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

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{4}{\left(x - 2\right) \left(x + 3\right)} + \frac{7}{\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{7 x - 14 - (4 x - 12)}{\left(x - 3\right) \left(x - 2\right) \left(x + 3\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x - 2}{\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\{\frac{2}{3}\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(-3, \frac{2}{3}\right) \cup \left(2, 3\right)