Consider the following LP model:

max $30$x$+40$x$$

st

$5$x$+4$x$<=200$

$3$x$+5$x$<=150$

$5$x$+4$x$>=100$

$8$x$+4$x$>=80$

x$>=0,$ x$>=0$

$($a$)$ Solve the problem with the graphical method $($i$.$e$.,$ find all CPF $($corner$-$point feasible$),$ find the optimal solution, and the value of Z at the optimal solution graphically$).$

$($b$)$ Is the feasible region bounded or unbounded? Does the answer change if the first constraint is removed?

$($c$)$ Now suppose the objective function is changed to Z $=8$ x$+4$ x$.$ Discuss how this may impact optimal solution and feasibility.