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# Integrated 2 Section 6-4

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Write and Graph Linear Inequalities

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### Integrated 2 Section 6-4

1. 1. SECTION 6-4 Write and Graph Linear Inequalities Tue, Dec 01
2. 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. 3. VOCABULARY 1. Open Half-plane: 2. Boundary: 3. Linear Inequality: 4. Solution to the Inequality: Tue, Dec 01
4. 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. 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. 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. 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. 8. VOCABULARY 5. Graph of the Inequality: 6. Closed Half-plane: 7.Test Point: Tue, Dec 01
9. 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. 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. 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. 12. GRAPHING A LINEAR INEQUALITY Tue, Dec 01
13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 38. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 Tue, Dec 01
39. 39. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m=3 Tue, Dec 01
40. 40. EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 Tue, Dec 01
41. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 54. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 Tue, Dec 01
55. 55. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m=− 2 Tue, Dec 01
56. 56. EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 Tue, Dec 01
57. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 70. WHERE TO SHADE Tue, Dec 01
71. 71. WHERE TO SHADE When y is isolated, there is a trick we can use: Tue, Dec 01
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 Tue, Dec 01
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 below y goes up when we get less (>, ≥), so shade above Tue, Dec 01
74. 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. 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. 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. 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. 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. 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. 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. 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. 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. 83. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
84. 84. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
85. 85. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
86. 86. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
87. 87. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
88. 88. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
89. 89. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
90. 90. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
91. 91. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
92. 92. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
93. 93. EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5 Tue, Dec 01
94. 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. 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. 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. 97. HOMEWORK Tue, Dec 01
98. 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