Question

If $ 2 < x < 3 $, then

• A. \( |x - 3| < |x - 2| \)
• B. \( (x - 3) > (x - 2) \)
• C. \( (x - 3)(x - 2) < 0 \)
• D. \( \frac{x - 3}{x - 2} > 0 \)

Hint:

For inequalities involving products of terms, analyze the signs of the individual terms within the given range.

Answer 1

The Correct Option is C

Step 1: Understanding the Problem
We are given that \( 2 < x < 3 \), and we need to check which of the options is true.
Step 2: Analyzing the Options
Option (a) is incorrect because \( |x - 3| \) is always greater than \( |x - 2| \) for \( 2 < x < 3 \).
Option (b) is incorrect because \( (x - 3) \) is less than \( (x - 2) \) for \( 2 < x < 3 \).
Option (c) is correct because the product \( (x - 3)(x - 2) \) is negative when \( 2 < x < 3 \) because one term is positive and the other is negative.
Option (d) is incorrect because the fraction \( \frac{x - 3}{x - 2} \) is negative for \( 2 < x < 3 \).
Step 3: Conclusion
Thus, the correct option is \( (x - 3)(x - 2) < 0 \).