Solve the inequality LaTeX: \displaystyle \frac{8}{x^{2} - 9}<\frac{4}{x^{2} + 2 x - 3}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{4}{\left(x - 1\right) \left(x + 3\right)} + \frac{8}{\left(x - 3\right) \left(x + 3\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 3\right) \left(x - 1\right) \left(x + 3\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{8 x - 8 - (4 x - 12)}{\left(x - 3\right) \left(x - 1\right) \left(x + 3\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{4 x + 4}{\left(x - 3\right) \left(x - 1\right) \left(x + 3\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-1\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-3, 1, 3\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-3, -1\right) \cup \left(1, 3\right)