Final answer:
To graph the solution set of the given system of inequalities, plot each inequality on a graph and shade the overlapping region. The solution set is determined to be unbounded, and the exact coordinates of the vertices are not provided here but can be found using algebraic methods or graphing tools.
Step-by-step explanation:
The task is to graph the solution set of the following system of inequalities and find the coordinates of all vertices to determine if the solution set is bounded or unbounded:
- x ≥ 4
- x + y ≥ 24
- x ≤ 2y + 12
Step 1: Plot each inequality on a graph.
- The inequality x ≥ 4 is a vertical line passing through (4, y) where y is any real number.
- The inequality x + y ≥ 24 forms a line when we rearrange it to y = -x + 24. This is a line with a negative slope and a y-intercept at (0, 24).
- The inequality x ≤ 2y + 12 can be rearranged to y = ½x - 6. This is a line with a positive slope and a y-intercept at (0, -6).
Step 2: Shade the region that satisfies all three inequalities. This is the intersection where the solutions of all three overlap.
To find the vertices, we look for intersection points of the lines. We notice that these lines do not form a closed region, indicating the solution set is unbounded (not confined within a certain area).