1. Math 20: Foundations FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. E. Above or Below the Line

2. Remembering Inequalities

3. What DO YOU Think? p. 293

4. 1. Graphing a Linear Inequality FM20.8 Demonstrate understanding of systems of linear inequalities in two variables.

5. 1. Graphing a Linear Inequality

8. What we do is use test points to determine which side of the line we are concerned about. Select a point on one side of the line and sub it into the inequality if the inequality is solved that is the side of the line we shade

10. The area we shade is called the Solution Region When we graph a line the Cartesian plane is cut into tow halves called Half Planes A Continuous line is a line that contains real numbers. All numbers are included so the line is solid (continuous).

11. Example 1

13. Example 2

14. x= results in a … Vertical Line y= results in a … Horizontal Line

15. Example 3

16. Summary p.302

19. Practice Ex. 6.1 (p.303) #1-12 #4-12

20. 2. Systems of Linear Inequalities FM20.8 Demonstrate understanding of systems of linear inequalities in two variables.

21. 2. Systems of Linear Inequalities

22. What combinations of morning and full-day students can the school accommodate and stay within the weekly snack budget?

23. Summary p.307

25. Practice Ex. 6.2 (p.307) #1-2

26. 3. Using the Graphs to Solve Problems FM20.8 Demonstrate understanding of systems of linear inequalities in two variables.

27. 3. Using the Graphs to Solve Problems What combinations of boats should the company make each day?

28. Is every point in the region a possible solution to the problem? How would the graph change if fewer than 25 boats were made each day? All whole points with whole number coordinates in the solution region are valid, but are they all reasonable?

29. Example 1

30. Example 2

31. Summary p.317

33. Practice Ex. 6.3 (p.317) #1-4 odds in each, 5-10 #4-12

34. 4. Creating Optimization Problems FM20.8 Demonstrate understanding of systems of linear inequalities in two variables.

35. 4. Creating Optimization Problems Optimization Problem – A problem where a quantity must be maximized or minimized following a set of guidelines or conditions. Constraint – A limiting condition of the optimization problem being modeled, represented by a linear inequality. Objective Function – In an optimization problem, the equation that represents the relationship between the two variables in the system of linear inequalities and the quantity to be optimized. Feasible Region – The solution region for a system of linear inequalities that is modeling an optimization problem.

36. Investigate the Math p.324

37. Example 1

38. Example 2

39. Summary p.329

41. Practice Ex. 6.4 (p.330) #1-8 #2-9

42. 5. Optimization Solutions FM20.8 Demonstrate understanding of systems of linear inequalities in two variables.

43. Explore the Math p.332 This is the same Problem from the beginning of last day dealing with the toy cars. So we don’t have to set it up an graph it again we can use our results from last day.

45. Summary p.333

46. Practice Ex. 6.5 (p.334) #1-3

47. 6. Determine the Max and Min FM20.8 Demonstrate understanding of systems of linear inequalities in two variables.

48. 6. Determine the Max and Min The process that we have been developing and will finish today is called linear programming Linear Programming is used to find the solution in the feasible region result in the optimal solutions of the objective functions

50. What happens if the optimal points (intersection points of constraints) do not land on a whole number coordinate? We solve the two inequalities like they where linear equations using the substitution or elimination strategies. This gives us the point where the two lines intersect.

51. Example 1

52. Summary p.341

54. Practice Ex. 6.6 (p.341) #1-15 #3-15,17