Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 16}<\frac{7}{x^{2} - 2 x - 8}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{2}{\left(x - 4\right) \left(x + 4\right)} - \frac{7}{\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 + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x + 4 - (7 x + 28)}{\left(x - 4\right) \left(x + 2\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 5 x - 24}{\left(x - 4\right) \left(x + 2\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{24}{5}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, -2, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, - \frac{24}{5}\right) \cup \left(-4, -2\right) \cup \left(4, \infty\right)