I. 4x2 + 3x - 27 = 0, II. 3y2 - 20y + 32 = 0

Concept: When two variables are each defined by their own quadratic equation, solve each equation for both roots using the quadratic formula, then compare the two ROOT SETS directly. If the largest root of one variable is still smaller than the smallest root of the other, the same strict inequality holds for every possible pairing of the roots.

1. Solve Equation I: 4x2 + 3x - 27 = 0. By the quadratic formula, x = [-3 ± √(32 - 4*4*(-27))] / (2*4) = [-3 ± √(441)] / 8 = [-3 ± 21] / 8, giving x = 2.25 or x = -3.

2. Solve Equation II: 3y2 - 20y + 32 = 0. By the quadratic formula, y = [20 ± √(202 - 4*3*32)] / (2*3) = [20 ± √(16)] / 6 = [20 ± 4] / 6, giving y = 4 or y = 8/3 (about 2.667).

3. Compare the root sets: x is in {-3, 2.25} and y is in {8/3, 4}. The maximum value of x (2.25) is still smaller than the minimum value of y (8/3), so every x value is less than every y value across all four pairings.

Cross-check by factoring: 4x2 + 3x - 27 = (4x - 9)(x + 3), giving x = 9/4 = 2.25 or x = -3; 3y2 - 20y + 32 = (3y - 8)(y - 4), giving y = 8/3 or y = 4 -- matching the quadratic-formula roots and confirming x < y in every case.