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Question #40289

Difficulty: 8/10 | 🟢 Free

The Question

Find the complete solution set for the inequality $\log_{0.5} \left( \frac{(x-4)^2}{x^2 - 9} \right) \ge 0$

Options

A.
$\left[ \frac{25}{8}, 4 \right) \cup (4, \infty)$
B.
$\left[ \frac{25}{8}, \infty \right)$
C.
$(-3, 3) \cup \left[ \frac{25}{8}, \infty \right)$
D.
$(-3, 3) \cup \left[ \frac{25}{8}, 4 \right) \cup (4, \infty)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Well-definedness|Wavy Curve (Method of Intervals)|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of logarithmic well-definedness with the wavy curve method, specifically targeting the trap of even multiplicity roots.

Students often solve the algebraic inequality $f(x) \le 1$ to get $(-3, 3) \cup [25/8, \infty)$ and forget to intersect it with the strict domain condition $f(x) > 0$ .

Furthermore, the even power $(x-4)^2$ creates a false sense of security; while it is non-negative, it equals zero at $x=4$ , making the logarithm undefined.

Failing to exclude $x=4$ or failing to reject the interval $(-3, 3)$ where $f(x) < 0$ leads directly to the incorrect options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Well-definedness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free