Solve $LaTeX: \displaystyle \frac{x}{x - 4} + \frac{3}{x - 3}=- \frac{3}{x^{2} - 7 x + 12}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x - 4\right) \left(x - 3\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x - 4\right) \left(x - 3\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x - 3\right) + 3 x - 12 = -3$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - 9=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 3\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 3$ and $LaTeX: \displaystyle x = -3$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle 3$ 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.