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Question
Solve for x, the inequality given below.
`(|x - 2| - 1)/(|x - 2| - 2) ≤ 0`
Solution
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).
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