 Solution

False: both primal and dual can be infeasible. For example, takeThe primal requires $0\le -1$, so is infeasible. Its dual requires the first coordinate of $A^{T}y$ to be at least $1$, namely $0\ge 1$, and is also infeasible. An infeasible primal therefore does not force an unbounded dual. If the dual is additionally assumed feasible, strong duality rules out a finite dual optimum; a feasible linear program with finite optimal value attains it, so in that qualified case the dual is unbounded below.