Solve the inequality $LaTeX: \displaystyle \frac{9}{x^{2} - 1}<\frac{4}{x^{2} - 10 x + 9}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle \frac{9}{\left(x - 1\right) \left(x + 1\right)} - \frac{4}{\left(x - 9\right) \left(x - 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 9\right) \left(x - 1\right) \left(x + 1\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{9 x - 81 - (4 x + 4)}{\left(x - 9\right) \left(x - 1\right) \left(x + 1\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{5 x - 85}{\left(x - 9\right) \left(x - 1\right) \left(x + 1\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{17\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-1, 1, 9\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-1, 1\right) \cup \left(9, 17\right)$