To solve the equation:

$3 - \frac{2}{x - 5} = 9$

Step-by-step solution:

1. Isolate the fraction term: Subtract 3 from both sides to isolate the fraction on one side: $-\frac{2}{x - 5} = 9 - 3$ Simplifying: $-\frac{2}{x - 5} = 6$

2. Remove the negative sign: Multiply both sides by $-1$: $\frac{2}{x - 5} = -6$

3. Solve for $x - 5$: Multiply both sides by $x - 5$ to eliminate the denominator: $2 = -6(x - 5)$

4. Distribute and solve for $x$: Distribute $-6$ on the right-hand side: $2 = -6x + 30$ Subtract 30 from both sides: $2 - 30 = -6x$ Simplifying: $-28 = -6x$

5. Solve for $x$: Divide both sides by $-6$: $x = \frac{-28}{-6} = \frac{14}{3}$

Final Answer:

$x = \frac{14}{3}$

Let me know if you want further details or have any questions!

Here are some related questions:

1. How do you solve equations with fractions?
2. What are the common mistakes when solving rational equations?
3. How do you solve an equation involving multiple fractions?
4. How does multiplying by the denominator help in solving rational equations?
5. Can we use another method to solve this type of equation?

Tip: Always check for extraneous solutions when dealing with rational equations.