Solve the inequality LaTeX: \displaystyle \frac{2}{x^{2} - 1}<\frac{6}{x^{2} - 8 x + 7}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{6}{\left(x - 7\right) \left(x - 1\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 7\right) \left(x - 1\right) \left(x + 1\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{2 x - 14 - (6 x + 6)}{\left(x - 7\right) \left(x - 1\right) \left(x + 1\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- 4 x - 20}{\left(x - 7\right) \left(x - 1\right) \left(x + 1\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-5\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-1, 1, 7\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -5\right) \cup \left(-1, 1\right) \cup \left(7, \infty\right)