Find the coordinates of the vertices of the figures formed by:

y<=x+2, x+y <=6, and y>= -2

A. (0,0), (2,4), (8,-2)
B. (-4,-2), (2,4), (8,-2)
C. (-4,-2), (4,2), (8,-2)

I have substituted all of the above in the equations and they all are true. I'm confused.

Thanks.

To find the vertices of the figures formed by the given inequalities, you can start by graphing each of the inequalities individually and then identifying the regions where they overlap. The overlapping regions will represent the solution to the system of inequalities.

Let's start with the first inequality, y <= x + 2. To graph this, you can start by considering the boundary line y = x + 2. This line has a slope of 1 and a y-intercept of 2. You can consider two points on this line, for example, when x = 0, y = 2, and when x = -2, y = 0. Plotting these points and drawing a line through them will give you the boundary line.

Now, to determine which side of the line represents the solution to the inequality, you can choose a test point that is not on the boundary line. For example, let's use the origin (0,0). Substitute the coordinates of this point into the inequality y <= x + 2: 0 <= 0 + 2. Since this inequality is true, you should shade the region below the boundary line.

Moving on to the second inequality, x + y <= 6. Similarly, you can consider the boundary line x + y = 6. To graph this line, set x to 0, y will be 6, and when y is 0, x will be 6.

Once again, choose a test point not on the boundary line, such as the origin (0,0), and substitute it into the inequality. 0 + 0 <= 6 is true, so the region below the line is shaded.

Lastly, for the third inequality, y >= -2, you need to graph the boundary line y = -2. This is a horizontal line passing through the point (0, -2). Since it's a "greater than or equal to" inequality, shade the region above the boundary line.

Now, you need to find the overlapping region of these shaded regions from all three inequalities. The vertices of this region will be the coordinates of the figures formed.

Looking at the options given, we can see that option A has the point (8,-2) as one of the vertices, so let's see if this option satisfies the other two points as well.

Substitute the coordinates of the point (2,4) into the three inequality equations:
For y <= x + 2, 4 <= 2 + 2, which is true.
For x + y <= 6, 2 + 4 <= 6, which is also true.
For y >= -2, 4 >= -2, which is true.

Therefore, option A, (0,0), (2,4), (8,-2), is the correct answer.