Consider the following mixed$-$integer linear program:

Max $1x_1+1x_2$

$s.t.$

$,7x_1+9x_{2}63$

$,9x_1+5x_{2}45$

$,3x_1+1x_{2}12$

$,x_1,x_{2}0$ and $x_2$ integer

a$.$ Graph the constraints for this problem. Indicate on your graph all feasible mixed$-$integer solutions.

b$.$ Find the optimal solution to the LP Relaxation. Round the value of $x_2$ down to find a feasible mixed$-$integer solution. Specify upper and lower bounds on the value of the optimal solution to the mixed$-$integer linear program.

c$.$ Find the optimal solution to the mixed$-$integer linear program.