1. Solving Systems of Equations Graphing Linear Inequalities

2. Objectives How do we graph an inequality Define a boundary line Graphing a boundary line Define the solution for a system of inequalities Find the solution of a system of inequalities

3. What is the solution of an inequality Solution of an inequality are all the ordered pairs (points) that make the inequality true.

4. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 y = xGraph Boundary line REMEMBER: Solution are all the ordered pairs (points) that make the inequality true.

5. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 1.Pick two points from each side of the graph (4,1) (1,3)

6. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 2.Check points if they make inequality true. (1,3)y ≥ x substitute into

7. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 2.Check points if they make inequality true. (1,3)(1,3)y ≥ x substitute into 3 ≥ 1

8. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 2.Check points if they make inequality true. (1,3)(1,3)y ≥ x substitute into 3 ≥ 1  

9. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 2.Check points if they make inequality true. (4,1)y ≥ x substitute into 

10. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 2.Check points if they make inequality true. (4,1)(4,1)y ≥ x substitute into  1 ≥ 4

11. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 2.Check points if they make inequality true. (4,1)(4,1)y ≥ x substitute into  1 ≥ 4X X

12. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (4,1) (1,3) 3.Shade the side where the correct point lies.  X

13. Graphing Inequalities Consider the inequality y ≥ x 6 4 2 2 14365 1 3 5 (1,3) 3.Shade the side where the correct point lies. 

14. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 x - 2y = 4Graph y = x - 2 1 2

15. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 x - 2y = 4Graph ¡¡TEST POINTS !! (0,1) (6,0) y = x - 2 1 2

16. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 (0,1) (6,0) (0,1) substitute into x - 2y ≤ 4

17. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 (0,1) (6,0) (0,1)(0,1) substitute into x - 2y ≤ 4 0 - 2(1) ≤ 4 -2 ≤ 4  

18. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 (0,1) (6,0) substitute into x - 2y ≤ 4 

19. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 (0,1) (6,0) (6,0)(6,0) substitute into x - 2y ≤ 4 6 - 2(0) ≤ 4 6 ≤ 4 X  X

20. Graphing Inequalities Consider the inequality x - 2y ≤ 4 3 1 2 14365 -2 2 (0,1) (6,0)  X ¡¡ SHADE CORRECT REGION !!

21. Examples 6 4 2 2 14365 1 3 5 3y - 2x ≥ 9 1. y = x + 3 2 3 GRAPH

22. Examples 6 4 2 2 14365 1 3 5 3y - 2x ≥ 9 1. y = x + 3 2 3 GRAPH TEST!! (0, 5) (3,0)  X 3(5) - 2(0) ≥ 9 15 - 0 ≥ 9 

23. Examples 2. x - 3y > -3 y = x + 1 1 3 6 4 2 2 14365 1 3 5 Graph TEST!! (0, 5) 0 - 3(5) > -3 0 - 15 > -3 X X

24. Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 3 1 2 13 2 -2-3

25. Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 3 1 2 13 2 -2-3 TEST: (0,0) Graph 0 + 0 ≥ -1 0 ≥ -1  (0,0) y = - x - 1

26. Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 3 1 2 13 2 -2-3 Graph TEST: (0,0) -2(0) + 0 < 2 0 < 2  y = 2x + 2 (0,0)

27. 3 1 2 13 2 -2-3 3 1 2 13 2 -2-3 x + y ≥ -1-2x + y < 2

28. Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 3 1 2 13 2 -2-3 SOLUTION: Lies where the two shaded regions intersect each other.

29. Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 1231 2 -2 Graph y = x - 2 2 3 324-21 TEST: (0,0) (0,0) -2(0) + 3(0) < -6 0 < -6X X

30. Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 1231 2 -2 Graph y = - x + 3 5 4 324-21 TEST: (0,0) 5(0) + 4(0) < 12 0 < 12 (0,0)  

31. Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 1231 2 -2 Graph 324-21 (0,0)  SOLUTION: Lies where the two shaded regions intersect each other.

32. Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 1231 2 -2 Graph 324-21 (0,0)  NOTE: All order pairs in dark region are true in both inequalities.

33. Solving a system of Inequalities Consider the system x - 4y ≤ 12 4y + x ≤ 12 (0,0) 10812642 -2 6 2 4 -4 -6 TEST: (0,0) (0) - 4(0) ≤ 12 0 - 0 ≤ 12 0 ≤ 12  Graph

34. Solving a system of Inequalities Consider the system x - 4y ≤ 12 4y + x ≤ 12 (0,0) 10812642 -2 6 2 4 -4 -6 TEST: (0,0) 4(0) + (0) ≤ 12 0 ≤ 12  Graph