We need to solve the equation $\frac{2x}{x-5} = 4 + \frac{10}{x-5}$ for $x$.

1. Problem Description

We need to solve the equation $\frac{2x}{x-5} = 4 + \frac{10}{x-5}$ for $x$.

2. Solution Steps

First, we multiply both sides of the equation by $(x-5)$ to eliminate the denominators.

$(x-5) \cdot \frac{2x}{x-5} = (x-5) \cdot (4 + \frac{10}{x-5})$

$2x = 4(x-5) + 10$

$2x = 4x - 20 + 10$

$2x = 4x - 10$

Now, we isolate the $x$ terms.

$2x - 4x = -10$

$-2x = -10$

Divide both sides by $-2$.

$x = \frac{-10}{-2}$

$x = 5$

However, we must check if $x=5$ is a valid solution because we multiplied by $x-5$, which is equal to zero when $x=5$. Substituting $x=5$ into the original equation, we get:

$\frac{2(5)}{5-5} = 4 + \frac{10}{5-5}$

$\frac{10}{0} = 4 + \frac{10}{0}$

Since we have division by zero, $x=5$ is not a valid solution. Therefore, there is no solution.

3. Final Answer

NO SOLUTION