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

Model Answer & Options

Source: Extra Practice

x ∈ (-∞, 2] ∪ (3, ∞)

x ∈ [2, 3)

x ∈ (-∞, 1] ∪ [2, 3)

x ∈ (3, ∞)

Explanation

The factor (x - 1)² is always non-negative for all real x. Therefore, the sign of the expression depends on (x - 2) / (x - 3), except at x = 1 where the expression is 0 (which satisfies ≥ 0). Solving (x - 2) / (x - 3) ≥ 0: critical points are 2 and 3. Testing intervals gives (-∞, 2] ∪ (3, ∞). Since x = 1 is already included in the interval (-∞, 2], the final solution is x ∈ (-∞, 2] ∪ (3, ∞).

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