Question

The solution set of the linear inequation \( \frac{3}{x-2} < 1 \) is:

• A. (2, 5)
• B. (2, 5)
• C. \( (-\infty, 2] \cup [5, \infty) \)
• D. \( (-\infty, 2) \cup (5, \infty) \)

Hint:

Never cross-multiply by a variable expression without knowing its sign, as it can flip the inequality.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To solve rational inequalities, bring all terms to one side to get \( \frac{f(x)}{g(x)} < 0 \).

Step 2: Key Formula or Approach:
\( \frac{3}{x-2} - 1 < 0 \implies \frac{3 - (x-2)}{x-2} < 0 \implies \frac{5-x}{x-2} < 0 \).
Multiply by -1 (reversing inequality): \( \frac{x-5}{x-2} > 0 \).

Step 3: Detailed Explanation:
The critical points are \( x=2 \) and \( x=5 \).
Test intervals:
\( (-\infty, 2) \): \( (negative)/(negative) = positive > 0 \) (True)
\( (2, 5) \): \( (negative)/(positive) = negative < 0 \) (False)
\( (5, \infty) \): \( (positive)/(positive) = positive > 0 \) (True)
The solution is \( (-\infty, 2) \cup (5, \infty) \).

Step 4: Final Answer:
The solution set is \( (-\infty, 2) \cup (5, \infty) \).