Solve $LaTeX: \displaystyle \frac{x}{x - 1} - \frac{1}{x - 4}=- \frac{3}{x^{2} - 5 x + 4}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x - 4\right) - x + 1 = -3$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - 5 x + 4=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 1$ and $LaTeX: \displaystyle x = 4$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle 1$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $LaTeX: \displaystyle 4$ is zero of the denominator it is not in the domain and must be rejected as a solution. Therefore the equation has no solutions.