Let $A$ be an $m$x$n$ matrix, $B$ $l$x$n$ matrix and $c\in R^n$. Prove that exactly one of the systems has a solution:
i)$$Ax\leq0,\:\: Bx=0,\:\: c^Tx>0,\:\: x\in R^n$$ ii)$$A^Tp+B^Tq=c,\:\:p\geq0,\:\:p\in R^m,\:\:q\in R^l$$
It's easy to prove that neither can be true at the same time by simply transposing and multiplying by x ii) and the other way around, but how do I show that exactly one must be true?
The hint which does not make much sense to me is to take $q=r-s$ where $r,s\geq0$.