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

To solve the inequality \( \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \), we need to address the conditions of the logarithmic function and the inequality.

Condition for the logarithmic function:

The expression inside the logarithm, \( \frac{x - 7}{2x - 3} \), must be positive:

\(\frac{x - 7}{2x - 3} > 0\)

This can be split into two cases: either both the numerator and denominator are positive, or both are negative.

Case 1: Both numerator and denominator are positive

\(x - 7 > 0\)→ \(x > 7\)

\(2x - 3 > 0\)→ \(x > \frac{3}{2}\)

Thus, for this case, \( x > 7 \).

Case 2: Both numerator and denominator are negative

\(x - 7 < 0\)→ \(x < 7\)

\(2x - 3 < 0\)→ \(x < \frac{3}{2}\)

Thus, for this case, \( x < \frac{3}{2} \).

Combining the cases:

The intervals derived from the two cases for \( x \) are:

• \( x > 7 \)
• \( x < \frac{3}{2} \)

Conclusion:

The integral solutions for \( x \) satisfy that it must be an integer less than \( \frac{3}{2} \) or greater than 7.

These are \( 1, 8, 9, 10, 11, 12, \ldots \)

Therefore, the number of integral solutions is 6, which is the correct answer.