The problem asks to combine the given fractions as indicated by the signs. Let's solve the expression step by step.

The expression given is:

$\frac{2x}{y^2 - x^2} - \frac{x}{y - x}$

Step 1: Factorize the Denominator $y^2 - x^2$

The denominator $y^2 - x^2$ is a difference of squares, which can be factored as: $y^2 - x^2 = (y - x)(y + x)$

So the first fraction becomes: $\frac{2x}{(y - x)(y + x)}$

The second fraction remains: $\frac{x}{y - x}$

Step 2: Rewrite with a Common Denominator

The least common denominator (LCD) between $(y - x)$ and $(y - x)(y + x)$ is $(y - x)(y + x)$.

Rewrite both fractions with this common denominator:

1. The first fraction already has the denominator $(y - x)(y + x)$, so it remains: $\frac{2x}{(y - x)(y + x)}$
2. For the second fraction, multiply numerator and denominator by $(y + x)$: $\frac{x}{y - x} = \frac{x(y + x)}{(y - x)(y + x)}$

Now the expression is: $\frac{2x}{(y - x)(y + x)} - \frac{x(y + x)}{(y - x)(y + x)}$

Step 3: Combine the Numerators

Since the denominators are the same, combine the numerators: $\frac{2x}{(y - x)(y + x)} - \frac{x(y + x)}{(y - x)(y + x)} = \frac{2x - x(y + x)}{(y - x)(y + x)}$

Simplify the numerator: $2x - x(y + x) = 2x - xy - x^2 = x(2 - y - x)$

So the combined fraction is: $\frac{x(2 - y - x)}{(y - x)(y + x)}$

Final Answer:

$\frac{x(2 - y - x)}{(y - x)(y + x)}$

Would you like me to explain any part in more detail?

5 Related Questions:

1. How do you handle expressions with different denominators in general?
2. What are some common factorization techniques used for algebraic expressions?
3. Can you explain the difference of squares and how it applies in this problem?
4. How do you check the validity of an algebraic simplification?
5. What are the real-world applications of simplifying rational expressions?

Tip:

Always factorize complex expressions first to identify common denominators and simplify the process.