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

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{8}{\left(x - 1\right) \left(x + 5\right)} + \frac{2}{\left(x - 1\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 5\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{2 x + 10 - (8 x + 8)}{\left(x - 1\right) \left(x + 1\right) \left(x + 5\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{2 - 6 x}{\left(x - 1\right) \left(x + 1\right) \left(x + 5\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{\frac{1}{3}\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-5, -1, 1\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(-1, \frac{1}{3}\right) \cup \left(1, \infty\right)$