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)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Basics & Logarithms · easy
If x = ∏(n=1)^(2026) n , then the value of the expression 1 (1)/(log₂ x) + (1)/(log₃ x) +…
Basics & Logarithms · easy
Number of integral values of x satisfying the inequality ((3)/(4) )^(6x+10-x²) < (27)/(64…
Basics & Logarithms · easy
Number of natural numbers for which the number log(4-x)(x²-14x+45) is defined is:
Basics & Logarithms · easy
The product of all solutions of the equation e^ 5(log(e)x)^(2)+3 =x^(8) , x>0 , is

Solve more, learn faster

Sign up free to solve more JEE Mathsquestions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath