Write and Graph Linear Inequalities

1. SECTION 6-4
Write and Graph Linear Inequalities

2. ESSENTIAL QUESTIONS
How do you write linear inequalities in two variables?
How do you graph linear inequalities in two variables
on the coordinate plane?
Where you’ll see this:
Business, market research, inventory

3. VOCABULARY
1. Open Half-plane:
2. Boundary:
3. Linear Inequality:
4. Solution to the Inequality:

4. VOCABULARY
1. Open Half-plane: A dashed boundary line separates
the plane
2. Boundary:
3. Linear Inequality:
4. Solution to the Inequality:

5. VOCABULARY
1. Open Half-plane: A dashed boundary line separates
the plane
2. Boundary: The line that separates half-planes
3. Linear Inequality:
4. Solution to the Inequality:

6. VOCABULARY
1. Open Half-plane: A dashed boundary line separates
the plane
2. Boundary: The line that separates half-planes
3. Linear Inequality: A sentence where instead of an =
sign, we use <, >, ≤, ≥, or ≠
4. Solution to the Inequality:

7. VOCABULARY
1. Open Half-plane: A dashed boundary line separates
the plane
2. Boundary: The line that separates half-planes
3. Linear Inequality: A sentence where instead of an =
sign, we use <, >, ≤, ≥, or ≠
4. Solution to the Inequality: ANY ordered pair that
makes the inequality true

8. VOCABULARY
5. Graph of the Inequality:
6. Closed Half-plane:
7.Test Point:

9. VOCABULARY
5. Graph of the Inequality: Includes graphing the
boundary line and the shaded half-plane that
includes the solution
6. Closed Half-plane:
7.Test Point:

10. VOCABULARY
5. Graph of the Inequality: Includes graphing the
boundary line and the shaded half-plane that
includes the solution
6. Closed Half-plane: A solid boundary line separates
the plane
7.Test Point:

11. VOCABULARY
5. Graph of the Inequality: Includes graphing the
boundary line and the shaded half-plane that
includes the solution
6. Closed Half-plane: A solid boundary line separates
the plane
7.Test Point: A point NOT on the boundary line that is
used to test whether to shade above or below the
boundary line

12. GRAPHING A LINEAR
INEQUALITY

13. GRAPHING A LINEAR
INEQUALITY
Begin by treating the inequality as an equation to
graph the boundary line and isolate y.

14. GRAPHING A LINEAR
INEQUALITY
Begin by treating the inequality as an equation to
graph the boundary line and isolate y.
If <, >, or ≠, the boundary line will be dashed.

15. GRAPHING A LINEAR
INEQUALITY
Begin by treating the inequality as an equation to
graph the boundary line and isolate y.
If <, >, or ≠, the boundary line will be dashed.
If ≤ or ≥, the boundary line will be solid.

16. GRAPHING A LINEAR
INEQUALITY
Begin by treating the inequality as an equation to
graph the boundary line and isolate y.
If <, >, or ≠, the boundary line will be dashed.
If ≤ or ≥, the boundary line will be solid.
Use a test point to determine shading OR

17. GRAPHING A LINEAR
INEQUALITY
Begin by treating the inequality as an equation to
graph the boundary line and isolate y.
If <, >, or ≠, the boundary line will be dashed.
If ≤ or ≥, the boundary line will be solid.
Use a test point to determine shading OR
If y is isolated, < and ≤ shade below, > and ≥
shade above

18. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0
(3, 5), (4, 0)

19. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0
(3, 5), (4, 0)
2(3) − 3(5) < 0

20. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0
(3, 5), (4, 0)
2(3) − 3(5) < 0
6 −15 < 0

21. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0
(3, 5), (4, 0)
2(3) − 3(5) < 0
6 −15 < 0
−9 < 0

22. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0
(3, 5), (4, 0)
2(3) − 3(5) < 0
6 −15 < 0
−9 < 0
(3, 5) is a solution

23. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0 2(4) − 3(0) < 0
(3, 5), (4, 0)
2(3) − 3(5) < 0
6 −15 < 0
−9 < 0
(3, 5) is a solution

24. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0 2(4) − 3(0) < 0
(3, 5), (4, 0)
8−0<0
2(3) − 3(5) < 0
6 −15 < 0
−9 < 0
(3, 5) is a solution

25. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0 2(4) − 3(0) < 0
(3, 5), (4, 0)
8−0<0
2(3) − 3(5) < 0 8<0
6 −15 < 0
−9 < 0
(3, 5) is a solution

26. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0 2(4) − 3(0) < 0
(3, 5), (4, 0)
8−0<0
2(3) − 3(5) < 0 8<0
6 −15 < 0 (4, 0) is not a solution
−9 < 0
(3, 5) is a solution

27. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
a. 2x − 3y < 0 2(4) − 3(0) < 0
(3, 5), (4, 0)
8−0<0
2(3) − 3(5) < 0 8<0
6 −15 < 0 (4, 0) is not a solution
−9 < 0 The boundary line is dashed
(3, 5) is a solution

28. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6
(-2, -6), (0, 0)

29. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6
(-2, -6), (0, 0)
4(−6) − (−2) ≥ −6

30. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6
(-2, -6), (0, 0)
4(−6) − (−2) ≥ −6
−24 + 2 ≥ −6

31. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6
(-2, -6), (0, 0)
4(−6) − (−2) ≥ −6
−24 + 2 ≥ −6
−22 ≥ −6

32. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6
(-2, -6), (0, 0)
4(−6) − (−2) ≥ −6
−24 + 2 ≥ −6
−22 ≥ −6
(-2, -6) is not a solution

33. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6 4(0) − 0 ≥ −6
(-2, -6), (0, 0)
4(−6) − (−2) ≥ −6
−24 + 2 ≥ −6
−22 ≥ −6
(-2, -6) is not a solution

34. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6 4(0) − 0 ≥ −6
(-2, -6), (0, 0)
0 − 0 ≥ −6
4(−6) − (−2) ≥ −6
−24 + 2 ≥ −6
−22 ≥ −6
(-2, -6) is not a solution

35. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6 4(0) − 0 ≥ −6
(-2, -6), (0, 0)
0 − 0 ≥ −6
4(−6) − (−2) ≥ −6 0 ≥ −6
−24 + 2 ≥ −6
−22 ≥ −6
(-2, -6) is not a solution

36. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6 4(0) − 0 ≥ −6
(-2, -6), (0, 0)
0 − 0 ≥ −6
4(−6) − (−2) ≥ −6 0 ≥ −6
−24 + 2 ≥ −6 (0, 0) is a solution
−22 ≥ −6
(-2, -6) is not a solution

37. EXAMPLE 1
Tell whether the given coordinates satisfy each
inequality by testing each point. Is the bondary line
solid or dashed?
b. 4y − x ≥ −6 4(0) − 0 ≥ −6
(-2, -6), (0, 0)
0 − 0 ≥ −6
4(−6) − (−2) ≥ −6 0 ≥ −6
−24 + 2 ≥ −6 (0, 0) is a solution
−22 ≥ −6 The boundary line is solid
(-2, -6) is not a solution

38. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5

39. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m=3

40. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1

41. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)

42. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

43. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

44. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

45. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

46. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

47. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

48. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed

49. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Check (0, 0):

50. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Check (0, 0): 0 > 3(0) − 5

51. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Check (0, 0): 0 > 3(0) − 5

52. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Check (0, 0): 0 > 3(0) − 5

53. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Check (0, 0): 0 > 3(0) − 5

54. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2

55. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m=−
2

56. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2

57. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)

58. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

59. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

60. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

61. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

62. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

63. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

64. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid

65. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid
Check (0, 0):

66. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid
3
Check (0, 0): 0 ≤ − (0) + 4
2

67. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid
3
Check (0, 0): 0 ≤ − (0) + 4
2

68. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid
3
Check (0, 0): 0 ≤ − (0) + 4
2

69. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Boundary line is solid
3
Check (0, 0): 0 ≤ − (0) + 4
2

70. WHERE TO SHADE

71. WHERE TO SHADE
When y is isolated, there is a trick we can use:

72. WHERE TO SHADE
When y is isolated, there is a trick we can use:
y goes down when we get less (<, ≤), so shade below

73. WHERE TO SHADE
When y is isolated, there is a trick we can use:
y goes down when we get less (<, ≤), so shade below
y goes up when we get less (>, ≥), so shade above

74. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.

75. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width

76. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y

77. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y
10 ≤ 2x + 2y

78. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y
10 ≤ 2x + 2y
-2x -2x

79. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y
10 ≤ 2x + 2y
-2x -2x
10 − 2x ≤ 2y

80. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y
10 ≤ 2x + 2y
-2x -2x
10 − 2x ≤ 2y
2 2

81. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y
10 ≤ 2x + 2y
-2x -2x 5− x ≤ y
10 − 2x ≤ 2y
2 2

82. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
x = length, y = width P = 2x + 2y
10 ≤ 2x + 2y
-2x -2x 5− x ≤ y
10 − 2x ≤ 2y
2 2 y ≥ −x + 5

83. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

84. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

85. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

86. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

87. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

88. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

89. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

90. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

91. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

92. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

93. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5

94. EXAMPLE 3
c. Does the “trick” tell us to shade above or below the
boundary line? How do you know?
d. Use the graph to name three possible combinations
of length and width for rectangle ABCD. Check to
make sure they satisfy the situation.

95. EXAMPLE 3
c. Does the “trick” tell us to shade above or below the
boundary line? How do you know?
You shade above, as y gets larger due to ≥
d. Use the graph to name three possible combinations
of length and width for rectangle ABCD. Check to
make sure they satisfy the situation.

96. EXAMPLE 3
c. Does the “trick” tell us to shade above or below the
boundary line? How do you know?
You shade above, as y gets larger due to ≥
d. Use the graph to name three possible combinations
of length and width for rectangle ABCD. Check to
make sure they satisfy the situation.
Any points on the line or the shaded region work. The
values must be positive in this situation.

97. HOMEWORK

98. HOMEWORK
p. 260 #1-37 odd
“Everyone has talent. What is rare is the courage
to follow the talent to the dark place where it
leads.” - Erica Jong