We are given the equation:

\[
\frac{x}{x - 2} - \frac{1}{x - 2} = \frac{8}{2x - 4}
\]

### Step 1: Simplify the denominator on the right side.

Notice that \( 2x - 4 \) can be factored:

\[
2x - 4 = 2(x - 2)
\]

So the equation becomes:

\[
\frac{x}{x - 2} - \frac{1}{x - 2} = \frac{8}{2(x - 2)}
\]

### Step 2: Multiply both sides by \( 2(x - 2) \) to eliminate the denominators.

Multiplying both sides of the equation by \( 2(x - 2) \) yields:

\[
2(x - 2) \left( \frac{x}{x - 2} - \frac{1}{x - 2} \right) = 2(x - 2) \cdot \frac{8}{2(x - 2)}
\]

This simplifies to:

\[
2(x - 1) = 8
\]

### Step 3: Solve for \( x \).

Now, divide both sides by 2:

\[
x - 1 = 4
\]

Finally, add 1 to both sides:

\[
x = 5
\]

### Solution:
The solution is \( x = 5 \).

---

Would you like any further details or have any questions? Here are some related questions to explore:

1. What happens if \( x - 2 = 0 \)?
2. How can this problem be solved graphically?
3. What would change if we had different denominators?
4. Can we verify the solution by plugging it back into the original equation?
5. How do you handle equations with different variable terms in the denominator?

**Tip:** When factoring, always check for common factors first to simplify the process!

Learn how to solve the rational equation x/(x-2) - 1/(x-2) = 8/(2x-4) by simplifying denominators and solving for x. This problem involves factoring and rational expressions, suitable for high school students in Grades 9-12.

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