Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 25}<\frac{4}{x^{2} + 13 x + 40}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{4}{\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 - (4 x - 20)}{\left(x - 5\right) \left(x + 5\right) \left(x + 8\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{x + 60}{\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\{-60\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(-60, -8\right) \cup \left(-5, 5\right)