The problem is to solve for $x$ in the equation $\frac{3x+5}{5x-2} = \frac{2}{3}$.

1. Problem Description

The problem is to solve for $x$ in the equation $\frac{3x+5}{5x-2} = \frac{2}{3}$.

2. Solution Steps

To solve for $x$, we can cross-multiply the equation:

$3(3x+5) = 2(5x-2)$

Expanding both sides, we get:

$9x + 15 = 10x - 4$

Now, we want to isolate $x$. Subtract $9x$ from both sides:

$15 = x - 4$

Add $4$ to both sides:

$15 + 4 = x$

$19 = x$

Therefore, $x = 19$.

3. Final Answer

$x = 19$