Solve the inequality LaTeX: \displaystyle \frac{4}{x^{2} - 9}<\frac{8}{x^{2} + 6 x - 27}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{8}{\left(x - 3\right) \left(x + 9\right)} + \frac{4}{\left(x - 3\right) \left(x + 3\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 3\right) \left(x + 3\right) \left(x + 9\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{4 x + 36 - (8 x + 24)}{\left(x - 3\right) \left(x + 3\right) \left(x + 9\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{12 - 4 x}{\left(x - 3\right) \left(x + 3\right) \left(x + 9\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\{-9, -3, 3\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -9\right) \cup \left(-3, 3\right) \cup \left(3, \infty\right)