For a rectangle $\mathcal R = \{ x \in R^n \mid l \preceq x \preceq u \}$ of maximum volume to be enclosed in polyhedron $\mathcal P = \{ x \mid Ax \preceq b \}$, according to Stephen Boyd's EE364a Homework 7 solutions, we simply need
$$\sum\limits_{j=1}^n(a_{ij}^+ u_j - a_{ij}^- l_j)\leq b_i, \quad i=1, \dots, m$$
where $a_{ij}^+ = \max(a_{ij}, 0)$ and $a_{ij}^- = \max(-a_{ij}, 0)$. How can I see this fact?