Find the set of all real values of x for which the inequality (2x - 3) / (x - 2) < 1 holds true.

Model Answer & Options

Source: Extra Practice

x ∈ (1, 2)

x ∈ (-∞, 1) ∪ (2, ∞)

x ∈ [1, 2]

x ∈ (2, 3)

Explanation

Do not cross-multiply, as the sign of (x-2) is unknown. Instead, subtract 1 from both sides: [(2x - 3) / (x - 2)] - 1 < 0. Simplify the fraction: (2x - 3 - x + 2) / (x - 2) < 0, which yields (x - 1) / (x - 2) 2, negative for 1 < x < 2, and positive for x 1. Option 3 is wrong because the inequality is strict (<) and x = 2 makes the denominator zero. Option 4 uses incorrect calculation steps.

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