To obtain the minimum value of z, the first term of the binomial equation should be as small as possible, while the second term should be as large as possible. Thus, you must choose the lowest boundary of x which is 0. Then, choose the highest boundary of y which is 8. So, we substitute this to the equation:
\(z=3x- \frac{1}{2}y\)
\(z=3(0)- \frac{1}{2}(8)=-4\)
So, the minimum value of z is -4.
