To simplify and combine the rational algebraic expression $\frac{2x}{x-1} + \frac{3}{x+1}$, we need to find a common denominator and then combine the terms.

Here's the step-by-step process:

1. Identify the denominators:

   • The denominators are $x-1$ and $x+1$.

2. Find the common denominator:

   • The common denominator is $(x-1)(x+1)$.

3. Rewrite each fraction with the common denominator:

   • For $\frac{2x}{x-1}$: $\frac{2x}{x-1} = \frac{2x(x+1)}{(x-1)(x+1)} = \frac{2x^2 + 2x}{(x-1)(x+1)}$
   • For $\frac{3}{x+1}$: $\frac{3}{x+1} = \frac{3(x-1)}{(x-1)(x+1)} = \frac{3x - 3}{(x-1)(x+1)}$

4. Combine the fractions:

   • Add the numerators over the common denominator: $\frac{2x^2 + 2x}{(x-1)(x+1)} + \frac{3x - 3}{(x-1)(x+1)} = \frac{2x^2 + 2x + 3x - 3}{(x-1)(x+1)}$
   • Simplify the numerator: $2x^2 + 2x + 3x - 3 = 2x^2 + 5x - 3$

5. Write the final expression:

   • The combined rational expression is: $\frac{2x^2 + 5x - 3}{(x-1)(x+1)}$

So, the simplified form of the given rational algebraic expression is: $\frac{2x^2 + 5x - 3}{(x-1)(x+1)}$

Would you like more details or have any questions?

Related Questions

1. How do you find the least common denominator (LCD) for rational expressions?
2. Can you explain how to factor polynomials in the numerator and denominator?
3. How do you simplify complex rational expressions?
4. What is the process for adding and subtracting rational expressions?
5. How do you solve rational equations involving multiple rational expressions?
6. Can you explain the process of simplifying a rational expression with multiple terms in the numerator and denominator?
7. How do you determine the excluded values for the given rational expression?
8. What are some common mistakes to avoid when simplifying rational expressions?

Tip

Always factor the polynomials completely to find the common denominators easily when working with rational expressions.