Solve $LaTeX: \displaystyle \frac{x}{x - 1} + \frac{3}{x - 4}=\frac{9}{x^{2} - 5 x + 4}$ .

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 $LaTeX: \displaystyle x \left(x - 4\right) + 3 x - 3 = 9$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - x - 12=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x + 3\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 4$ and $LaTeX: \displaystyle x = -3$ . 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 $LaTeX: \displaystyle -3$ is not zero of the denominator it is a solution.