Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 25}<\frac{3}{x^{2} - 3 x - 10}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{6}{\left(x - 5\right) \left(x + 5\right)} - \frac{3}{\left(x - 5\right) \left(x + 2\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x + 2\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x + 12 - (3 x + 15)}{\left(x - 5\right) \left(x + 2\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x - 3}{\left(x - 5\right) \left(x + 2\right) \left(x + 5\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{1\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-5, -2, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-5, -2\right) \cup \left(1, 5\right)