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The chain W ≥ X ≥ Y ≥ Z does not support the conclusion W > Z. Replacing exactly one of its three relations by which of the following would make W > Z definitely follow?

Competition practiceInequalities
  • AReplace W ≥ X by W = X
  • BReplace X ≥ Y by X ≤ Y
  • CReplace W ≥ X by X ≥ W
  • DReplace Y ≥ Z by Y > Z

Step-by-Step Solution

A run of relations composes to a strict result exactly when the links all point the same way and at least one of them is strict. W ≥ X ≥ Y ≥ Z has the right direction throughout but not a single strict link, which is why only W ≥ Z follows. Turning Y ≥ Z into Y > Z supplies the missing strict link and gives W ≥ X ≥ Y > Z, hence W > Z. Replacing W ≥ X by W = X keeps every link non-strict, so the chain still yields only W ≥ Z. Replacing X ≥ Y by X ≤ Y reverses the middle link and breaks the run, leaving W and Z uncomparable. Replacing W ≥ X by X ≥ W does the same at the front of the chain.
Answer: D) Replace Y ≥ Z by Y > Z

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