Solve the inequality $LaTeX: \displaystyle \frac{8}{x^{2} - 1}<\frac{9}{x^{2} - 5 x + 4}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{8}{\left(x - 1\right) \left(x + 1\right)} - \frac{9}{\left(x - 4\right) \left(x - 1\right)}< 0$ . 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 $LaTeX: \displaystyle \frac{8 x - 32 - (9 x + 9)}{\left(x - 4\right) \left(x - 1\right) \left(x + 1\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{- x - 41}{\left(x - 4\right) \left(x - 1\right) \left(x + 1\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{-41\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-1, 1, 4\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -41\right) \cup \left(-1, 1\right) \cup \left(4, \infty\right)$