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

Answer

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Answer

Final Answer: $- 0.5$

Steps

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: $(3 x + 8) (x - 4) = 9 x (x + 7)$

Step 3 ：Simplify the equation and solve for x: $3 x^{2} - 4 x + 8 x - 32 = 9 x^{2} + 63 x$

Step 4 ：Rearrange the equation: $6 x^{2} + 59 x + 32 = 0$

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: $- 0.5$