Inequalities — Given A > B ≥ C and C < D < E, which of the following is definitely true? Choose the single option that must hold in all cases consistent with the chain.

Introduction / Context:This problem checks deterministic reasoning with chained inequalities. From a given set of relations between five variables A, B, C, D, and E, we must identify the statement that is guaranteed to be true for every numeric assignment that satisfies the chain. The core skill is reading chained comparisons and propagating them without making unsupported leaps.

Given Data / Assumptions:

• A > B ≥ C
• C < D < E (that is, D > C and E > D)
• All letters represent real numbers; no additional relations are given.

Concept / Approach:Break the long chain into safe implications. From A > B and B ≥ C, it follows that A > C. From C < D and D < E, we know D > C and E > C as well as E > D. However, the chain gives no direct comparison between A and D or E, nor between D and B (beyond both being on opposite sides of C). Therefore, only relations supported purely by transitivity on shared elements (especially anchored at C) are certain.

Step-by-Step Solution:

Verification / Alternative check:Construct counterexamples for the dubious options. Let C = 0, B = 0, A = 1, D = 0.1, E = 0.2. Then D ≥ B is true (0.1 ≥ 0), but choose C = 0, B = 5, A = 6, D = 4, E = 7 and D ≥ B is false (4 ≥ 5 fails). So D ≥ B is not definite. Similarly, A > D can be true or false depending on chosen values consistent with the chain. Meanwhile, E > C always holds because E > D > C is baked into the chain.

Why Other Options Are Wrong:

• A ≤ C: Contradicts A > C, which follows from A > B ≥ C.
• D ≥ B: Not guaranteed; depends on actual placements of B and D above C.
• A > D: Not guaranteed; A may be only slightly above B while D can be far above C.

Common Pitfalls:

• Assuming comparability between variables not directly linked by the chain (e.g., leaping from A > B and D > C to A > D without evidence).
• Forgetting that “≥” allows equality; B could equal C.

Final Answer:E > C