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The equation in the image is:

\[
\frac{x - 2}{x - 4} - \frac{x - 4}{x - 6} = \frac{x - 1}{x - 3} - \frac{x - 3}{x - 5}
\]

### Step 1: Simplify Each Side

**Left Side:**

\[
\frac{x - 2}{x - 4} - \frac{x - 4}{x - 6}
\]

To simplify this, find a common denominator:

- The common denominator is \((x - 4)(x - 6)\).

\[
\frac{(x - 2)(x - 6) - (x - 4)(x - 4)}{(x - 4)(x - 6)}
\]

Expanding the numerators:

\[
(x - 2)(x - 6) = x^2 - 6x - 2x + 12 = x^2 - 8x + 12
\]

\[
(x - 4)(x - 4) = x^2 - 8x + 16
\]

Subtracting the numerators:

\[
\frac{x^2 - 8x + 12 - (x^2 - 8x + 16)}{(x - 4)(x - 6)} = \frac{x^2 - 8x + 12 - x^2 + 8x - 16}{(x - 4)(x - 6)} = \frac{-4}{(x - 4)(x - 6)}
\]

So, the left side simplifies to:

\[
\frac{-4}{(x - 4)(x - 6)}
\]

**Right Side:**

\[
\frac{x - 1}{x - 3} - \frac{x - 3}{x - 5}
\]

Similarly, find the common denominator:

- The common denominator is \((x - 3)(x - 5)\).

\[
\frac{(x - 1)(x - 5) - (x - 3)(x - 3)}{(x - 3)(x - 5)}
\]

Expanding the numerators:

\[
(x - 1)(x - 5) = x^2 - 5x - x + 5 = x^2 - 6x + 5
\]

\[
(x - 3)(x - 3) = x^2 - 6x + 9
\]

Subtracting the numerators:

\[
\frac{x^2 - 6x + 5 - (x^2 - 6x + 9)}{(x - 3)(x - 5)} = \frac{x^2 - 6x + 5 - x^2 + 6x - 9}{(x - 3)(x - 5)} = \frac{-4}{(x - 3)(x - 5)}
\]

So, the right side simplifies to:

\[
\frac{-4}{(x - 3)(x - 5)}
\]

### Step 2: Set the Simplified Expressions Equal

Now, we have:

\[
\frac{-4}{(x - 4)(x - 6)} = \frac{-4}{(x - 3)(x - 5)}
\]

Since the numerators are equal, we can equate the denominators:

\[
(x - 4)(x - 6) = (x - 3)(x - 5)
\]

### Step 3: Expand and Solve the Quadratic Equation

Expand both sides:

\[
x^2 - 6x - 4x + 24 = x^2 - 5x - 3x + 15
\]

Simplify:

\[
x^2 - 10x + 24 = x^2 - 8x + 15
\]

Subtract \(x^2\) from both sides:

\[
-10x + 24 = -8x + 15
\]

Add \(10x\) to both sides:

\[
24 = 2x + 15
\]

Subtract 15 from both sides:

\[
9 = 2x
\]

Divide by 2:

\[
x = \frac{9}{2} = 4.5
\]

### Conclusion

The solution to the equation is \( x = 4.5 \).

---

Would you like more details, or do you have any questions about this solution?

Here are some related questions you might consider:

1. What happens if you substitute \( x = 4.5 \) back into the original equation?
2. How would you handle cases where the equation involves more complex fractions?
3. What are the conditions where the solution might not exist (e.g., division by zero)?
4. Can you explore other methods to solve similar equations, such as cross-multiplication?
5. How does the solution change if the equation has an additional constant term?

**Tip:** Always check the domain of the equation before simplifying to ensure there are no restrictions (like division by zero).

Learn how to simplify and solve fractional equations using common denominators with a detailed step-by-step solution. This problem involves fractions and quadratic equations, suitable for advanced high school students.

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