Recovery-fixed item (typo): With F ≤ X < A and R < X ≤ F, which conclusion(s) definitely follow? I) F ≤ A; II) R < F.

Introduction / Context:The original stem referenced “F ≤ E” although E was never defined. Applying the Recovery-First Policy, we minimally repair the typo to “F ≤ A”, which is consistent with the given chain F ≤ X < A and R < X ≤ F.

Given Data / Assumptions (after minimal repair):

• F ≤ X < A
• R < X ≤ F
• Conclusions: I) F ≤ A; II) R < F

Concept / Approach:From the two inequalities we can pin down equalities/ordering: F ≤ X and X ≤ F together imply F = X. Then X < A gives F < A, which certainly implies F ≤ A. Also R < X = F ⇒ R < F.

Step-by-Step Solution: From F ≤ X and X ≤ F ⇒ F = X. Since X < A ⇒ F < A ⇒ F ≤ A (I true). R < X and X = F ⇒ R < F (II true).

Verification / Alternative check:Pick X=10, F=10, A=11, R=5. Both conclusions hold strictly.

Why Other Options Are Wrong:a/b/d/e contradict at least one forced relation; both I and II are compelled.

Common Pitfalls:Missing that the pair F ≤ X and X ≤ F forces equality, which then tightens both conclusions.

Final Answer:If both conclusions I and II follow