Solve the inequality LaTeX: \displaystyle \frac{6}{x^{2} - 25}<\frac{7}{x^{2} - 12 x + 35}

Getting zero on one side and factoring gives LaTeX: \displaystyle \frac{6}{\left(x - 5\right) \left(x + 5\right)} - \frac{7}{\left(x - 7\right) \left(x - 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 7\right) \left(x - 5\right) \left(x + 5\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{6 x - 42 - (7 x + 35)}{\left(x - 7\right) \left(x - 5\right) \left(x + 5\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{- x - 77}{\left(x - 7\right) \left(x - 5\right) \left(x + 5\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{-77\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-5, 5, 7\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(-\infty, -77\right) \cup \left(-5, 5\right) \cup \left(7, \infty\right)