Solve each equation and check for extraneous solutions. Enter the largest solution, and round to three decimal places. \[ \frac{3 x+8}{x+7}=\frac{9 x}{x-4} \]

Step 1 ：Given the equation \(\frac{3 x+8}{x+7}=\frac{9 x}{x-4}\)

Step 2 ：Cross-multiply to get rid of the fractions: \((3x + 8)(x - 4) = 9x(x + 7)\)

Step 3 ：Simplify the equation and solve for x: \(3x^2 - 4x + 8x - 32 = 9x^2 + 63x\)

Step 4 ：Rearrange the equation: \(6x^2 + 59x + 32 = 0\)

Step 5 ：Solve the quadratic equation to find the solutions: \(x = -\frac{32}{3}, -\frac{1}{2}\)

Step 6 ：Check for extraneous solutions by substituting the solutions back into the original equation. Both solutions are valid.

Step 7 ：The largest solution is \(-\frac{1}{2}\)

Step 8 ：Final Answer: \(\boxed{-0.5}\)