To solve the equation $\frac{p}{x} = \frac{q}{x - 2}$, we can use cross-multiplication:

$p(x - 2) = qx$

Now, distribute the $p$ on the left-hand side:

$p(x - 2) = px - 2p$

So, the equation becomes:

$px - 2p = qx$

Next, let's move all terms involving $x$ to one side of the equation. Subtract $qx$ from both sides:

$px - qx = 2p$

Factor out $x$ on the left-hand side:

$x(p - q) = 2p$

Finally, solve for $x$ by dividing both sides by $(p - q)$ (assuming $p \neq q$):

$x = \frac{2p}{p - q}$

Thus, the solution is:

$x = \frac{2p}{p - q}$

Would you like more details or further clarification?

Here are 5 related questions:

1. What happens if $p = q$?
2. How would the solution change if the equation had additional terms?
3. Can you solve a similar equation for specific values of $p$ and $q$?
4. What if the variables were squared, i.e., $\frac{p}{x^2} = \frac{q}{(x-2)^2}$?
5. How would you solve this if $p$ or $q$ were negative?

Tip: Always check for restrictions in the denominator to avoid dividing by zero (e.g., $x \neq 0$ and $x \neq 2$ in this problem).