It looks like there might be a typographical error in the equation you've provided, as it seems to have two equal signs. To clarify, let's solve two separate equations based on what I think you're trying to convey.

1. **For the equation** $4x - 1 = \frac{1}{24}x$
2. **For the equation** $4x - 1 = \frac{1}{2}$

Let's solve them one by one.

### 1. Solve $4x - 1 = \frac{1}{24}x$

Step 1: Move all terms involving $x$ to one side:

$$4x - \frac{1}{24}x = 1$$

Step 2: Combine the $x$ terms. To do this, express $4x$ with a common denominator:

$$\frac{96}{24}x - \frac{1}{24}x = 1$$

This simplifies to:

$$\frac{95}{24}x = 1$$

Step 3: Multiply both sides by the reciprocal of $\frac{95}{24}$:

$$x = 1 \cdot \frac{24}{95} = \frac{24}{95}$$

### 2. Solve $4x - 1 = \frac{1}{2}$

Step 1: Add 1 to both sides:

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

Step 2: Convert 1 to a fraction with a common denominator:

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

Step 3: Divide both sides by 4:

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

### Summary

The solutions are:
1. For $4x - 1 = \frac{1}{24}x$, $x = \frac{24}{95}$.
2. For $4x - 1 = \frac{1}{2}$, $x = \frac{3}{8}$.

If the original equation was intended differently, please clarify and I can assist you further!