Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 33 / 1 / a / Solution

True. Suppose a dual feasible solution $y$ exists. For every primal feasible $x$, weak duality gives $c^{T}x\le b^{T}y$, a finite upper bound independent of $x$. This contradicts primal unboundedness. Therefore an unbounded primal has an infeasible dual. Here “unbounded” arithmetic means a nonempty feasible set with objective values unbounded above, rather than merely an unbounded feasible region.