To simplify the expression \( \frac{1}{x} + \frac{2x}{x+y} \), follow these steps:

1. **Identify the denominators:**
   - The denominators are \( x \) and \( x + y \).

2. **Find the common denominator:**
   - The common denominator is \( x(x + y) \).

3. **Rewrite each fraction with the common denominator:**
   - For \( \frac{1}{x} \):
     \[
     \frac{1}{x} = \frac{1 \cdot (x+y)}{x(x+y)} = \frac{x+y}{x(x+y)}
     \]
   - For \( \frac{2x}{x+y} \):
     \[
     \frac{2x}{x+y} = \frac{2x \cdot x}{(x+y) \cdot x} = \frac{2x^2}{x(x+y)}
     \]

4. **Combine the fractions:**
   - Add the numerators over the common denominator:
     \[
     \frac{x+y}{x(x+y)} + \frac{2x^2}{x(x+y)} = \frac{x+y + 2x^2}{x(x+y)}
     \]

5. **Write the final expression:**
   - The combined rational expression is:
     \[
     \frac{2x^2 + x + y}{x(x+y)}
     \]

So, the simplified form of the given rational algebraic expression is:
\[
\frac{2x^2 + x + y}{x(x + y)}
\]

Would you like more details or have any questions?

### Related Questions

1. How do you find the least common denominator (LCD) for rational expressions with multiple variables?
2. Can you explain how to factor polynomials with more than one variable?
3. How do you simplify complex rational expressions with different variable terms?
4. What is the process for adding and subtracting rational expressions with unlike denominators?
5. How do you solve rational equations that involve multiple rational expressions with different variables?
6. Can you explain the process of simplifying a rational expression when the terms include sums and products of variables?
7. How do you determine the excluded values for the given rational expression?
8. What are some techniques to simplify rational expressions with polynomial numerators?

### Tip

When working with rational expressions, always identify and combine like terms in the numerator and ensure you simplify the expression as much as possible.

Learn how to simplify the rational expression 1/x + 2x/(x+y) step-by-step. This problem involves rational expressions and common denominators, suitable for high school students. Find detailed solutions and explanations.

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