Question

The number of integral solutions of \( \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \) is:

• A. 5
• B. 7
• C. 8
• D. 6

Hint:

When solving logarithmic inequalities, always ensure that the argument inside the logarithm is positive and satisfies the given constraints.

Answer 1

The Correct Option is D

We are given the inequality: \[ \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \] This implies that: \[ \frac{x - 7}{2x - 3} \geq 1 \] Now, solving this inequality: \[ \frac{x - 7}{2x - 3} \geq 1 \quad \Rightarrow \quad x - 7 \geq 2x - 3 \] Simplifying: \[ -7 + 3 \geq 2x - x \quad \Rightarrow \quad x \leq -4 \] Thus, the solution for \( x \) is: \[ x \leq -4 \] Additionally, we need to consider the condition for \( x \) that the logarithm is defined, i.e., the argument inside the logarithm must be positive: \[ x - 7>0 \quad \Rightarrow \quad x>7 \] Thus, the feasible region for \( x \) is: \[ x>7 \] Next, we consider the second part of the problem: \[ \frac{x - 7}{2x - 3} \quad \text{and solve the inequality as outlined.} \] Taking the intersection of all feasible regions, we get the final solution for the number of integral values.