Compare greater roots (definition repaired): Let x be the greater root of x^2 + 10x + 24 = 0 and y be the greater root of 4y^2 − 17y + 18 = 0. Choose the correct relation.

Introduction / Context:
We repair the stem by explicitly defining the comparison target: the larger roots of each quadratic. Then we compute those roots and compare. If x is strictly less than y, the statement x ≤ y is true as well.

Given Data / Assumptions:

• x^2 + 10x + 24 = 0; let x be the greater root.
• 4y^2 − 17y + 18 = 0; let y be the greater root.

Concept / Approach:
Factor or use the quadratic formula to find each equation’s larger root. Compare them exactly to avoid sign mistakes, then choose the most accurate relation provided among the options.

Step-by-Step Solution:

Verification / Alternative check:
Substitute the computed roots back into the original equations to confirm correctness before comparing.

Why Other Options Are Wrong:
x ≥ y and x > y contradict the numerical comparison; “cannot be established” is incorrect because both larger roots are uniquely determined.

Common Pitfalls:
Forgetting that the “greater root” for a quadratic with both negative roots is the one closer to zero.

Final Answer:
x ≤ y