Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 16}<\frac{3}{x^{2} + 12 x + 32}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{3}{\left(x + 4\right) \left(x + 8\right)} + \frac{2}{\left(x - 4\right) \left(x + 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x + 4\right) \left(x + 8\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x + 16 - (3 x - 12)}{\left(x - 4\right) \left(x + 4\right) \left(x + 8\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{28 - x}{\left(x - 4\right) \left(x + 4\right) \left(x + 8\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{28\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-8, -4, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -8\right) \cup \left(-4, 4\right) \cup \left(28, \infty\right)