To solve the equation
\[ \frac{x - 1}{x + 1} = \frac{9}{7} \]
we can cross-multiply:
\[ 7(x - 1) = 9(x + 1). \]
Now, distribute both sides:
\[ 7x - 7 = 9x + 9. \]
Next, we will isolate \(x\) by moving all \(x\) terms to one side and constant terms to the other:
\[ 7x - 9x = 9 + 7, \] which simplifies to:
\[ -2x = 16. \]
Now, divide both sides by -2:
\[ x = -8. \]
Thus, the solution is
\[ \boxed{-8}. \]