Solve the inequality LaTeX: \displaystyle \frac{3}{x^{2} - 9}<\frac{2}{x^{2} + 5 x + 6}

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