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, move all terms to one side, find critical points, and check the signs in the intervals created.

Step 2: Detailed Explanation:
1. \( \frac{3}{x-2} - 1 < 0 \implies \frac{3 - (x-2)}{x-2} < 0 \implies \frac{5-x}{x-2} < 0 \).
2. Critical points are \( x=2 \) and \( x=5 \).
3. Test intervals: - For \( x < 2 \) (e.g., \( x=0 \)): \( \frac{5-0}{0-2} = -2.5 < 0 \) (True). - For \( 2 < x < 5 \) (e.g., \( x=3 \)): \( \frac{5-3}{3-2} = 2 > 0 \) (False). - For \( x > 5 \) (e.g., \( x=6 \)): \( \frac{5-6}{6-2} = -0.25 < 0 \) (True).
The solution set is \( (-\infty, 2) \cup (5, \infty) \).

Step 3: Final Answer:
The solution set is \( (-\infty, 2) \cup (5, \infty) \).