1. Lesson 7.5, page 755 Systems of Inequalities Objective: To graph linear inequalities, systems of inequalities, and solve linear programming problems.

2. Review -- Graphing a Line 1. Put in y = mx + b form. 2. Plot the y-intercept. 3. Use the slope and rise/run to plot at least 2 more points. 4. Draw the line.

3. Review Practice Graphing Lines y x 10 -10 Graph y = 3x + 5. Graph 2x + 3y = 6. y x 10 -10

4. Review Linear Inequality in Two Variables A linear inequality divides the xy-plane into 2 parts. Either the points on one side of the line make the inequality true or points on the other side do. Once the line is determined, select any point on either side to test in the original inequality to determine if that point is a solution or not. If the point makes the inequality true, all points on that side are also in the solution set. If the point makes the inequality false, all points on the other side of the line are in the solution set.

5. Steps for Graphing Inequalities FOR INEQUALITIES: dashed line > or < solid line Write the inequality as an equation. Put in slope-intercept form and graph the line, dashed or solid. Test a point, not on the line to see if it makes a true statement. If (0, 0) is not on the line, use it. It is the easiest point to test. If true, shade on the side of the line that contains the test point. If false, shade on the side of the line that does not contain the test point.

6. Example Graph y > x  4. We begin by graphing the related equation y = x  4. We use a dashed line because the inequality symbol is >. This indicates that the points on the line itself are not in the solution set. Determine which half-plane satisfies the inequality and shade. Test point (0,0) y > x  4 0 ? 0  4 0 >  4 True y x 10 -10

7. Example Graph: 4x + 2y > 8 1. Graph the related equation, using a solid line. 2. Determine which half-plane to shade. 4x + 2y > 8 4(0) + 2(0) >? 8 0 > 8 is false. We shade the region not containing (0, 0). y x 10 -10

8. Check Points 1 & 2 4x - 2y > 8 y > -3x 4 y x 10 -10 y x 10 -10

9. Example 3, page 758 Check Point 3 y > 1 x < -2 y x 10 -10 y x 10 -10

10. To Graph a Linear Inequality: A Recap Graph the related equation. If the inequality symbol is, draw the line dashed. If the inequality symbol is  or , draw the line solid. Because an inequality has many possible solutions, the graph consists of a half-plane on one side of the line and, if the line is solid, the line as well. To determine which half-plane to shade, test a point not on the line in the original inequality. If that point is a solution, shade the half-plane containing that point. If not, shade the opposite half-plane.

11. Nonlinear Inequalities Consider the graph of a circle. The plan is divided into the area inside the circle, and that outside the circle. Solve it as you would a linear system. Consider the graph of other nonlinear inequalities (parabolas, ellipses, hyperbolas). Again, the graph would show that the points that make the inequality true would be found inside OR outside of the graph.

12. Graphing a Non-Linear Inequality See Example 4, page 759 Check Point 4 x 2 + y 2 > 16 y x 10 -10

13. System of Linear Inequalities If the 2 lines intersect at one point, the plane is divided into 4 areas. The solution could be found in one of these areas. Often graphing and looking for overlapping areas is easier than looking at points in each region.

14. Steps for Graphing a System of Inequalities 1. Graph each inequality and indicate which part should be shaded. 2. Shade the area which is common to all graphs or the area where the shading overlaps. 3. Pick any point in the commonly shaded area and check it in all inequalities.

15. Systems of Linear Inequalities Graph the solution set of the system. First, we graph x + y  3 using a solid line. Choose a test point (0, 0) and shade the correct plane. Next, we graph x  y > 1 using a dashed line. Choose a test point and shade the correct plane. The solution set of the system of equations is the region shaded both red and green, including part of the line x + y  3.

16. Check Point 5, page 759 x – 3y < 6 2x + 3y > -6 y x 10 -10

17. Check Point 6, page 761 y > x 2 – 4 x + y < 2 y x 10 -10

18. See Example 7, page 762. Check Point 7: Graph the solution set of the system: y x 10 -10