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Write and Graph Linear Inequalities

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- 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

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