Write and Graph Linear Inequalities

1. SECTION 6-4
Write and Graph Linear Inequalities
Tue, Dec 01

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
Tue, Dec 01

3. VOCABULARY
1. Open Half-plane:
2. Boundary:
3. Linear Inequality:
4. Solution to the Inequality:
Tue, Dec 01

4. VOCABULARY
1. Open Half-plane: A dashed boundary line separates
the plane
2. Boundary:
3. Linear Inequality:
4. Solution to the Inequality:
Tue, Dec 01

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:
Tue, Dec 01

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:
Tue, Dec 01

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
Tue, Dec 01

8. VOCABULARY
5. Graph of the Inequality:
6. Closed Half-plane:
7.Test Point:
Tue, Dec 01

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:
Tue, Dec 01

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:
Tue, Dec 01

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
Tue, Dec 01

12. GRAPHING A LINEAR
INEQUALITY
Tue, Dec 01

13. GRAPHING A LINEAR
INEQUALITY
Begin by treating the inequality as an equation to
graph the boundary line and isolate y.
Tue, Dec 01

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.
Tue, Dec 01

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.
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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)
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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)
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

38. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
Tue, Dec 01

39. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m=3
Tue, Dec 01

40. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
Tue, Dec 01

41. EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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):
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

54. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
Tue, Dec 01

55. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m=−
2
Tue, Dec 01

56. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
Tue, Dec 01

57. EXAMPLE 2
Graph the following inequalities.
3
b. y ≤ − x + 4
2
3
m = − Down 3, right 2
2
y-int: (0, 4)
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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):
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

70. WHERE TO SHADE
Tue, Dec 01

71. WHERE TO SHADE
When y is isolated, there is a trick we can use:
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

74. EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

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
Tue, Dec 01

83. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

84. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

85. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

86. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

87. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

88. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

89. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

90. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

91. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

92. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

93. EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01

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.
Tue, Dec 01

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.
Tue, Dec 01

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.
Tue, Dec 01

97. HOMEWORK
Tue, Dec 01

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
Tue, Dec 01