1. Bell Work… Graph the following equations. 1. 2. 6 4 2 1 3 5 6 4 2 1 3

2. Solving Systems of Equations
Graphing Linear Inequalities

3. 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

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

5. Graphing Inequalities
Consider the inequality
y ≥ x
y = x
Graph
REMEMBER: Solution are all
the ordered pairs (points) that
make the inequality true. 6
Before we start graphing inequalities I wanted to make sure which line we graphing and what this line is formally called. Their book calls it a boundary line, so I wanted to make sure that they receive this information. I didn’t focus to much on graphing the line as I was confident that they knew how to graph lines manually from the previous lessons in the unit. (NEXT) 5 4 3
Boundary line 2 1 1 2 3 4 5 6

6. Graphing Inequalities
Consider the inequality
y ≥ x
Pick two points from each
side of the graph 6
It is at this slide where I truly begin to give directions on how to graph an inequality. Notice I tell them to choose two points, one on each side of the boundary line. I wanted them to see that by picking two points, only one will satisfy the system. Once they started to understand this concept I allowed them to only test one of the points and shade the correct side depending whether the point satisfies the system. (NEXT) 5 4
(1,3) 3 2
(4,1) 1 1 2 3 4 5 6

7. Graphing Inequalities
Consider the inequality
y ≥ x
Check points if they make
inequality true.
(1,3)
y ≥ x 6
substitute into
I begin to demonstrate the steps of substitution of the point into the inequality. 5 4
(1,3) 3 2
(4,1) 1 1 2 3 4 5 6

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

9. Graphing Inequalities
Consider the inequality
y ≥ x
Check points if they make
inequality true.
(1,3)
y ≥ x 6
substitute into
The point checks out. But in the next slide we test the other, to demonstrate that only one side would satisfy the inequality.
3 ≥ 1  5 4 
(1,3) 3 2
(4,1) 1 1 2 3 4 5 6

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

11. Graphing Inequalities
Consider the inequality
y ≥ x
Check points if they make
inequality true.
(4,1)
y ≥ x 6
substitute into
This point does not satisfy the inequality. So we shade the side containing point (1,3)
1 ≥ 4 5 4 
(1,3) 3 2
(4,1) 1 1 2 3 4 5 6

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

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

14. Graphing Inequalities
Consider the inequality
y ≥ x
Shade the side where the correct
point lies. 6
Click on the “Slide Show” to see the full demonstration on how I wanted to graph an inequality.
On the following slide I give students another example on how to graph an inequality. 5 4 
(1,3) 3 2 1 1 2 3 4 5 6

15. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
x - 2y = 4
Graph
y = x - 2 1 2 3 2 1 1 2 3 4 5 6 -1 -2

16. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
x - 2y = 4
Graph
y = x - 2 1 2
¡¡TEST POINTS !! 3 2 1
(0,1)
(6,0) 1 2 3 4 5 6 -1 -2

17. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
(0,1)
x - 2y ≤ 4
substitute into 3 2 1
(0,1)
(6,0) 1 2 3 4 5 6 -1 -2

18. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
(0,1)
x - 2y ≤ 4
substitute into
0 - 2(1) ≤ 4 
-2 ≤ 4 3 2  1
(0,1)
(6,0) 1 2 3 4 5 6 -1 -2

19. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
(6,0)
x - 2y ≤ 4
substitute into 3 2  1
(0,1)
(6,0) 1 2 3 4 5 6 -1 -2

20. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
(6,0)
x - 2y ≤ 4
substitute into
6 - 2(0) ≤ 4 X
6 ≤ 4 3 2  1
(0,1)
(6,0) X 1 2 3 4 5 6 -1 -2

21. Graphing Inequalities
Consider the inequality
x - 2y ≤ 4
¡¡ SHADE CORRECT REGION !! 3 2  1
(0,1)
(6,0) X 1 2 3 4 5 6 -1 -2

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

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

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

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

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

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

28. 3 1 2 -1 -2 -3 3 1 2 -1 -2 -3
x + y ≥ -1
-2x + y < 2

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

30. Solving a system of Inequalities
Consider the system
-2x + 3y < -6
Graph 3 1 2
5x + 4y < 12
y = x - 2 2 3 X
(0,0)
TEST: (0,0) 3 2 4 -1 -2 1
-2(0) + 3(0) < -6 -1
0 < -6 X -2

31. Solving a system of Inequalities
Consider the system
-2x + 3y < -6 3 1 2
5x + 4y < 12
Graph
y = x + 3 5 4 
(0,0)
TEST: (0,0) 3 2 4 -1 -2 1
5(0) + 4(0) < 12 -1 
0 < 12 -2

32. Solving a system of Inequalities
Consider the system
-2x + 3y < -6 3 1 2
5x + 4y < 12
Graph
SOLUTION: 
(0,0)
Lies where the two shaded
regions intersect each
other. 3 2 4 -1 -2 1 -1 -2

33. Solving a system of Inequalities
Consider the system
-2x + 3y < -6 3 1 2
5x + 4y < 12
Graph
NOTE: 
(0,0)
All order pairs in dark
region are true in both
inequalities. 3 2 4 -1 -2 1 -1 -2

34. Solving a system of Inequalities
Consider the system 10 8 12 6 4 2 -2 -4 -6
x - 4y ≤ 12
Graph
4y + x ≤ 12
TEST: (0,0)
(0,0)
(0) - 4(0) ≤ 12
0 - 0 ≤ 12 
0 ≤ 12

35. Solving a system of Inequalities
Consider the system 10 8 12 6 4 2 -2 -4 -6
x - 4y ≤ 12
4y + x ≤ 12
Graph
TEST: (0,0)
(0,0)
4(0) + (0) ≤ 12 
0 ≤ 12

36. HOMEWORK…
Finish pg. 289 #8-16 (solve the system of inequalities by graphing)
#19 and 20.

37. Problem Model
Patricio’s family, on average, drives their SUV more than twice as many miles as they drive their car. His family’s car emits 0.75 pounds of CO2 per mile and the SUV emits 1.25 pounds of CO2 per mile. Patricio is concern with the environment and convinces his family to limit the total CO2 emissions to less than 600 pounds per month. How many miles can they drive their car and SUV to meet this limit? x
= SUV miles x
> 2y y
= Car miles
0.75y +
1.25x
<
600

38. Problem Model
> 2y x
0.75y
1.25y
600
< +

39. Problem Model
The science club can spend at most $400 on a field trip to a dinosaur exhibit. It has enough chaperones to allow at most 100 students to go on the trip. The exhibit costs $3.00 for students 12 and under and $6.00 for students 12 and over. How many students 12 years and under can go if 20 students over 12 go? x
= Students 12 and under x + y ≤
100 y
= Students 12 and over 3x + 4y ≤
400

40. Problem Model x + y ≤
100 3x + 4y ≤
400

41. Now you try…
The Math Club want to advertise their fundraiser each week in the school paper. They know that a front-page ad is more effective than an ad inside the paper. They have a total of $30 budget for advertising. It costs $2 for each front-page ad and $1 for each inside-page ad. If the club wants to advertise at least 20 times, what are the different possibilities for the number of front-page and inside-page ads. x
= front-page ads x + y ≤ 20 y
= inside-page ads 2x + 1y ≤ 30

42. Now you try… x + y ≤ 20 2x + 1y ≤ 30