Figure 2 - Test point
Figure 2 - Test point
Next decide whether the solution is half-plane I or half plane II. To do this, choose any point not on the boundary. For example, choose the point (0, 0) in Figure 2 - this choice, if not on the boundary, is usually the best because of the ease of the arithmetic involved. Notice from Figure 2 that the point (0, 0) is in half-plane I. If (0, 0) makes the inequality true, then the solution is the half-plane containing (0, 0); if (0, 0) makes the inequality false, then the solution set is the half-plane not containing (0, 0). Checking by substituting (0, 0) into 2x+3y<=12, you have
2*(0)+3*(0)<=12 true
Therefore the solution set is the area shown as half-plane I. This is the shaded portion of Figure 3.
Figure 3 - Graph of 2x+3y<=12
Notice that the solution set is the closed half-plane I, since it includes the boundary. This is shown on the graph by using a solid line for the boundary. If the boundary is not included (when the inequality symbols are < or >), a dashed line is used to indicate the boundary.