Solve the Logarithmic Inequality: log₁/₂(5) - log₁/₂(4) ≤ log₁/₂(x) - log₁/₂(3)

$x=\text{?}$

$\log_{\frac{1}{2}}5-\log_{\frac{1}{2}}4\le\log_{\frac{1}{2}}x-\log_{\frac{1}{2}}3$

To solve the inequality involving logarithms with base $\frac{1}{2}$, we will perform the following steps:

• Step 1: Apply the subtraction property of logarithms.
• Step 2: Simplify and solve the inequality.

Let's go through the steps:

Step 1: Simplify both sides using the logarithm subtraction rule:

Left side: $\log_{\frac{1}{2}}5 - \log_{\frac{1}{2}}4 = \log_{\frac{1}{2}}\left(\frac{5}{4}\right)$

Right side: $\log_{\frac{1}{2}}x - \log_{\frac{1}{2}}3 = \log_{\frac{1}{2}}\left(\frac{x}{3}\right)$

This gives us the inequality:

$\log_{\frac{1}{2}}\left(\frac{5}{4}\right) \le \log_{\frac{1}{2}}\left(\frac{x}{3}\right)$

Step 2: Since $\frac{1}{2}$ is less than 1, the inequality sign flips when we remove the logarithms.

This gives: $\frac{5}{4} \ge \frac{x}{3}$

Multiplying both sides by 3 to solve for $x$:

$3 \cdot \frac{5}{4} \ge x$

$\frac{15}{4} \ge x$

Thus, $x \le \frac{15}{4}$, which simplifies to $x \le 3.75$.

Since we assumed $x > 0$, the final solution is:

$0 < x \le 3.75$