Let x = (√2 + 1)/(√2 − 1) and y = (√2 − 1)/(√2 + 1). Evaluate 3 (xy + 1) in terms of x and y.

Introduction:
This problem leverages conjugate pairs and simplification. Recognize that x and y are reciprocals because they are formed by swapping numerator and denominator with conjugates of √2 ± 1.

Given Data / Assumptions:

• x = (√2 + 1)/(√2 − 1)
• y = (√2 − 1)/(√2 + 1)
• Expression: 3(xy + 1)

Concept / Approach:
Compute xy. Since y is the reciprocal of x, xy = 1. Substitute into 3(xy + 1) and compare with expressions involving x and y to find identity matches.

Step-by-Step Solution:
xy = [(√2 + 1)/(√2 − 1)] * [(√2 − 1)/(√2 + 1)] = 13(xy + 1) = 3(1 + 1) = 6Compute x + y: it simplifies to 6 (by rationalization or numeric evaluation), hence 3(xy + 1) = x + y.

Verification / Alternative check:
Numerically, x ≈ 5.828 and y ≈ 0.172, so x + y ≈ 6 and 3(xy + 1) = 6.

Why Other Options Are Wrong:
x − y: ≈ 5.6568, not 6.2(x + y): ≈ 12, not 6.None of above: incorrect since x + y matches.

Common Pitfalls:
Missing that xy = 1 and overcomplicating the algebra; arithmetic slip when rationalizing surds.

Final Answer:
x + y