Can the following linear program have a negative optimal solution?

Question

Can the following linear program have a negative optimal solution? $$\max \ x_{m+1}$$ $$a_{i}^{T}x+x_{m+1}\leq b_{i} \hspace{5mm},i=1,\dots,n$$ $$x=(x_{1},\dots,x_m)^{T}\in \mathbb{R}^m,x_{m+1}\in \mathbb{R}$$

Answer 1

For example, for $m=1$ and \begin{eqnarray} x+x_2&\le&-1,\\ -x+x_2&\le&-1 \end{eqnarray} will make $x_2\le -1-|x|$ with the optimal value $\max x_2=-1$.