To graph the feasible region for this system of inequalities, we need to plot each of the inequalities and identify where all of them are true, which will be the feasible region. Here are the steps:
1. Plot the inequality x ≥ 0: This is the region to the right of the y-axis (including the y-axis). It is also know as x-axis and half of the plane right to it.
2. Plot the inequality y ≥ 0: This is the region above the x-axis (including the x-axis). It is x-axis and half of the plane above it.
For these two steps, your graph will be bounded by the x and y axis and will be the first quadrant.
3. Plot the inequality x + y ≤ 10: This inequality can be written as y ≤ 10 - x. To graph the inequality, you would first graph the line y = 10 - x. This is a line that intercepts the y-axis at 10 and goes downward, crossing the x-axis at 10 as well. The inequality y ≤ 10 - x represents all the points below this line and on the line itself.
4. Lastly, plot the inequality 2x + 3y ≤ 30: This inequality can be written as y ≤ (30 - 2x) / 3. To graph the equality, you would graph the line y = (30 - 2x) / 3 which intercepts the y-axis at 10 and crosses the x-axis at 15. The inequality represents the points below this line.
Your feasible region is where all these shaded regions overlap, which is the region below and including both lines (y = 10 - x and y = (30 - 2x)/3) and above the x and y-axis in the first quadrant.
Please note that without a diagram, it's a bit tough to show the final resulting area, but hopefully the text description gives you the understanding of how to plot this.
Also, according to the specific characteristics of the inequalities, the feasible region is a polygon with vertices at the points where the lines intersect. In this case, these points are (0,0), (0,10), (10,0) and the intersection of the lines y = 10 - x and y = (30 - 2x) / 3. By solving the equations together, you will get this intersection point as (6,4).
So these four points: (0,0), (0,10), (10,0), (6,4) are the corners of the polygon that represents the feasible region for this set of inequalities.