Source: Extra Practice
Find the solution set for: (x² - 5x + 6) / (x² - 1) ≤ 0.
Options
Option A is correct
x ∈ (-1, 1) ∪ [2, 3]
Option B
x ∈ [-1, 1] ∪ [2, 3]
Option C
x ∈ (-∞, -1) ∪ (1, 2] ∪ [3, ∞)
Option D
x ∈ (1, 2] ∪ [3, ∞)
Explanation
First, factor the numerator and denominator: [(x - 2)(x - 3)] / [(x - 1)(x + 1)] ≤ 0. The critical points are -1, 1, 2, and 3. Testing intervals: (-∞, -1) yields (+), (-1, 1) yields (-), (1, 2) yields (+), [2, 3] yields (-), and (3, ∞) yields (+). The inequality is satisfied in (-1, 1) and [2, 3]. Note that x cannot be -1 or 1 as they are in the denominator, while 2 and 3 are included because they make the expression zero.