Find the minimum and maximum values of $z=9 x+8 y$ (if possible) for the following set of constraints. Use graphical methods to solve. \[ \begin{array}{c} 3 x+4 y \geq 12 \\ x+4 y \geq 8 \\ x \geq 0, y \geq 0 \end{array} \] Use the graphing tool to graph the system. Graph the region that represents the correct solutio only once. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The minimum value is $\square$. (Round to the nearest tenth as needed.) B. There is no minimum value. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The maximum value is $\square$. (Round to the nearest tenth as needed.) B. There is no maximum value.

Step 1 ：The constraints are: \(3x + 4y \geq 12\), \(x + 4y \geq 8\), \(x \geq 0\), and \(y \geq 0\).

Step 2 ：Rewrite the first constraint as \(y \geq 3 - \frac{3}{4}x\). This is a line with a negative slope that intersects the y-axis at 3.

Step 3 ：Rewrite the second constraint as \(y \geq 2 - \frac{1}{4}x\). This is also a line with a negative slope, but it intersects the y-axis at 2.

Step 4 ：The third and fourth constraints mean that x and y must both be greater than or equal to 0, which confines our solution to the first quadrant.

Step 5 ：The feasible region is the area that satisfies all these constraints. It is the area above the line \(y \geq 3 - \frac{3}{4}x\) and above the line \(y \geq 2 - \frac{1}{4}x\) in the first quadrant.

Step 6 ：The function to be maximized or minimized is \(z = 9x + 8y\). The maximum and minimum values of z will occur at the vertices of the feasible region.

Step 7 ：The vertices of the feasible region are (0,3), (4,0) and (2,2).

Step 8 ：Substitute these points into the function z: At (0,3), \(z = 9(0) + 8(3) = 24\). At (4,0), \(z = 9(4) + 8(0) = 36\). At (2,2), \(z = 9(2) + 8(2) = 34\).

Step 9 ：So, the minimum value of z is 24 and the maximum value of z is 36.

Step 10 ：Therefore, the answer is: \(\boxed{\text{The minimum value is 24.}}\) and \(\boxed{\text{The maximum value is 36.}}\)