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

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Let |x| = y. Then

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

On equating (y – 1) and (y – 2) equal to zero, we have the critical points as y = 1, 2. Now using the real number line, we see that the expression \(\frac{y-1}{y-2}\) is greater than equal to zero (positive) only when, y < 1 or y > 2.

⇒ |x| < 1 or |x| > 2 

⇒ –1 < x < 1 or (x < –2 or x > 2) 

\(x\)∈ [–1, 1] ∪ (–∞, –2) ∪ (2, ∞)

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