Question

Solve the equation. Remember to check for extraneous solutions. $$\frac{3}{x+4}+\frac{4}{x}=\frac{-5}{x^{2}+4 x}$$

Short answer

The solution to the rational equation is \(x = -3\).

Step-by-step solution

Simplification

Multiply the entire equation by \(x(x+4)\) to get rid of the fractions and simplify the equation. This yields: \(3x + 4(x+4) = -5\)

Solve for the variable

Combine like terms to isolate the variable: \[3x + 4x +16 = -5\]. Combining terms gives: \[7x +16 = -5\]. Subtract 16 from both sides to isolate x: \[7x = -21\]. Then, divide by 7 to solve for x: \[x = -3\]

Check for Extraneous Solutions

Substitute \(x = -3\) back into the original equation: \[\frac{3}{-3+4} + \frac{4}{-3} = \frac{-5}{(-3)^{2}+4 (-3)}\]. Simplifying, get: \[3 - \frac{4}{3} = -\frac{1}{3}\]. Thus, \(x = -3\) is a valid solution and not extraneous.