Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 16}<\frac{2}{x^{2} - 12 x + 32}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{6}{\left(x - 4\right) \left(x + 4\right)} - \frac{2}{\left(x - 8\right) \left(x - 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 8\right) \left(x - 4\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x - 48 - (2 x + 8)}{\left(x - 8\right) \left(x - 4\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{4 x - 56}{\left(x - 8\right) \left(x - 4\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{14\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, 4, 8\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-4, 4\right) \cup \left(8, 14\right)