Detailed Solution<\/h3>\r\n <\/div>\r\n\r\n

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Step 1: Graph the Constraints

First, we graph the constraints on the $x$-$y$ plane.\r\n\r\nConstraint 1: $x+y \le 6$. The boundary line is $x+y = 6$. The region satisfying the inequality is below this line.\r\nConstraint 2: $x \ge 2$. The boundary line is $x = 2$. The region satisfying the inequality is to the right of this line.\r\nConstraint 3: $y \le 3$. The boundary line is $y = 3$. The region satisfying the inequality is below this line.\r\nConstraint 4: $x \ge 0$ and $y \ge 0$. This restricts the solution to the first quadrant.

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Step 2: Identify the Feasible Region

The feasible region is the intersection of all the regions defined by the constraints. It is a polygon with vertices at the intersection points of the boundary lines.

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Step 3: Find the Vertices of the Feasible Region

The vertices of the feasible region are the points where the boundary lines intersect. We find these points by solving the equations of the intersecting lines.\r\n\r\nIntersection of $x=2$ and $x+y=6$: $2+y=6 \Rightarrow y=4$. So, the point is $(2,4)$.\r\nIntersection of $x=2$ and $y=3$: The point is $(2,3)$.\r\nIntersection of $y=3$ and $x+y=6$: $x+3=6 \Rightarrow x=3$. So, the point is $(3,3)$.

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Step 4: Evaluate the Objective Function at Each Vertex

We evaluate the objective function $Z = 2x + 3y$ at each vertex of the feasible region:\r\n\r\nAt $(2,3)$: $Z = 2(2) + 3(3) = 4 + 9 = 13$.\r\nAt $(2,4)$: $Z = 2(2) + 3(4) = 4 + 12 = 16$.\r\nAt $(3,3)$: $Z = 2(3) + 3(3) = 6 + 9 = 15$.

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Step 5: Determine the Maximum Value

The maximum value of $Z$ is the largest value obtained in the previous step. In this case, the maximum value is $16$, which occurs at the point $(2,4)$.

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\r\n Final Answer: The maximum value of $Z$ is 16 at $(x, y) = (2, 4)$<\/span>\r\n <\/p>\r\n <\/div>\r\n <\/div>

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