Solve the inequality LaTeX: \displaystyle \frac{4}{x^{2} - 16}<\frac{3}{x^{2} - 6 x + 8}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{4}{\left(x - 4\right) \left(x + 4\right)} - \frac{3}{\left(x - 4\right) \left(x - 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x - 2\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{4 x - 8 - (3 x + 12)}{\left(x - 4\right) \left(x - 2\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{x - 20}{\left(x - 4\right) \left(x - 2\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{20\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, 2, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-4, 2\right) \cup \left(4, 20\right)