Solve the inequality: (x - 1)³(x + 2) / (x - 3) ≤ 0.

Model Answer & Options

Source: Extra Practice

x ∈ [-2, 1] ∪ (3, ∞)

x ∈ (-∞, -2] ∪ [1, 3)

x ∈ [-2, 3)

x ∈ (-∞, -2] ∪ (1, 3)

Explanation

Critical points are -2, 1, and 3. Powers of (x-1) and (x+2) are odd (3 and 1), so the sign changes at every critical point. Testing intervals: x < -2 (e.g., -3): (-)(-)/(-) = - (True). -2 < x < 1 (e.g., 0): (-)(+)/(-) = + (False). 1 < x 3 (e.g., 4): (+)(+)/(+) = + (False). Combining intervals where the expression is ≤ 0 and excluding x=3 (denominator), we get (-∞, -2] ∪ [1, 3).

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