Solve the inequality LaTeX: \displaystyle \frac{9}{x^{2} - 4}<\frac{6}{x^{2} + 7 x - 18}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{6}{\left(x - 2\right) \left(x + 9\right)} + \frac{9}{\left(x - 2\right) \left(x + 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 2\right) \left(x + 2\right) \left(x + 9\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{9 x + 81 - (6 x + 12)}{\left(x - 2\right) \left(x + 2\right) \left(x + 9\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x + 69}{\left(x - 2\right) \left(x + 2\right) \left(x + 9\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-23\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-9, -2, 2\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-23, -9\right) \cup \left(-2, 2\right)