The document discusses systems of linear inequalities and includes examples of writing, graphing, and solving systems of linear inequalities. It provides step-by-step workings for graphing two linear inequalities on the same coordinate plane and interpreting their intersection as the solution set to the corresponding system of inequalities. It also gives an example of writing a system of inequalities to model a real-world situation involving costs and time constraints.

1. SECTION 8-7
Systems of Inequalities

2. ESSENTIAL QUESTIONS
• How do you write a system of linear inequalities for a given
graph?
• How do you graph the solution set of a system of linear
inequalities?
• Where you’ll see this:
• Sports, entertainment, retail, finance

3. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

4. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

5. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

6. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

7. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

8. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

9. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

10. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

11. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.
y < 2x + 3 y ≥ −3x + 4

12. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.

13. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.

14. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.

15. EXAMPLE 1
Graph the following linear inequalities on separate
coordinates, then again on the same coordinates.

16. EXAMPLE 2
Write a system of linear inequalities for the graph shown.

17. EXAMPLE 2
Write a system of linear inequalities for the graph shown.
 y ≥ −4x + 6

 1
y < x − 3
 2

18. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.

19. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.
x = DJ, y = band

20. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.
x = DJ, y = band
Time:

21. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.
x = DJ, y = band
Time: x + y ≥ 4

22. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.
x = DJ, y = band
Time: x + y ≥ 4
Money:

23. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.
x = DJ, y = band
Time: x + y ≥ 4
Money: 150x + 300y <1200

24. EXAMPLE 3
Matt Mitarnowski was planning a wedding reception
that will last at least 4 hours. The couple wants to
hire a band that charges $300 per hour and a DJ that
charges $150 per hour. They want to keep the music
expenses under $1200. The band and DJ charge by
the whole hour only.
a. Write a system of linear inequalites that models the
situation.
x = DJ, y = band
x + y ≥ 4
Time: x + y ≥ 4 
150x + 300y <1200
Money: 150x + 300y <1200

25. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200

26. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x+y≥4

27. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x+y≥4
−x −x

28. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x+y≥4
−x −x
y ≥ −x + 4

29. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x+y≥4 150x + 300y <1200
−x −x
y ≥ −x + 4

30. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4

31. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200

32. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300

33. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

34. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

35. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

36. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

37. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

38. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

39. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

40. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

41. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

42. EXAMPLE 3
b. Solve the system by graphing.
x + y ≥ 4

150x + 300y <1200
x + y ≥ 4 150x + 300y <1200
−x −x −150x −150x
y ≥ −x + 4 300y < −150x +1200
300 300
1
y <− x+4
2

43. EXAMPLE 3
b. Solve the system by graphing.

44. EXAMPLE 3
b. Solve the system by graphing.
(1, 3), (2, 2), (2, 3),
(3, 1), (4, 1), (5, 1)

45. HOMEWORK

46. HOMEWORK
p. 364 #1-21 odd, 23-30 all
“Try to learn something about everything and everything
about something.” - Thomas H. Huxley