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Assume every relation in the line below is true, and read ≤ as “is at most”.
Statement: T > U ≤ V < W
Conclusions: I. T > V
II. U < W
Which conclusion follows?

Competition practiceInequalities
  • AOnly II follows
  • BOnly I follows
  • CBoth I and II follow
  • DNeither I nor II follows

Step-by-Step Solution

Between T and V the two links point opposite ways: T > U says T is above U, while U ≤ V says U is at or below V. V may sit anywhere relative to T, so I does not follow. From U to W the links agree in sense: U ≤ V and V < W give U < W, since a non-strict link followed by a strict one is strict. So only II follows. 'Only I follows' is exactly what a solver gets who misreads U ≤ V as V ≤ U: that reading gives T > U ≥ V, hence T > V, while it also leaves U above V and so breaks the link to W, killing II. 'Both' comes from composing across the reversal at U — arguing that because T is above U and V sits just beyond U, T must be above V — while accepting the genuinely true II. 'Neither' comes from rejecting II on the grounds that the non-strict U ≤ V can only yield U ≤ W, never U < W.
Answer: A) Only II follows

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