Solve $LaTeX: \displaystyle \frac{x}{x - 9} - \frac{5}{x - 8}=\frac{5}{x^{2} - 17 x + 72}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x - 9\right) \left(x - 8\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x - 9\right) \left(x - 8\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x - 8\right) - 5 x + 45 = 5$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - 13 x + 40=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x - 5\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 8$ and $LaTeX: \displaystyle x = 5$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle 8$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $LaTeX: \displaystyle 5$ is not zero of the denominator it is a solution.