Basics & LogarithmseasyFree

Basics & Logarithms: Does Belong

JEE Maths question with a full step-by-step solution.

Question
If ∣x∣−1∣x∣−2≥0\dfrac{|x|-1}{|x|-2} \geq 0, x∈Rx \in \mathbb{R}, then xx does not belong to:
A(−∞,−2)(-\infty, -2)
B[−1,1][-1, 1]
C(2,∞)(2, \infty)
D(1,2)(1, 2)correct
Solution
Step 1: Case x≥0x \geq 0
x−1x−2≥0  ⟹  x∈[0,1]∪(2,∞)\frac{x-1}{x-2} \geq 0 \implies x \in [0, 1] \cup (2, \infty)
 Step 2: Case x<0x < 0
−x−1−x−2≥0=x+1x+2≥0  ⟹  x∈(−∞,−2)∪[−1,0)\frac{-x-1}{-x-2} \geq 0 = \frac{x+1}{x+2} \geq 0 \implies x \in (-\infty, -2) \cup [-1, 0)
 Step 3: Combine
x∈(−∞,−2)∪[−1,1]∪(2,∞)x \in (-\infty, -2) \cup [-1, 1] \cup (2, \infty)
 So xx does not belong to (1,2)(1, 2). Answer: (4)
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