Solve the inequality $LaTeX: \displaystyle \frac{3}{x^{2} - 1}<\frac{3}{x^{2} + 4 x + 3}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{3}{\left(x + 1\right) \left(x + 3\right)} + \frac{3}{\left(x - 1\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 3\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{3 x + 9 - (3 x - 3)}{\left(x - 1\right) \left(x + 1\right) \left(x + 3\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{12}{\left(x - 1\right) \left(x + 1\right) \left(x + 3\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \emptyset$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-3, -1, 1\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -3\right) \cup \left(-1, 1\right)$