Solve the inequality LaTeX: \displaystyle \frac{8}{x^{2} - 16}<\frac{5}{x^{2} - 10 x + 24}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{8}{\left(x - 4\right) \left(x + 4\right)} - \frac{5}{\left(x - 6\right) \left(x - 4\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 6\right) \left(x - 4\right) \left(x + 4\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{8 x - 48 - (5 x + 20)}{\left(x - 6\right) \left(x - 4\right) \left(x + 4\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x - 68}{\left(x - 6\right) \left(x - 4\right) \left(x + 4\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{\frac{68}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-4, 4, 6\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-4, 4\right) \cup \left(6, \frac{68}{3}\right)