2. Lesson 3.3 Systems of Inequalities Objective: To graph linear inequalities, systems of inequalities, and solve linear programming problems.

3. 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.

4. Warm - Up Graphing Lines y x 10 -10 Graph y = 3x + 5. Graph 2x + 3y = 6. y x 10 -10

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. Example y > 1 x < -2 y x 10 -10 y x 10 -10

9. 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.

10. 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.

11. 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.

12. Graphing Method Example: Graph the inequalities on the same plane: x + y 4. Before we graph them simultaneously, let’s look at them separately. Graph of x + y

13. Graphing Method Now lets graph : 2x - y > 4.

14. Coolness Discovered! Wow! The solution to the system is the hashtag region - where the two shaded areas coincide.

15. So what were the steps? Graph first inequality Shade lightly (or use colored pencils) Graph second inequality Shade lightly (or use colored pencils) Shade darkly over the common region of intersection. That is your solution!

16. 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.

17. Practice Problem 4x+ 2y > 8 y < -2x-3 y x 10 -10

18. Practice Problem y > x – 4 x + y < 2 y x 10 -10

19. What about THREE inequalities? Graph x ≥ 0, y ≥ 0, and 4x + 3y ≤ 24 First off, let’s look at x ≥ 0 and y ≥ 0 separately.

20. Graphing THREE inequalities Now let’s look at x ≥ 0 and y ≥ 0 together. Clearly, the solution set is the first quadrant.

21. Graphing THREE inequalities So therefore, after we graph the third inequality, we know the solution region will be trapped inside the first quadrant. So let’s look at 4x + 3y ≤ 24 by itself.

22. Graphing THREE inequalities Now we can put all of our knowledge together. The solution region is the right triangle in the first quadrant.

23. Homework Section 3.3 - pg.171-173  #1-3, 8-12, 16, 21, 25, 29, 31, 34, 35, 36