Solve the inequality LaTeX: \displaystyle \frac{3}{x^{2} - 16}<\frac{8}{x^{2} - 5 x - 36}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{3}{\left(x - 4\right) \left(x + 4\right)} - \frac{8}{\left(x - 9\right) \left(x + 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 9\right) \left(x - 4\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{3 x - 27 - (8 x - 32)}{\left(x - 9\right) \left(x - 4\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{5 - 5 x}{\left(x - 9\right) \left(x - 4\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{1\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, 4, 9\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -4\right) \cup \left(1, 4\right) \cup \left(9, \infty\right)