Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 4}<\frac{9}{x^{2} + 6 x - 16}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{9}{\left(x - 2\right) \left(x + 8\right)} + \frac{5}{\left(x - 2\right) \left(x + 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 2\right) \left(x + 2\right) \left(x + 8\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{5 x + 40 - (9 x + 18)}{\left(x - 2\right) \left(x + 2\right) \left(x + 8\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{22 - 4 x}{\left(x - 2\right) \left(x + 2\right) \left(x + 8\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{\frac{11}{2}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-8, -2, 2\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -8\right) \cup \left(-2, 2\right) \cup \left(\frac{11}{2}, \infty\right)