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 goal is to graph these inequalities to define the feasible region and subsequently identify its corner points.
To facilitate graphing, the constraint equations are converted to slope-intercept form:- Constraint 1: \( 2x - y = -5 \) becomes \( y = 2x + 5 \).
- Constraint 2: \( 3x + y = 3 \) becomes \( y = -3x + 3 \).
- Constraint 3: \( 2x - 3y = 12 \) becomes \( y = \frac{2x - 12}{3} \).
By plotting these lines and considering the non-negativity constraints \( x \geq 0 \) and \( y \geq 0 \), the feasible region is determined. The corner points of this region, which represent the vertices formed by the intersections of the constraint lines, are found by solving the systems of equations and verifying that the solutions lie within the feasible region. The identified corner points are:
\[(0, 3), (0, 5), (1, 0), (6, 0)\]