Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 25}<\frac{2}{x^{2} + 13 x + 40}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{2}{\left(x + 5\right) \left(x + 8\right)} + \frac{5}{\left(x - 5\right) \left(x + 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x + 5\right) \left(x + 8\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{5 x + 40 - (2 x - 10)}{\left(x - 5\right) \left(x + 5\right) \left(x + 8\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x + 50}{\left(x - 5\right) \left(x + 5\right) \left(x + 8\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{50}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-8, -5, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(- \frac{50}{3}, -8\right) \cup \left(-5, 5\right)