Solve the equation. \[ \frac{9}{x-9}+1=\frac{x}{x+9} \]

To solve the equation \[ \frac{9}{x-9}+1=\frac{x}{x+9}, \] we start by simplifying the left side. We can rewrite \(1\) as \(\frac{x-9}{x-9}\): \[ \frac{9}{x-9} + \frac{x-9}{x-9} = \frac{9 + (x-9)}{x-9} = \frac{x}{x-9}. \] Now, the equation becomes \[ \frac{x}{x-9} = \frac{x}{x+9}. \] Next, we can cross-multiply to eliminate the fractions: \[ x(x + 9) = x(x - 9). \] This simplifies to: \[ x^2 + 9x = x^2 - 9x. \] Subtract \(x^2\) from both sides: \[ 9x = -9x. \] Adding \(9x\) to both sides gives: \[ 18x = 0. \] Dividing both sides by 18 gives: \[ x = 0. \] To ensure this solution is valid, we need to check if it creates any division by zero in the original equation. The original equation has the terms \(x - 9\) and \(x + 9\). For \(x = 0\): - \(x - 9 = 0 - 9 = -9\) (not zero) - \(x + 9 = 0 + 9 = 9\) (not zero) Thus, \(x = 0\) does not lead to any undefined expressions. Finally, we conclude that the solution to the equation is: \[ \boxed{0}. \]