Consider the optimization problem

We can convert this problem into an equivalent standard form linear programming problem

by introducing the new variables

x i $=$ x$+$i $$ x$$i where x$+$i $>=0,$ x$$i $>=0,$ i $=1,2,...,$ n

and

$|$ x i $|=$ x$+$i $+$ x$$i $,$ i $=1,2,...,$ n

Then we can apply the simplex method to solve the equivalent problem. Explain, in two or

three sentences, why we will always have that only either x$+$i or x$$i can be positive but never

both x$+$i and x$$i can be positive. In other words, we will always have, x$+$ix$$i$=0.$