Solve: |x|−1/ |x|−2 ≥ 0 x ∈ R, x ≠ ±2

Question

Solve: \(\frac{|x|-1}{|x|-2}\) ≥ 0 x ∈ R, x ≠ ±2

Answer 1

We have, 

\(\frac{|x|-1}{|x|-2}\) ≥ 0

⇒ \(\frac{y-1}{y-2}\) ≥ 0, where y = |x|

⇒ y ≤ 1 or y > 2

⇒ |x| ≤ 1 or |x| > 2

⇒ (−1 ≤ x ≤ 1)or (x < −2 or x > 2)

⇒ x ∈ [−1,1] or x ∈ (−∞, −2) ∪ (2, ∞)

⇒ x ∈ [−1,1] ∪ (−∞, −2) ∪ (2, ∞)

Hence, solution set of the given in equation is [−1,1] ∪ (−∞, −2) ∪ (2, ∞)