Let x be the greater real root of x^2 + x − 20 = 0 and y be the greater real root of y^2 − y − 30 = 0. Compare x and y and choose the correct relation.

Introduction / Context: The original statement was ambiguous because each quadratic has two roots. Applying the Recovery-First Policy, we define x and y explicitly as the greater real roots. We then compare these values by solving each quadratic via factoring.

Given Data / Assumptions:

• x solves x^2 + x − 20 = 0 and is the greater root.
• y solves y^2 − y − 30 = 0 and is the greater root.
• We compare x to y.

Concept / Approach: Factor each quadratic to get roots quickly. For ax^2 + bx + c with small integers, try integer factor pairs of c that sum to b. Identify the larger root in each case and compare.

Step-by-Step Solution:

Verification / Alternative check: Quadratic formula gives identical results; factoring is sufficient here.

Why Other Options Are Wrong: x > y or x ≥ y contradict the computed values; equality does not hold.

Common Pitfalls: Not clarifying which root is used; comparing mismatched (e.g., larger with smaller) roots gives inconsistent relations.

Final Answer: If x < y