Source: Extra Practice

Solve the inequality 1∣x−3∣<12\frac{1}{|x - 3|} < \frac{1}{2}, where x≠3x \ne 3.

Options

Option A is correct

(-∞, 1) ∪ (5, ∞)

Option B

(1, 5)

Option C

(3, 5)

Option D

(-∞, 3) ∪ (3, 5)

Explanation

Given 1/|x - 3| < 1/2. Since the modulus is always positive for x ≠ 3, we can cross-multiply: 2 2. This implies x - 3 > 2 or x - 3 5 or x < 1. In interval notation, this is (-∞, 1) ∪ (5, ∞). Option B is for |x - 3| < 2. Options C and D do not cover both regions correctly.