We are asked to solve the equation $\frac{4}{x-2} + \frac{1}{2} = \frac{3}{4}$ for $x$.

1. Problem Description

We are asked to solve the equation $\frac{4}{x-2} + \frac{1}{2} = \frac{3}{4}$ for $x$.

2. Solution Steps

First, we want to isolate the term with $x$. Subtract $\frac{1}{2}$ from both sides of the equation:

$\frac{4}{x-2} + \frac{1}{2} - \frac{1}{2} = \frac{3}{4} - \frac{1}{2}$

$\frac{4}{x-2} = \frac{3}{4} - \frac{2}{4}$

$\frac{4}{x-2} = \frac{1}{4}$

Next, we can cross-multiply to solve for $x$:

$4 \cdot 4 = 1 \cdot (x-2)$

$16 = x - 2$

Add $2$ to both sides of the equation:

$16 + 2 = x - 2 + 2$

$18 = x$

Therefore, $x = 18$.

3. Final Answer

$x = 18$