0 (excluded for strict <). Solution = (-1, 1) U (1, 4).

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