To solve the equation
\[ \frac{3(x - 5)}{4} - x = 1 - \frac{x}{8}, \]
let's start by clearing the fractions. We can do this by multiplying each term in the equation by 8 (the least common multiple of the denominators, 4 and 8):
\[ 8 \left( \frac{3(x - 5)}{4} \right) - 8x = 8 \left( 1 - \frac{x}{8} \right). \]
This simplifies to:
\[ 6(x - 5) - 8x = 8 - x. \]
Next, distribute 6 on the left side:
\[ 6x - 30 - 8x = 8 - x. \]
Now, combine like terms on the left:
\[ -2x - 30 = 8 - x. \]
Next, add \(x\) to both sides:
\[ -2x + x - 30 = 8. \]
This simplifies to:
\[ -x - 30 = 8. \]
Now, add 30 to both sides:
\[ -x = 38. \]
Finally, multiply by -1 to solve for \(x\):
\[ x = -38. \]
Thus, the solution is:
\[ \boxed{-38}. \]