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Question #40518

Difficulty: 8/10 | 🟢 Free

The Question

Find the complete solution set for $x$ satisfying the inequality: $\frac{|\log_2(x-2)| + |\log_2(x-2) - 2| - 2}{|x^2 - 6x + 8| - 3} \le 0$

Options

A.
$(2, 5) \cup (5, 6]$
B.
$(2, 6]$
C.
$[3, 5) \cup (5, 6]$
D.
$(2, 5)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Wavy Curve (Method of Intervals)|Triangle Inequality Mind Games|Modulus Function Symmetry|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Mismatch' between the Method of Intervals and domain/range constraints.

Students often blindly apply the Wavy Curve method, finding where the denominator is negative, which yields $(2, 5)$ (Option D).

However, the Triangle Inequality guarantees the numerator is always non-negative and exactly zero on $[3, 6]$ .

A critical trap is failing to realize that when the numerator is zero, the rational expression is zero (hence $\le 0$ ) regardless of a positive denominator, extending the solution to include $(5, 6]$ .

Furthermore, students must exclude $x=5$ where the denominator vanishes, punishing those who carelessly write $(2, 6]$ (Option B).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Wavy Curve (Method of Intervals)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free