Solve the inequality LaTeX: \displaystyle \frac{9}{x^{2} - 4}<\frac{7}{x^{2} - 9 x + 14}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{9}{\left(x - 2\right) \left(x + 2\right)} - \frac{7}{\left(x - 7\right) \left(x - 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 7\right) \left(x - 2\right) \left(x + 2\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{9 x - 63 - (7 x + 14)}{\left(x - 7\right) \left(x - 2\right) \left(x + 2\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{2 x - 77}{\left(x - 7\right) \left(x - 2\right) \left(x + 2\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{\frac{77}{2}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-2, 2, 7\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-2, 2\right) \cup \left(7, \frac{77}{2}\right)