Exercise 4 Page 255

When we graph one inequality, the shaded region is the solution set. Therefore, when graphing two inequalities in one coordinate plane, the region where both of the shadings overlap is the solution to the system of inequalities, because this is where both of them are satisfied.

Example

Let's consider the following system of inequalities. y ≤ - 0.5 x+1 y < 2x-1 We want to graph it. First, we are going to draw the boundary lines for both inequalities.

Let's now take a point that does not lie on either of the lines and see if it is a solution. If it is a solution to the inequality, we will shadow the half-plane that contains it. If it is not, we will shadow the other half-plane. Let ( 0, 0) be that point.

Point (0,0) makes only the first inequality true. As a result, the solution set to the first inequality the half-plane that contains it. Conversely, the solution set to the second inequality the half-plane that does not contain it.

The area where the shaded regions overlap represents the common solutions to our system of inequalities. As a result, any point in this region is a solution. For example, point (2,-2) is a solution.