We need to solve the equation $\frac{3(x-1)}{4} - 2 = 5x - \frac{x-2}{2}$ for $x$.

1. Problem Description

We need to solve the equation $\frac{3(x-1)}{4} - 2 = 5x - \frac{x-2}{2}$ for $x$.

2. Solution Steps

First, we multiply both sides of the equation by the least common multiple of the denominators, which is

4. $4 \cdot \left(\frac{3(x-1)}{4} - 2\right) = 4 \cdot \left(5x - \frac{x-2}{2}\right)$

$3(x-1) - 8 = 20x - 2(x-2)$

Now, we expand the terms:

$3x - 3 - 8 = 20x - 2x + 4$

$3x - 11 = 18x + 4$

Now, we subtract $3x$ from both sides:

$-11 = 15x + 4$

Next, we subtract 4 from both sides:

$-15 = 15x$

Finally, we divide both sides by 15:

$x = -1$

3. Final Answer

$x = -1$