To solve the equation
\[ \frac{5(x - 2)}{6} - x = 1 - \frac{x}{8}, \]
we will begin by eliminating the fractions. To do this, we can find a common multiple of the denominators 6 and 8, which is 24. We will multiply every term in the equation by 24:
\[ 24 \left( \frac{5(x - 2)}{6} \right) - 24x = 24 \left( 1 - \frac{x}{8} \right). \]
This simplifies each term as follows:
- For the left side: \[ 24 \cdot \frac{5(x - 2)}{6} = 4 \cdot 5(x - 2) = 20(x - 2) = 20x - 40. \]
- The term \(-24x\) remains as is.
Combining these gives: \[ 20x - 40 - 24x. \]
So, the left side becomes: \[ 20x - 24x - 40 = -4x - 40. \]
- For the right side: \[ 24 \cdot 1 - 24 \cdot \frac{x}{8} = 24 - 3x. \]
Putting it all together, the equation becomes: \[ -4x - 40 = 24 - 3x. \]
Next, we will rearrange this equation to solve for \(x\). First, add \(4x\) to both sides: \[ -40 = 24 + x. \]
Now, subtract 24 from both sides: \[ -40 - 24 = x \Rightarrow -64 = x. \]
Thus, the solution to the equation is: \[ \boxed{-64}. \]