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# 3 8 linear inequalities (optional)

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### 3 8 linear inequalities (optional)

1. 1. Linear Inequalities (Optional) Frank Ma © 2011
2. 2. The graphs of the linear equations are straight lines. Linear Inequalities
3. 3. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. Linear Inequalities
4. 4. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities
5. 5. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities
6. 6. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities y = x
7. 7. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. y = x
8. 8. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. y = x
9. 9. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. Specifically, the line divides the plane into two half-planes. y = x
10. 10. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. y = x
11. 11. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. y = x To match which half-plane fits which inequality, we sample a point,
12. 12. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. y = x To match which half-plane fits which inequality, we sample a point, any point, from these two regions.
13. 13. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. y = x To match which half-plane fits which inequality, we sample a point, any point, from these two regions. Let’s take the point (0, 5),
14. 14. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. y = x Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. To match which half-plane fits which inequality, we sample a point, any point, from these two regions. Let’s take the point (0, 5),
15. 15. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. y = x Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. To match which half-plane fits which inequality, we sample a point, any point, from these two regions. Let’s take the point (0, 5), since 5 > 0, it fits the relation y > x.
16. 16. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. y = x Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. To match which half-plane fits which inequality, we sample a point, any point, from these two regions. Let’s take the point (0, 5), since 5 > 0, it fits the relation y > x. So the side that contains (0, 5) is y > x,
17. 17. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. y = x y > x Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. To match which half-plane fits which inequality, we sample a point, any point, from these two regions. Let’s take the point (0, 5), since 5 > 0, it fits the relation y > x. So the side that contains (0, 5) is y > x,
18. 18. The graphs of the linear equations are straight lines. Example A. Graph of the linear equations y = x. x –3 0 3 y –3 0 3 Linear Inequalities The points on the line fit the description that y = x. So for any point that off the line, y = x. y = x y > x y < x Specifically, the line divides the plane into two half-planes. One of them fits the relation that y < x and the other fits y > x. To match which half-plane fits which inequality, we sample a point, any point, from these two regions. Let’s take the point (0, 5), since 5 > 0, it fits the relation y > x. So the side that contains (0, 5) is y > x, and the other half-plane must be y < x.
19. 19. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. Linear Inequalities
20. 20. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. Linear Inequalities
21. 21. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Linear Inequalities
22. 22. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities
23. 23. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0).
24. 24. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12.
25. 25. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12.
26. 26. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12. Use intercepts to graph.
27. 27. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12. Use intercepts to graph. x y 0 0
28. 28. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12. Use intercepts to graph. x y 0 4 0
29. 29. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12. Use intercepts to graph. x y 0 4 6 0
30. 30. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12. Use intercepts to graph. x y 0 4 6 0
31. 31. In general, the points that fit the linear inequality Ax + By > C or Ax + By < C corresponds to one of the two half-planes on each side of the line Ax + By = C. To identify which half-plane fits the given inequality, sample any point in a half-plane by plugging in the values of x and y. If the point satisfies the inequality, then the entire half-plane that contains the sample point fits that inequality. Otherwise, the other side fits the inequality. Linear Inequalities Remark: If (0, 0) is not on the line, sample (0, 0). Example B. Shade 2x + 3y > 12. Graph 2x + 3y = 12. Use intercepts to graph. Draw a dotted line because the line itself is not part of the answer. x y 0 4 6 0 2x + 3y = 12
32. 32. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). 2x + 3y = 12
33. 33. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). 2x + 3y = 12
34. 34. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 2x + 3y = 12
35. 35. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y = 12
36. 36. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. 2x + 3y = 12
37. 37. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line.
38. 38. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that.
39. 39. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 To find the region that fits a system of two linear inequalities, graph the equations first. Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that.
40. 40. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 To find the region that fits a system of two linear inequalities, graph the equations first. In general, we get two intersecting lines that divide the plane into 4 regions. Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that.
41. 41. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 To find the region that fits a system of two linear inequalities, graph the equations first. In general, we get two intersecting lines that divide the plane into 4 regions. Then we sample to determine the two half-planes that fit the two inequalities. Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that.
42. 42. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 To find the region that fits a system of two linear inequalities, graph the equations first. In general, we get two intersecting lines that divide the plane into 4 regions. Then we sample to determine the two half-planes that fit the two inequalities. The overlapped region of the two half-planes is the region that fits the system. Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that.
43. 43. Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). Plug in (0, 0) into the inequality, we’ve 0 + 0 > 12 which is false. 2x + 3y > 12 2x + 3y = 12 To find the region that fits a system of two linear inequalities, graph the equations first. In general, we get two intersecting lines that divide the plane into 4 regions. Then we sample to determine the two half-planes that fit the two inequalities. The overlapped region of the two half-planes is the region that fits the system. To give the complete solution, we need to locate the tip of the region by solving the system. Hence the half-plane that fits 2x + 3y > 12 must be the other side of the line. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that.
44. 44. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { Linear Inequalities
45. 45. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. Linear Inequalities
46. 46. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. Linear Inequalities
47. 47. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 0 Linear Inequalities Find the intercepts.
48. 48. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Linear Inequalities Find the intercepts.
49. 49. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Linear Inequalities Find the intercepts.
50. 50. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Linear Inequalities Find the intercepts.
51. 51. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Linear Inequalities Find the intercepts.
52. 52. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Test (0, 0), Linear Inequalities Find the intercepts.
53. 53. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Test (0, 0), it does not fit. Linear Inequalities Find the intercepts.
54. 54. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Test (0, 0), it does not fit. Hence, the other side fits the inequality 2x – y < –2. Linear Inequalities Find the intercepts.
55. 55. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Test (0, 0), it does not fit. Hence, the other side fits the inequality 2x – y < –2. Shade it. Linear Inequalities Find the intercepts.
56. 56. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { We are looking for the region that satisfies both inequalities. For 2x – y < –2, graph the equation 2x – y = –2. x y 0 2 –1 0 Since the inequality includes “ = ”, use a solid line for the graph. Test (0, 0), it does not fit. Hence, the other side fits the inequality 2x – y < –2. Shade it. Linear Inequalities Find the intercepts.
57. 57. For x + y < 5, graph x = y = 5 Linear Inequalities
58. 58. For x + y < 5, graph x = y = 5 x y 0 0 Linear Inequalities
59. 59. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Linear Inequalities
60. 60. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Linear Inequalities
61. 61. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Linear Inequalities
62. 62. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Linear Inequalities
63. 63. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), Linear Inequalities
64. 64. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Linear Inequalities
65. 65. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Linear Inequalities
66. 66. For x + y < 5, graph x = y = 5 x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Shade it. Linear Inequalities
67. 67. x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Shade it. Linear Inequalities For x + y < 5, graph x = y = 5
68. 68. x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Shade it. The region that fits the system is the region has both shading. Linear Inequalities For x + y < 5, graph x = y = 5
69. 69. x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Shade it. The region that fits the system is the region has both shading. Linear Inequalities For x + y < 5, graph x = y = 5
70. 70. x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Shade it. The region that fits the system is the region has both shading. Linear Inequalities 2x – y = –2 x + y = 5{ For x + y < 5, graph x = y = 5 To find the tip of the region, we solve the system of equations for the point of intersection.
71. 71. x y 0 5 5 0 Since the inequality is strict, use a dotted line for the graph. Test (0, 0), it fits x + y < 5. Hence, this side fits x + y < 5. Shade it. The region that fits the system is the region has both shading. Linear Inequalities 2x – y = –2 x + y = 5{ Add these equations to remove the y. For x + y < 5, graph x = y = 5 To find the tip of the region, we solve the system of equations for the point of intersection.
72. 72. Linear Inequalities 2x – y = –2 x + y = 5+)
73. 73. Linear Inequalities 2x – y = –2 x + y = 5+) 3x = 3
74. 74. Linear Inequalities 2x – y = –2 x + y = 5+) 3x = 3 x = 1
75. 75. Linear Inequalities Set x = 1 in x + y = 5, 2x – y = –2 x + y = 5+) 3x = 3 x = 1
76. 76. Linear Inequalities Set x = 1 in x + y = 5, we get 1 + y = 5  y = 4. 2x – y = –2 x + y = 5+) 3x = 3 x = 1
77. 77. Linear Inequalities Set x = 1 in x + y = 5, we get 1 + y = 5  y = 4. 2x – y = –2 x + y = 5+) 3x = 3 x = 1 Hence the tip of the region is (1, 4). (1, 4)
78. 78. Exercise A. Shade the following inequalities in the x and y coordinate system. 1. x – y > 3 2. 2x ≤ 6 3. –y – 7 ≥ 0 4. 0 ≤ 8 – 2x 5. y < –x + 4 6. 2x/3 – 3 ≤ 6/5 7. 2x < 6 – 2y 8. 4y/5 – 12 ≥ 3x/4 9. 2x + 3y > 3 10. –6 ≤ 3x – 2y 11. 3x + 2 > 4y + 3x 12. 5x/4 + 2y/3 ≤ 2 Linear Inequalities 16.{–x + 2y ≥ –12 2x + y ≤ 4 Exercise B. Shade the following regions. Label the tip. 13.{x + y ≥ 3 2x + y < 4 14. 15.{x + 2y ≥ 3 2x – y > 6 {x + y ≤ 3 2x – y > 6 17. {3x + 4y ≥ 3 x – 2y < 6 18. { x + 3y ≥ 3 2x – 9y ≥ –4 19.{–3x + 2y ≥ –1 2x + 3y ≤ 5 20. {2x + 3y > –1 3x + 4y ≥ 2 21. {4x – 3y ≤ 3 3x – 2y > –4 { x – y < 3 x – y ≤ –1 3 2 2 3 1 2 1 4 22. { x + y ≤ 1 x – y < –1 1 2 1 5 3 4 1 6 23.