Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 4}<\frac{8}{x^{2} - 11 x + 18}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{2}{\left(x - 2\right) \left(x + 2\right)} - \frac{8}{\left(x - 9\right) \left(x - 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 9\right) \left(x - 2\right) \left(x + 2\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x - 18 - (8 x + 16)}{\left(x - 9\right) \left(x - 2\right) \left(x + 2\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 6 x - 34}{\left(x - 9\right) \left(x - 2\right) \left(x + 2\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{17}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-2, 2, 9\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, - \frac{17}{3}\right) \cup \left(-2, 2\right) \cup \left(9, \infty\right)