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# Int Math 2 Section 6-4 1011

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Write and Graph Linear Inequalities

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• ### Int Math 2 Section 6-4 1011

1. 1. SECTION 6-4Write and Graph Linear Inequalities
2. 2. ESSENTIAL QUESTIONSHow do you write linear inequalities in two variables?How do you graph linear inequalities in two variableson the coordinate plane?Where you’ll see this: Business, market research, inventory
3. 3. VOCABULARY1. Open Half-plane:2. Boundary:3. Linear Inequality:4. Solution to the Inequality:
4. 4. VOCABULARY1. Open Half-plane: A dashed boundary line separates the plane2. Boundary:3. Linear Inequality:4. Solution to the Inequality:
5. 5. VOCABULARY1. Open Half-plane: A dashed boundary line separates the plane2. Boundary: The line that separates half-planes3. Linear Inequality:4. Solution to the Inequality:
6. 6. VOCABULARY1. Open Half-plane: A dashed boundary line separates the plane2. Boundary: The line that separates half-planes3. Linear Inequality: A sentence where instead of an = sign, we use <, >, ≤, ≥, or ≠4. Solution to the Inequality:
7. 7. VOCABULARY1. Open Half-plane: A dashed boundary line separates the plane2. Boundary: The line that separates half-planes3. 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. 8. VOCABULARY5. Graph of the Inequality:6. Closed Half-plane:7.Test Point:
9. 9. VOCABULARY5. Graph of the Inequality: Includes graphing the boundary line and the shaded half-plane that includes the solution6. Closed Half-plane:7.Test Point:
10. 10. VOCABULARY5. Graph of the Inequality: Includes graphing the boundary line and the shaded half-plane that includes the solution6. Closed Half-plane: A solid boundary line separates the plane7.Test Point:
11. 11. VOCABULARY5. Graph of the Inequality: Includes graphing the boundary line and the shaded half-plane that includes the solution6. Closed Half-plane: A solid boundary line separates the plane7.Test Point: A point NOT on the boundary line that is used to test whether to shade above or below the boundary line
12. 12. GRAPHING A LINEAR INEQUALITY
13. 13. GRAPHING A LINEAR INEQUALITYBegin by treating the inequality as an equation tograph the boundary line and isolate y.
14. 14. GRAPHING A LINEAR INEQUALITYBegin by treating the inequality as an equation tograph the boundary line and isolate y.If <, >, or ≠, the boundary line will be dashed.
15. 15. GRAPHING A LINEAR INEQUALITYBegin by treating the inequality as an equation tograph the boundary line and isolate y.If <, >, or ≠, the boundary line will be dashed.If ≤ or ≥, the boundary line will be solid.
16. 16. GRAPHING A LINEAR INEQUALITYBegin by treating the inequality as an equation tograph 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. 17. GRAPHING A LINEAR INEQUALITYBegin by treating the inequality as an equation tograph 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 ORIf y is isolated, < and ≤ shade below, > and ≥shade above
18. 18. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0)
19. 19. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0
20. 20. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0 6 −15 < 0
21. 21. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0 6 −15 < 0 −9 < 0
22. 22. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary 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. 23. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary 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. 24. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary 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. 25. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary 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. 26. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary 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. 27. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary 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. 28. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0)
29. 29. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0)4(−6) − (−2) ≥ −6
30. 30. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0)4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6
31. 31. EXAMPLE 1 Tell whether the given coordinates satisfy eachinequality by testing each point. Is the boundary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0)4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6 −22 ≥ −6
32. 32. EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the boundary 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. 33. EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the boundary 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. 34. EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the boundary 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. 35. EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the boundary 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. 36. EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the boundary 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. 37. EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the boundary 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. 38. EXAMPLE 2 Graph the following inequalities.a. y > 3x − 5
39. 39. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5m=3
40. 40. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5m = 3 Up 3, right 1
41. 41. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5m = 3 Up 3, right 1 y-int: (0, -5)
42. 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. 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. 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. 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. 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. 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. 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. 49. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5)Boundary line is dashedCheck (0, 0):
50. 50. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5)Boundary line is dashedCheck (0, 0): 0 > 3(0) − 5
51. 51. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5)Boundary line is dashedCheck (0, 0): 0 > 3(0) − 5
52. 52. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5)Boundary line is dashedCheck (0, 0): 0 > 3(0) − 5
53. 53. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5)Boundary line is dashedCheck (0, 0): 0 > 3(0) − 5
54. 54. EXAMPLE 2 Graph the following inequalities. 3b. y ≤ − x + 4 2
55. 55. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m=− 2
56. 56. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2
57. 57. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4)
58. 58. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
59. 59. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
60. 60. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
61. 61. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
62. 62. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
63. 63. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
64. 64. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
65. 65. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solidCheck (0, 0):
66. 66. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3Check (0, 0): 0 ≤ − (0) + 4 2
67. 67. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3Check (0, 0): 0 ≤ − (0) + 4 2
68. 68. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3Check (0, 0): 0 ≤ − (0) + 4 2
69. 69. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3Check (0, 0): 0 ≤ − (0) + 4 2
71. 71. WHERE TO SHADEWhen y is isolated, there is a trick we can use:
72. 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. 73. WHERE TO SHADE When y is isolated, there is a trick we can use:y goes down when we get less (<, ≤), so shade belowy goes up when we get greater (>, ≥), so shade above
74. 74. EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm.a. Write a linear inequality that represents the situation.
75. 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. 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. 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. 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. 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. 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. 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. 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. 83. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
84. 84. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
85. 85. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
86. 86. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
87. 87. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
88. 88. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
89. 89. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
90. 90. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
91. 91. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
92. 92. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
93. 93. EXAMPLE 3b. Graph the solution to the inequality. y ≥ −x + 5
94. 94. EXAMPLE 3c. 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. 95. EXAMPLE 3c. 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. 96. EXAMPLE 3c. 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. 97. PROBLEM SET
98. 98. PROBLEM SET 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