The objective function is subject to the following constraints:
1. \( 2x - y \geq -5 \)
2. \( 3x + y \geq 3 \)
3. \( 2x - 3y \leq 12 \)
4. \( x \geq 0, y \geq 0 \)
The task is to graph these inequalities to define the feasible region and identify its corner points.
Converting the inequalities to their corresponding equality forms for graphing:
- Constraint 1: \( 2x - y = -5 \), which can be written in slope-intercept form as \( y = 2x + 5 \).
- Constraint 2: \( 3x + y = 3 \), which can be written as \( y = -3x + 3 \).
- Constraint 3: \( 2x - 3y = 12 \), which can be written as \( y = \frac{2x - 12}{3} \).
After plotting these lines and considering the non-negativity constraints (\( x \geq 0, y \geq 0 \)), the feasible region is formed. The corner points of this region are the intersection points of these constraint lines that lie within the feasible area. Solving the systems of equations for these intersections yields the following corner points:
\[(0, 3), (0, 5), (1, 0), (6, 0)\]