Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 1}<\frac{6}{x^{2} + 3 x + 2}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{6}{\left(x + 1\right) \left(x + 2\right)} + \frac{6}{\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 + 2\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x + 12 - (6 x - 6)}{\left(x - 1\right) \left(x + 1\right) \left(x + 2\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{18}{\left(x - 1\right) \left(x + 1\right) \left(x + 2\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \emptyset , or the zeros of the denominator LaTeX: \displaystyle \left\{-2, -1, 1\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -2\right) \cup \left(-1, 1\right)