# Solve (|x|-1)/(|x|-2)≥0, x ∈ R, x ≠ ± 2.

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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, ∞)