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

Options

A. `1/(4x^2 - 1)`
B. `(8x)/(4x^2 - 1)`
C. `(-2)/(2x^2 - 1)`
D. `(4x)/(4x^2 - 1)`

Answer

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

### Step-by-step Solution:

1. **Write the expression for \( A - B \)**:
\[ A - B = \frac{2x + 1}{2x - 1} - \frac{2x - 1}{2x + 1} \]

2. **Find a common denominator**:
The common denominator for the two fractions is \( (2x - 1)(2x + 1) \).

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

4. **Expand the numerators**:
- For \( (2x + 1)(2x + 1) \):
\[ (2x + 1)^2 = 4x^2 + 4x + 1 \]
- For \( (2x - 1)(2x - 1) \):
\[ (2x - 1)^2 = 4x^2 - 4x + 1 \]

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

6. **Simplify the numerator**:
\[ = \frac{4x^2 + 4x + 1 - 4x^2 + 4x - 1}{(2x - 1)(2x + 1)} \]
\[ = \frac{8x}{(2x - 1)(2x + 1)} \]

7. **Final result**:
\[ A - B = \frac{8x}{(2x - 1)(2x + 1)} \]