Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 1}<\frac{8}{x^{2} - 5 x + 4}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{8}{\left(x - 4\right) \left(x - 1\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right) \left(x + 1\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x - 8 - (8 x + 8)}{\left(x - 4\right) \left(x - 1\right) \left(x + 1\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 6 x - 16}{\left(x - 4\right) \left(x - 1\right) \left(x + 1\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{8}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-1, 1, 4\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, - \frac{8}{3}\right) \cup \left(-1, 1\right) \cup \left(4, \infty\right)