If `A = (x +1)/(x - 1) and B = (x - 2)/(x + 1)`, then A + B is

Options

A. `(2x^(2)-x+3)/((x^(2)-1))`
B. `(2(x^(2)+1))/((x^(2)-1))`
C. `((x^(2)+1))/((x^(2)-1))`
D. None of these

Answer

To solve the problem where \( A = \frac{x + 1}{x - 1} \) and \( B = \frac{x - 2}{x + 1} \), we need to find \( A + B \).

### Step-by-Step Solution:

1. **Write down the expressions for A and B:**
\[ A = \frac{x + 1}{x - 1}, \quad B = \frac{x - 2}{x + 1} \]

2. **Find a common denominator:**
The denominators are \( x - 1 \) and \( x + 1 \). The least common multiple (LCM) of these denominators is:
\[ (x - 1)(x + 1) \]

3. **Rewrite A and B with the common denominator:**
\[ A + B = \frac{(x + 1)(x + 1)}{(x - 1)(x + 1)} + \frac{(x - 2)(x - 1)}{(x - 1)(x + 1)} \]

4. **Combine the fractions:**
\[ A + B = \frac{(x + 1)^2 + (x - 2)(x - 1)}{(x - 1)(x + 1)} \]

5. **Expand the numerators:**
- For \( (x + 1)^2 \):
\[ (x + 1)^2 = x^2 + 2x + 1 \]
- For \( (x - 2)(x - 1) \):
\[ (x - 2)(x - 1) = x^2 - x - 2 \]

6. **Combine the expanded numerators:**
\[ A + B = \frac{(x^2 + 2x + 1) + (x^2 - x - 2)}{(x - 1)(x + 1)} \]
Simplifying the numerator:
\[ = \frac{2x^2 + 2x - 1}{(x - 1)(x + 1)} \]

7. **Final expression:**
\[ A + B = \frac{2x^2 + 2x - 1}{x^2 - 1} \]