Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 4}<\frac{2}{x^{2} - 4 x - 12}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{6}{\left(x - 2\right) \left(x + 2\right)} - \frac{2}{\left(x - 6\right) \left(x + 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 6\right) \left(x - 2\right) \left(x + 2\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x - 36 - (2 x - 4)}{\left(x - 6\right) \left(x - 2\right) \left(x + 2\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{4 x - 32}{\left(x - 6\right) \left(x - 2\right) \left(x + 2\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{8\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-2, 2, 6\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-2, 2\right) \cup \left(6, 8\right)