Solve the inequality LaTeX: \displaystyle \frac{5}{x^{2} - 4}<\frac{6}{x^{2} - 2 x - 8}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{5}{\left(x - 2\right) \left(x + 2\right)} - \frac{6}{\left(x - 4\right) \left(x + 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x - 2\right) \left(x + 2\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{5 x - 20 - (6 x - 12)}{\left(x - 4\right) \left(x - 2\right) \left(x + 2\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- x - 8}{\left(x - 4\right) \left(x - 2\right) \left(x + 2\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-8\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-2, 2, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -8\right) \cup \left(-2, 2\right) \cup \left(4, \infty\right)