Solve the equation $\frac{6}{x-4} + 5 = \frac{2}{2x-8}$.

1. Problem Description

Solve the equation $\frac{6}{x-4} + 5 = \frac{2}{2x-8}$.

2. Solution Steps

The given equation is $\frac{6}{x-4} + 5 = \frac{2}{2x-8}$.

We can simplify the equation as follows:

$\frac{6}{x-4} + 5 = \frac{2}{2(x-4)}$

$\frac{6}{x-4} + 5 = \frac{1}{x-4}$

Now, we subtract $\frac{1}{x-4}$ from both sides of the equation:

$\frac{6}{x-4} - \frac{1}{x-4} + 5 = 0$

$\frac{5}{x-4} + 5 = 0$

Subtract 5 from both sides:

$\frac{5}{x-4} = -5$

Multiply both sides by $(x-4)$:

$5 = -5(x-4)$

Divide both sides by -5:

$-1 = x-4$

Add 4 to both sides:

$x = -1 + 4$

$x = 3$

Now, we need to check if $x=3$ is a valid solution by substituting it into the original equation. The original equation is $\frac{6}{x-4} + 5 = \frac{2}{2x-8}$.

If $x=3$, then the left side is:

$\frac{6}{3-4} + 5 = \frac{6}{-1} + 5 = -6 + 5 = -1$

The right side is:

$\frac{2}{2(3)-8} = \frac{2}{6-8} = \frac{2}{-2} = -1$

Since both sides are equal, $x=3$ is a valid solution.

3. Final Answer

A. x = 3