Question

Solve for x, the inequality given below.

`(|x - 2| - 1)/(|x - 2| - 2) ≤ 0`

Answer 1

Given that, `(|x - 2| - 1)/(|x - 2| - 2) ≤ 0`

Put |x – 2| = y

∴ `(y - 1)/(y - 2) ≤ 0`

⇒ y – 1 > 0, y – 2 < 0

⇒ y > 1, y < 2

⇒ 1 < y < 2

⇒ 1 < |x – 2| < 2

⇒ 1 < |x – 2|, |x – 2| < 2

⇒ x – 2 < –1 or x – 2 > 1 and –2 < x – 2 < 2

⇒ x < 1 or x > 3 and –2 + 2 < x < 2 + 2

⇒ x < 1 or x > 3 and 0 < x < 4

Hence, the required solution is (0, 1) ∪ (3, 4).