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

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{4}{\left(x - 1\right) \left(x + 4\right)} + \frac{3}{\left(x - 1\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 4\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{3 x + 12 - (4 x + 4)}{\left(x - 1\right) \left(x + 1\right) \left(x + 4\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{8 - x}{\left(x - 1\right) \left(x + 1\right) \left(x + 4\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{8\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-4, -1, 1\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -4\right) \cup \left(-1, 1\right) \cup \left(8, \infty\right)$