Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 25}<\frac{3}{x^{2} - 11 x + 30}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{6}{\left(x - 5\right) \left(x + 5\right)} - \frac{3}{\left(x - 6\right) \left(x - 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 6\right) \left(x - 5\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x - 36 - (3 x + 15)}{\left(x - 6\right) \left(x - 5\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x - 51}{\left(x - 6\right) \left(x - 5\right) \left(x + 5\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{17\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-5, 5, 6\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-5, 5\right) \cup \left(6, 17\right)