Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 9}<\frac{4}{x^{2} - 6 x - 27}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{2}{\left(x - 3\right) \left(x + 3\right)} - \frac{4}{\left(x - 9\right) \left(x + 3\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 9\right) \left(x - 3\right) \left(x + 3\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x - 18 - (4 x - 12)}{\left(x - 9\right) \left(x - 3\right) \left(x + 3\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 2 x - 6}{\left(x - 9\right) \left(x - 3\right) \left(x + 3\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-3\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-3, 3, 9\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -3\right) \cup \left(-3, 3\right) \cup \left(9, \infty\right)