Graph a linear inequality in 2 variables'

1
Section 4.4

• Graph a linear inequality in 2 variables.
• Use mathematical models involving linear
  inequalities.
• Graph a system of linear inequalities.

2

• What is a solution to a linear equation?

3

• What is a solution to a linear equation?
• (1,1) is a solution to y 3x 2 because
• 1 3(1) 2

4

• What is a solution to a linear equation?
• (1,1) is a solution to y 3x 2 because
• 1 3(1) 2
• What is a solution to a linear inequality?

5

• What is a solution to a linear equation?
• (1,1) is a solution to y 3x 2 because
• 1 3(1) 2
• What is a solution to a linear inequality?
• (3,2) is a solution to x y gt 1 because
• 3 2 gt 1

6

• If we graph the inequality the plane is separated
  into 3 parts.
• What are these parts?

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Pink Box on 277
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(No Transcript)
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Graph 2x 3y 6
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Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with

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Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with
• Find intercepts to graph the line
• 2x 3(0) 6

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Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with
• Find intercepts to graph the line
• 2x 3(0) 6
• 2x 6

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Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with
• Find intercepts to graph the line
• 2x 3(0) 6
• 2x 6
• x 3 is x-intercept

14
Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with
• Find intercepts to graph the line
• 2x 3(0) 6
• 2x 6
• x 3 is x-intercept
• 2(0) 3y 6

15
Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with
• Find intercepts to graph the line
• 2x 3(0) 6
• 2x 6
• x 3 is x-intercept
• 2(0) 3y 6
• -3y 6

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Graph 2x 3y 6

• 2x 3y 6 replace inequality
  with
• Find intercepts to graph the line
• 2x 3(0) 6
• 2x 6
• x 3 is x-intercept
• 2(0) 3y 6
• -3y 6
• y -2 is y-intercept

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(3,0) and (0,-2) are points on the line
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(3,0) and (0,-2) are points on the line
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Test Point

• Choose a point from one of the half-planes.
• See if it is a solution. Easiest choice is often
  the origin.

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Test Point

• Choose a point from one of the half-planes.
• See if it is a solution. Easiest choice is often
  the origin.
• (0,0) test point 2x 3y 6 inequality

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Test Point

• Choose a point from one of the half-planes.
• See if it is a solution. Easiest choice is often
  the origin.
• (0,0) test point 2x 3y 6 inequality
• 2(0) 3(0) 6

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Test Point

• Choose a point from one of the half-planes.
• See if it is a solution. Easiest choice is often
  the origin.
• (0,0) test point 2x 3y 6 inequality
• 2(0) 3(0) 6
• 0 0 6

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Test Point

• Choose a point from one of the half-planes.
• See if it is a solution. Easiest choice is often
  the origin.
• (0,0) test point 2x 3y 6 inequality
• 2(0) 3(0) 6
• 0 0 6 False

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Test Point

• Choose a point from one of the half-planes.
• See if it is a solution. Easiest choice is often
  the origin.
• (0,0) test point 2x 3y 6 inequality
• 2(0) 3(0) 6
• 0 0 6 False
• The half-plane containing the origin is not the
  solution set for this inequality. Thus the other
  half-plane is the solution set.

25
Graph y gt 3x
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Graph y gt 3x

• Replace the gt with an and graph with a dotted
  line.

27
Graph y gt 3x

• Replace the gt with an and graph with a dotted
  line.
• y 3x 0 has m 3 and y-intercept 0.

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Graph y gt 3x

• Replace the gt with an and graph with a dotted
  line.
• y 3x 0 has m 3 and y-intercept 0.
• Graph this line.

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Test Point

• Notice we cant use the origin as our test point.

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Test Point

• Notice we cant use the origin as our test point.
• (0,1) is our test point.

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Test Point

• Notice we cant use the origin as our test point.
• (0,1) is our test point.
• y gt 3x

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Test Point

• Notice we cant use the origin as our test point.
• (0,1) is our test point.
• y gt 3x
• 1 gt 3(0)

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Test Point

• Notice we cant use the origin as our test point.
• (0,1) is our test point.
• y gt 3x
• 1 gt 3(0) True

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Test Point

• Notice we cant use the origin as our test point.
• (0,1) is our test point.
• y gt 3x
• 1 gt 3(0) True
• The point (0,1) is a solution to the inequality
  so the half-plane containing (0,1) is the
  solution set to the inequality.

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System of linear inequalities

• Just as 2 or more equations make up a system of
  equations, 2 or more inequalities makes up a
  system of inequalities.

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System of linear inequalities

• Just as 2 or more equations make up a system of
  equations, 2 or more inequalities makes up a
  system of inequalities.
• A solution to a system of inequalities must
  satisfy all inequalities in the system.

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Graph x y gt 1 -x y 1

• x y gt 1

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Graph x y gt 1 -x y 1

• x y gt 1
• y gt -x 1 has slope -1 and y-intercept 1

39
Graph x y gt 1 -x y 1

• x y gt 1
• y gt -x 1 has slope -1 and y-intercept 1
• -x y 1

40
Graph x y gt 1 -x y 1

• x y gt 1
• y gt -x 1 has slope -1 and y-intercept 1
• -x y 1
• y x 1 has slope 1 and y-intercept 1

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Graph x y gt 1 -x y 1

• x y gt 1
• y gt -x 1 has slope -1 and y-intercept 1
• -x y 1
• y x 1 has slope 1 and y-intercept 1
• Graph these lines with appropriate solid/dotted
  lines.

42
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.

43
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.

44
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1

45
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1
• 3 0 gt 1

46
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1
• 3 0 gt 1 True

47
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1 -x y 1
• 3 0 gt 1 True

48
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1 -x y 1
• 3 0 gt 1 True -3 0 1

49
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1 -x y 1
• 3 0 gt 1 True -3 0 1
  False

50
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (3,0) on both inequalities.
• x y gt 1 -x y 1
• 3 0 gt 1 True -3 0 1
  False
• Although (3,0) satisfies one inequality it fails
  the other inequality and is NOT a solution to the
  system.

51
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (0,2) on both inequalities.

52
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 1

53
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 1
• 0 2 gt 1 True

54
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 1 -x y 1
• 0 2 gt 1 True

55
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 1 -x y 1
• 0 2 gt 1 True 0 2 1
  True

56
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 1 -x y 1
• 0 2 gt 1 True 0 2 1
  True
• The point (0,2) satisfies all inequalities in the
  system and is a solution to the system.

57
Graph x y gt 1 -x y 1

• Note that this time we have 4 regions to choose a
  test point from.
• So we shade in the region of the graph containing
  (0,2) .

58
Graph -x y gt 3 -x y lt 1
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Graph -x y gt 3 -x y lt 1

• -x y gt 3

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Graph -x y gt 3 -x y lt 1

• -x y gt 3
• y gt x 3 has slope 1 and y-intercept 3

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Graph -x y gt 3 -x y lt 1

• -x y gt 3
• y gt x 3 has slope 1 and y-intercept 3
• -x y lt 1

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Graph -x y gt 3 -x y lt 1

• -x y gt 3
• y gt x 3 has slope 1 and y-intercept 3
• -x y lt 1
• y lt x 1 has slope 1 and y-intercept 1

63
Graph -x y gt 3 -x y lt 1

• -x y gt 3
• y gt x 3 has slope 1 and y-intercept 3
• -x y lt 1
• y lt x 1 has slope 1 and y-intercept 1
• Graph these lines with appropriate solid/dotted
  lines.

64
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.

65
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.

66
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3

67
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3
• 0 0 gt 3

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Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3
• 0 0 gt 3 False

69
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3 -x y lt 1
• 0 0 gt 3 False

70
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3 -x y lt 1
• 0 0 gt 3 False 0 0 lt 1

71
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3 -x y lt 1
• 0 0 gt 3 False 0 0 lt 1
  True

72
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,0) on both inequalities.
• x y gt 3 -x y lt 1
• 0 0 gt 3 False 0 0 lt 1
  True
• NOT a solution

73
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.

74
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3

75
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3
• 0 2 gt 3

76
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3
• 0 2 gt 3 False

77
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3 -x y lt 1
• 0 2 gt 3 False

78
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3 -x y lt 1
• 0 2 gt 3 False 0 2 lt 1

79
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3 -x y lt 1
• 0 2 gt 3 False 0 2 lt 1
  False

80
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,2) on both inequalities.
• x y gt 3 -x y lt 1
• 0 2 gt 3 False 0 2 lt 1
  False
• NOT a solution

81
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.

82
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3

83
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3
• 0 4 gt 3

84
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3
• 0 4 gt 3 True

85
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3 -x y lt 1
• 0 4 gt 3 True

86
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3 -x y lt 1
• 0 4 gt 3 True 0 4 lt 1

87
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3 -x y lt 1
• 0 4 gt 3 True 0 4 lt 1
  False

88
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• Test (0,4) on both inequalities.
• x y gt 3 -x y lt 1
• 0 4 gt 3 True 0 4 lt 1
  False
• NOT a solution

89
Graph -x y gt 3 -x y lt 1

• Note that this time we have 3 regions to choose a
  test point from.
• All 3 parts failed.
• There is NO solution to the system of
  inequalities.