Solve the following equation. \[ \frac{x}{9}+1=-2-\frac{x}{5} \]

To solve the equation \[ \frac{x}{9} + 1 = -2 - \frac{x}{5}, \] first, we can eliminate the fractions by finding a common denominator, which in this case is 45 (the least common multiple of 9 and 5). Multiply every term in the equation by 45: \[ 45 \left(\frac{x}{9}\right) + 45(1) = 45(-2) - 45 \left(\frac{x}{5}\right). \] This simplifies to: \[ 5x + 45 = -90 - 9x. \] Next, combine like terms. Adding \(9x\) to both sides gives: \[ 5x + 9x + 45 = -90. \] This simplifies to: \[ 14x + 45 = -90. \] Now, subtract 45 from both sides: \[ 14x = -90 - 45, \] which simplifies to: \[ 14x = -135. \] Finally, divide both sides by 14 to solve for \(x\): \[ x = \frac{-135}{14}. \] Thus, the solution is \[ x = -\frac{135}{14}. \]