To solve the equation \( x - \frac{1}{2} = \frac{8}{x} - 1 \), follow these steps:

1. **Isolate the fractions**: First, let's get rid of the fractions by finding a common denominator or by multiplying through by \( x \). In this case, multiplying through by \( x \) will help simplify the equation.

\[
x \left( x - \frac{1}{2} \right) = x \left( \frac{8}{x} - 1 \right)
\]

2. **Distribute \( x \)** on both sides:

\[
x^2 - \frac{x}{2} = 8 - x
\]

3. **Clear the fraction**: Multiply every term by 2 to eliminate the fraction:

\[
2x^2 - x = 16 - 2x
\]

4. **Move all terms to one side** to set the equation to zero:

\[
2x^2 - x + 2x - 16 = 0
\]

Simplify the equation:

\[
2x^2 + x - 16 = 0
\]

5. **Solve the quadratic equation**: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 1 \), and \( c = -16 \).

\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-16)}}{2 \cdot 2}
\]

Simplify inside the square root:

\[
x = \frac{-1 \pm \sqrt{1 + 128}}{4}
\]

\[
x = \frac{-1 \pm \sqrt{129}}{4}
\]

6. **Find the solutions**:

\[
x = \frac{-1 + \sqrt{129}}{4} \quad \text{or} \quad x = \frac{-1 - \sqrt{129}}{4}
\]

So, the solutions to the equation \( x - \frac{1}{2} = \frac{8}{x} - 1 \) are:

\[
x = \frac{-1 + \sqrt{129}}{4} \quad \text{and} \quad x = \frac{-1 - \sqrt{129}}{4}
\]