Solve the inequality LaTeX: \displaystyle \frac{7}{x^{2} - 1}<\frac{4}{x^{2} - 8 x - 9}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{7}{\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{7 x - 63 - (4 x - 4)}{\left(x - 9\right) \left(x - 1\right) \left(x + 1\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x - 59}{\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\{\frac{59}{3}\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, \frac{59}{3}\right)