Comparison of x and y from two equations (use mapping below): I. 17x^2 + 48x = 9 II. 13y^2 = 32y − 12 Mapping: 1 → x > y, 2 → x < y, 3 → x = y, 4 → Relationship cannot be determined.

Introduction / Context:
Each variable is governed by a quadratic; we must list all possible roots for x and y, then check whether a single inequality relation holds for every admissible pair (x, y).

Given Data / Assumptions:

• I: 17x^2 + 48x − 9 = 0
• II: 13y^2 − 32y + 12 = 0

Concept / Approach:
Compute discriminants to find exact roots. Then compare the sets: if every x is less than every y, conclude x < y conclusively.

Step-by-Step Solution:
I: D = 48^2 − 4*17*(−9) = 2304 + 612 = 2916 = 54^2x = [−48 ± 54]/(2*17) ⇒ x ∈ {3/17 ≈ 0.1765, −3}II: D = (−32)^2 − 4*13*12 = 1024 − 624 = 400 = 20^2y = [32 ± 20]/(2*13) ⇒ y ∈ {2, 6/13 ≈ 0.4615}Compare every pair: −3 < 0.4615 and −3 < 2; 0.1765 < 0.4615 and 0.1765 < 2.

Verification / Alternative check:
Note the largest x is about 0.1765 while the smallest y is about 0.4615, ensuring x < y for all combinations.

Why Other Options Are Wrong:
“x > y” or “x = y” never occurs with these root sets; “indeterminate” is wrong because the inequality is consistent.

Common Pitfalls:
Missing one root or miscomputing the discriminant can flip the conclusion; compute precisely.

Final Answer:
x < y