Solve the inequality: 1 / (x - 1) < 2.

Model Answer & Options

Source: Extra Practice

x ∈ (1, 1.5)

x ∈ (-∞, 1) ∪ (1.5, ∞)

x ∈ (1.5, ∞)

x ∈ (-∞, 1.5)

Explanation

Do not cross-multiply as (x-1) can be negative. Rearrange: 1/(x-1) - 2 < 0 → (1 - 2(x-1))/(x-1) < 0 → (1 - 2x + 2)/(x-1) < 0 → (3 - 2x)/(x-1) 0. Critical points are 1 and 1.5. Testing intervals: for x < 1, (+)/( - ) = - (False); for 1 < x 0. If x=0, -3/-1=3>0 (True). If x=1.2, -0.6/0.2=-3 (False). If x=2, 1/1=1>0 (True). So solution is (-∞, 1) ∪ (1.5, ∞).

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