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

1. Problem Description

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

2. Solution Steps

First, we want to isolate the $x$ terms on one side of the equation and the constant terms on the other side.

We can subtract $\frac{3}{4}x$ from both sides:

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

We can add $\frac{1}{2}$ to both sides:

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

Simplify the left side:

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

Rewrite $2x$ as $\frac{8}{4}x$:

$$-\frac{1}{2} = \frac{8}{4}x - \frac{3}{4}x$$

Combine the $x$ terms on the right side:

$$-\frac{1}{2} = \frac{5}{4}x$$

Now, we can multiply both sides by $\frac{4}{5}$ to isolate $x$:

$$-\frac{1}{2} \cdot \frac{4}{5} = x$$

Simplify the left side:

$$-\frac{4}{10} = x$$

$$-\frac{2}{5} = x$$

3. Final Answer

$x = -\frac{2}{5}$