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!