Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set
$$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geq 0, Mx \leqslant b\right\}$$
and
$$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}$$
(adopting the convention: $\inf \emptyset = +\infty$ and $\sup \emptyset = -\infty$).
Show that $\alpha \geqslant \beta$.