Solve the inequality LaTeX: \displaystyle \frac{3}{x^{2} - 9}<\frac{4}{x^{2} - 2 x - 3}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{3}{\left(x - 3\right) \left(x + 3\right)} - \frac{4}{\left(x - 3\right) \left(x + 1\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 3\right) \left(x + 1\right) \left(x + 3\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{3 x + 3 - (4 x + 12)}{\left(x - 3\right) \left(x + 1\right) \left(x + 3\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- x - 9}{\left(x - 3\right) \left(x + 1\right) \left(x + 3\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-9\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-3, -1, 3\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -9\right) \cup \left(-3, -1\right) \cup \left(3, \infty\right)