Solve the inequality $LaTeX: \displaystyle \frac{4}{x^{2} - 1}<\frac{7}{x^{2} + 6 x - 7}$

Getting zero on one side and factoring gives $LaTeX: \displaystyle - \frac{7}{\left(x - 1\right) \left(x + 7\right)} + \frac{4}{\left(x - 1\right) \left(x + 1\right)}< 0$ . This gives the least common denominator as $LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 7\right)$ . Building each fraction to get the common denominator gives $LaTeX: \displaystyle \frac{4 x + 28 - (7 x + 7)}{\left(x - 1\right) \left(x + 1\right) \left(x + 7\right)} < 0$ . Simplifying gives $LaTeX: \displaystyle \frac{21 - 3 x}{\left(x - 1\right) \left(x + 1\right) \left(x + 7\right)}<0$ . The inequality can change signs at the zeros of the numerator, $LaTeX: \displaystyle \left\{7\right\}$ , or the zeros of the denominator $LaTeX: \displaystyle \left\{-7, -1, 1\right\}$ . Making a sign chart gives: This gives the solution $LaTeX: \displaystyle \left(-\infty, -7\right) \cup \left(-1, 1\right) \cup \left(7, \infty\right)$