\be
	\item Use the simplex algorithm to solve each of the following LPs.
		\be
		\item \begin{align*}
		 \mbox{maximize } z &=3x_1+2x_2  \\
		\mbox{ subject to} \notag\\
		x_1 \phantom{ +20x_2} &\leq 4 \notag\\
		x_1+3x_2 &\leq 15  \notag\\
		2x_1+x_2 &\leq 10 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        \item \begin{align*}
		 \mbox{maximize } z &=4x_1+3x_2  \\
		\mbox{ subject to} \notag\\
		x_1 \phantom{ +20x_2} &\leq 4 \notag\\
		-2x_1+x_2 &\leq 12  \notag\\
		x_1+2x_2 &\leq 14 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        \item
		\begin{align*}
		 \mbox{maximize } z &=4x_1 + x_2 +5x_3  \\
		\mbox{ subject to} \notag\\
		2x_1+x_2+3x_3 &\leq 14 \notag\\
		6x_1+3x_2 +3x_3 &\leq 22  \notag\\
		2x_1+3x_2  \x &\leq 14 \notag \\
		x_1,x_2,x_3 &\geq 0\notag
		\end{align*}
		\item \begin{align*}
		 \mbox{minimize } z &=3x_1- 2x_2   \\
		\mbox{ subject to} \notag\\
		x_1-2x_2 &\leq 2  \notag\\
		x_1+x_2&\leq 4 \notag\\
		x_1,x_2&\geq 0\notag
		\end{align*}	
		\ee
	\item Try to use the simplex algorithm to solve the minimization LP
	\begin{align*}
		 \mbox{minimize } z &=-x_1+4x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +3x_2 &\geq 8 \notag\\
		x_1+x_2 &\geq 4  \notag\\
		x_1-x_2 &\leq 2 \notag \\
		x_1,x_2 &\geq 0.\notag
		\end{align*}
		What happens? Specify the attribute of this problem that makes solving it more challenging than solving the
	minimization problem in Exercise 1.
	\item A constraint is said to be \emph{binding}\index{binding constraint} at the optimal solution of an LP if there
	is equality in that constraint in the optimal solution. For the first two LPs in Exercise 1, determine
	which constraints are binding.
\ee

