Find the graphical solution of each inequality. $$ 4 x-8<0 $$

Consider the linear programming problem $$ \begin{array}{rr} \text { Maximize } & P=2 x+7 y \\ \text { subject to } & 2 x+y \geq 8 \\ & x+y \geq 6 \\ x & \geq 0, y \geq 0 \end{array} $$ a. Sketch the feasible set \(S\). b. Find the corner points of \(S\). c. Find the values of \(P\) at the corner points of \(S\) found in part (b). d. Show that the linear programming problem has no (optimal) solution. Does this contradict Theorem 1 ?

Solve each linear programming problem by the method of corners. $$ \begin{array}{ll} \text { Minimize } & C=10 x+15 y \\ \text { subject to } & x+y \leq 10 \\ 3 x+y & \geq 12 \\ -2 x+3 y & \geq 3 \\ x & \geq 0, y \geq 0 \end{array} $$

MANUFACTURING-PRODUCTION SCHEDULING Ace Novelty manufactures "Giant Pandas" and "Saint Bernards." Each Panda requires \(1.5 \mathrm{yd}^{2}\) of plush, $30 \mathrm{ft}^{3}$ of stuffing, and pieces of trim; each Saint Bernard requires \(2 \mathrm{yd}^{2}\) of plush, \(35 \mathrm{ft}^{3}\) of stuffing, and 8 pieces of trim. The profit for each Panda is \(\$ 10\) and the profit for each Saint Bernard is. \(\$ 15\). If \(3600 \mathrm{yd}^{2}\) of plush, $66,000 \mathrm{ft}^{3}$ of stuffing and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize profit?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} x+y \leq 6 \\ 0 \leq x \leq 3 \\ y \geq 0 \end{array} $$

Find the graphical solution of each inequality. $$ 3 y+2>0 $$