To solve the equation 

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

let's start by simplifying and finding a common denominator.

### Step 1: Recognize the Common Denominator

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

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

So, the equation becomes:

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

### Step 2: Combine into a Single Fraction

Rewrite \(1\) as a fraction with the same denominator:

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

Substitute this into the equation:

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

Combine the fractions on the right:

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

Simplify the numerator:

\[
x + 2(x - 2) = x + 2x - 4 = 3x - 4.
\]

So the equation is now:

\[
\frac{1}{x - 2} = \frac{3x - 4}{2(x - 2)}.
\]

### Step 3: Cross Multiply to Solve for \(x\)

Cross multiply to solve for \(x\):

\[
2(x - 2) \cdot 1 = (3x - 4)(x - 2).
\]

Expand both sides:

\[
2(x - 2) = 3x^2 - 10x + 8.
\]

### Step 4: Rearrange into a Quadratic Equation

Rearrange the equation into standard quadratic form:

\[
3x^2 - 10x + 8 - 2x + 4 = 0.
\]

Simplify further:

\[
3x^2 - 12x + 12 = 0.
\]

### Step 5: Solve the Quadratic Equation

Divide everything by 3:

\[
x^2 - 4x + 4 = 0.
\]

Factor the quadratic:

\[
(x - 2)^2 = 0.
\]

So,

\[
x - 2 = 0 \implies x = 2.
\]

### Step 6: Check for Extraneous Solutions

Substitute \(x = 2\) back into the original equation:

\[
\frac{1}{2 - 2} = \frac{2}{2(2 - 2)} + 1.
\]

This simplifies to:

\[
\frac{1}{0} = \text{undefined}.
\]

Since dividing by zero is undefined, \(x = 2\) is not a valid solution. Therefore, there are no solutions to this equation.

Would you like to know more details or have any questions?

### Follow-up Questions

1. Why is dividing by zero undefined?
2. How do we identify extraneous solutions?
3. Can there be cases where quadratic equations have no valid solutions?
4. How do we factor different types of quadratic expressions?
5. What are other methods to solve equations with fractions?

### Tip

Always check for values that could make a denominator zero, as they may indicate extraneous solutions in equations involving fractions.

Discover how to solve the rational equation 1/(x-2) = x/(2x-4) + 1 with a step-by-step solution. This problem explores key algebraic concepts such as factoring, cross multiplication, and extraneous solutions. Suitable for high school students in Grades 9-12.

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