Solve the inequality LaTeX: \displaystyle \frac{9}{x^{2} - 16}<\frac{7}{x^{2} - 12 x + 32}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{9}{\left(x - 4\right) \left(x + 4\right)} - \frac{7}{\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{9 x - 72 - (7 x + 28)}{\left(x - 8\right) \left(x - 4\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{2 x - 100}{\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\{50\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, 50\right)