Solve $LaTeX: \displaystyle \frac{x}{x - 6} + \frac{2}{x - 4}=- \frac{4}{x^{2} - 10 x + 24}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x - 6\right) \left(x - 4\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x - 6\right) \left(x - 4\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x - 4\right) + 2 x - 12 = -4$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - 2 x - 8=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x + 2\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 4$ and $LaTeX: \displaystyle x = -2$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle 4$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $LaTeX: \displaystyle -2$ is not zero of the denominator it is a solution.