Question

If \( A = \frac{x+1}{x-1} \) and \( B = \frac{x-1}{x+1} \), then \( A + B \) is:

• A. None of these
• B. \( \frac{2(x^2 + 1)}{(x - 1)^2} \)
• C. \( \frac{2(x^2 - 1)}{x^2 + 1} \)
• D. \( \frac{x^2 + 1}{x^2 - 1} \)

Hint:

When adding fractions, always find a common denominator and simplify the expression carefully.

Answer 1

The Correct Option is B

We are given the expressions for \( A \) and \( B \). First, let's add \( A \) and \( B \): \[ A + B = \frac{x+1}{x-1} + \frac{x-1}{x+1}. \] We need to find a common denominator, which is \( (x-1)(x+1) = x^2 - 1 \). So: \[ A + B = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)}. \] Expanding the numerator: \[ (x+1)^2 + (x-1)^2 = x^2 + 2x + 1 + x^2 - 2x + 1 = 2x^2 + 2. \] So: \[ A + B = \frac{2(x^2 + 1)}{x^2 - 1}. \] Thus, the correct answer is \( \frac{2(x^2 + 1)}{x^2 - 1} \), which matches option (B).