Solve $LaTeX: \displaystyle \frac{x}{x - 1} - \frac{4}{x - 2}=- \frac{4}{x^{2} - 3 x + 2}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x - 2\right) \left(x - 1\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x - 2\right) \left(x - 1\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x - 2\right) - 4 x + 4 = -4$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - 6 x + 8=0$ . 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 $LaTeX: \displaystyle 2$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $LaTeX: \displaystyle 4$ is not zero of the denominator it is a solution.