Exercise 3 Page 255

To graph a system of inequalities, we have to graph each of them separately in the same coordinate plane. If we graph the inequalities in different colors, the area where the shaded regions overlap are the common solutions to the system of inequalities.

Example

Let's consider the following system of inequalities. Here is what we are going to do.

1. Graph the boundary line for each inequality. We will use a dashed line for $>$ and a solid line for $\le .$
2. Test any point that is not on the boundary line to see if it is a solution.
3. If the point is a solution, shade the half-plane that contains it. If it is not, then shade the other half-plane.

We can follow with Step $1.$ We need to remember that one of the lines will be dashed, line $y=\text{-}x+2.$

Let's now take a point and see if it is a solution to the first inequality. Let $(0,0)$ be that point. The point makes the inequality false. As a result, it is not a solution. This means we need to shadow the half-plane that does not contain it. We can now test the same point to see if it is a solution to the second inequality. Notice that we can use point $(0,0)$ because it lies on neither of the lies. Point $(0,0)$ makes the second inequality true. As a result, we will be shadowing the half-plane that contains it.

The area where the shaded regions overlap represents the common solutions to our system of inequalities.