1. 6.6 Solving System of Linear Inequalities: System of Linear Inequalities: Two or more linear inequality equations. Solution of a System of Linear Equations: Any ordered pair that makes all the equations in a system true.

2. Always Remember:

3. GOAL:

4. Y> x + 1 Y< x + 1 > : Dashed line and shade above < : Dashed line and shade below

5. Y≤ x + 1 Y ≥ x + 1 ≤ : Solid line and Shade below ≥ : Solid line and shade above

6. Y≤ x + 1 Y> x + 1 Y< x + 1 Y ≥ x + 1

7. SOLVING A SYSTEM BY GRAPHING: To solve a system of inequalities we must: 1) Write the equations in slope-intercept form (y □ mx+b) 2) Graph the equations and shade 3) Find an ordered pair point inside the shaded intersection region 4) Check

8. Ex: What is the solution of the system? Use a graph to check your answer. http://www.meta-calculator.com/online/

9. SOLUTION: 1) Write the equations in slope-intercept form (y□mx+b)

10. SOLUTION:2) Graph the equations Dashed line and shade down Solid line and shade up

11. SOLUTION:2) In the same graph:

12. SOLUTION:3) Find the solution Looking at the graph, we see that any point in the double shaded region will be a solution, say: (0, 3)

13. SOLUTION:4) Check We know that (0,3) is the solution from our graph.

14. YOU TRY IT: What is the solution of the system? Use a graph to check your answer.

15. SOLUTION: 1) Write the equations in slope-intercept form (y=mx+b)

16. SOLUTION:2) Graph the equations Dashed line and shade down

17. SOLUTION:2) In the same graph:

18. SOLUTION:3) Find the solution (2,-1) Looking at the graph, we see that any point in the double shaded region will be a solution, say:

19. SOLUTION:4) Check We know that (2,-1) is the solution from our graph.

20. Real-World: You are planning what to do after school. You can spend at most 6 hrs. daily playing the trumpet and doing homework. You want to spend less than 2 hrs. playing the trumpet. You must spend at least 1.5 hrs. on homework. What is a graph showing how much you can spend your time?

21. Real-World(SOLUTION): You are planning what to do after school. You can spend at most 6 hrs. daily playing the trumpet and doing homework. You want to spend less than 2 hrs. playing the trumpet. You must spend at least 1.5 hrs. on homework. What is a graph showing how much you can spend your time? At most 6 hrs  x + y ≤ 6 Less than 2 hrs trumpet  x < 2 hrs At least 1.5 hrs homework  y ≥1.5 hrs

22. SOLUTION: x + y ≤ 6  y ≤ - x + 6 1 23 4 1 2 3 4 5 6 7 8 9 10 5 6

23. SOLUTION: x < 2 1 23 4 1 2 3 4 5 6 7 8 9 10 5 6

24. SOLUTION: Y ≥1.5 1 23 4 1 2 3 4 5 6 7 8 9 10 5 6

25. SOLUTION: y ≤ - x + 6 123 4 1 2 3 4 5 6 7 8 9 10 5 6 x < 2 Y ≥1.5 All together in the same graph:

26. SOLUTION: y ≤ - x + 6 1 23 4 1 2 3 4 5 6 7 8 9 10 5 6 x < 2 Y ≥1.5 Any point in the red shaded area are solutions to the problem

27. YOU TRY IT: You have a job mowing the lawn for $10 per hour. You also have another job playing for parties and you charge $12 per hour. You need to earn at least $350 to buy a new instrument but you cannot work more than 35 hrs per week. You must work a minimum of 10 hrs playing at parties. What is a graph showing how many hours per week you can work at each job?

28. YOU TRY IT: (SOLUTION) mowing the lawn  earn at least $350  X playing at parties  y no more than 35 hrs per week  10x + 12y ≥ 350 minimum of 10 hrs playing  x + y ≤ 35 y ≥ 10 Given info:

29. SOLUTION: 10 20 3040 10 20 30 40 50 60 50 6 10x + 12y ≥ 350 

30. SOLUTION: 10 20 3040 10 20 30 40 50 60 50 6 x + y ≤ 35  y ≤ - x + 35

31. SOLUTION: 10 20 3040 10 20 30 40 50 60 50 6 y ≥ 10

32. SOLUTION: All in one graph 10 20 3040 10 20 30 40 50 60 50 6 y ≥ 10 y ≤ - x + 35

33. SOLUTION: All in one graph 10 20 3040 10 20 30 40 50 60 50 6 y ≤ - x + 35 y ≥ 10 Any point inside the red region is a solution

34. VIDEOS: Graphing Inequalities https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/graphing-linear- inequalities/v/solving-and-graphing-linear- inequalities-in-two-variables-1 https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/graphing-linear- inequalities/v/graphing-inequalities

35. CLASSWORK: Page 398-401 Problems: As many as needed to master the concept.