To simplify the given rational expression $\frac{6}{x-1} + \frac{3}{x+1}$, we need to combine the fractions by finding a common denominator and adding the numerators.

Steps to Simplify

1. Identify the common denominator:

   • The denominators are $(x-1)$ and $(x+1)$.
   • The least common denominator (LCD) is $(x-1)(x+1)$.

2. Rewrite each fraction with the common denominator: $\frac{6}{x-1} = \frac{6(x+1)}{(x-1)(x+1)}$ $\frac{3}{x+1} = \frac{3(x-1)}{(x-1)(x+1)}$

3. Add the fractions: $\frac{6(x+1)}{(x-1)(x+1)} + \frac{3(x-1)}{(x-1)(x+1)} = \frac{6(x+1) + 3(x-1)}{(x-1)(x+1)}$

4. Expand and combine the numerators: $6(x+1) + 3(x-1) = 6x + 6 + 3x - 3$ $= 9x + 3$

5. Write the final simplified expression: $\frac{9x + 3}{(x-1)(x+1)}$

Answers

1. Identify the correct variable term for the numerator:

   • Answer: $9x$

2. Identify the correct constant term for the numerator:

   • Answer: $3$

Summary

The simplified form of the expression is: $\frac{9x + 3}{(x-1)(x+1)}$

Would you like further clarification or have any questions about this explanation?

Related Questions

1. How do you find the least common denominator for two or more fractions?
2. What are the steps to simplify complex rational expressions?
3. How do you add rational expressions with different denominators?
4. What are common mistakes to avoid when simplifying rational expressions?
5. How can factoring be used to simplify rational expressions?
6. How do you multiply and divide rational expressions?
7. How do you solve equations involving rational expressions?
8. What real-world problems can be modeled using rational expressions?

Tip

When adding or subtracting rational expressions, always start by identifying the least common denominator and rewriting each expression with that denominator before combining them.