To solve the equation
\[ \frac{5(x - 1)}{6} - x = 1 - \frac{x}{9}, \]
we will start by eliminating the fractions. The least common multiple of the denominators (6 and 9) is 18. We'll multiply every term in the equation by 18 to clear the fractions:
\[ 18 \cdot \left( \frac{5(x - 1)}{6} \right) - 18 \cdot x = 18 \cdot \left( 1 - \frac{x}{9} \right). \]
This simplifies to:
\[ 3 \cdot 5(x - 1) - 18x = 18 - 2x. \]
Now simplify each term:
\[ 15(x - 1) - 18x = 18 - 2x. \]
Distributing the 15 gives:
\[ 15x - 15 - 18x = 18 - 2x. \]
Combining like terms on the left side:
\[ -3x - 15 = 18 - 2x. \]
Next, we will add \(3x\) to both sides to isolate terms involving \(x\):
\[ -15 = 18 + x. \]
Now, we subtract 18 from both sides:
\[ -15 - 18 = x \implies -33 = x. \]
So the solution to the equation is:
\[ \boxed{-33}. \]
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