- 2. The solutions of inequalities in x are segments of the real line. 2D Linear Inequalities
- 3. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. 2D Linear Inequalities
- 4. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities
- 5. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities y = x The graph of y = x is the diagonal line.
- 6. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. y = x The graph of y = x is the diagonal line.
- 7. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x The graph of y = x is the diagonal line.
- 8. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x Specifically, the line y = x divides the plane into two half-planes. The graph of y = x is the diagonal line.
- 9. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x Specifically, the line y = x divides the plane into two half-planes. One of them fits the relation that y < x and the other fits x < y. The graph of y = x is the diagonal line.
- 10. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x Specifically, the line y = x divides the plane into two half-planes. One of them fits the relation that y < x and the other fits x < y. To identify which half-plane matches which inequality, sample any point in the half planes. The graph of y = x is the diagonal line.
- 11. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x Specifically, the line y = x divides the plane into two half-planes. One of them fits the relation that y < x and the other fits x < y. To identify which half-plane matches which inequality, sample any point in the half planes. For example, let's select (0, 5) to test the inequalities. (0, 5)The graph of y = x is the diagonal line.
- 12. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x Specifically, the line y = x divides the plane into two half-planes. One of them fits the relation that y < x and the other fits x < y. To identify which half-plane matches which inequality, sample any point in the half planes. For example, let's select (0, 5) to test the inequalities. Since 0 < 5 so the side that contains (0, 5) is x < y. The graph of y = x is the diagonal line. x < y (0, 5)
- 13. The solutions of inequalities in x are segments of the real line. The solutions of inequalities in x and y are regions of the plane. Example A. Use the graph of y = x to identify the regions associated with y > x and y < x. 2D Linear Inequalities Points on the line fit the condition that y = x. Points not on the line fit the condition y = x. y = x x > y Specifically, the line y = x divides the plane into two half-planes. One of them fits the relation that y < x and the other fits x < y. To identify which half-plane matches which inequality, sample any point in the half planes. For example, let's select (0, 5) to test the inequalities. Since 0 < 5 so the side that contains (0, 5) is x < y. It follows that the other side is x > y. The graph of y = x is the diagonal line. x < y (0, 5)
- 14. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. 2D Linear Inequalities
- 15. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. 2D Linear Inequalities Example B. Shade 2x + 3y > 12.
- 16. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. 2D Linear Inequalities Example B. Shade 2x + 3y > 12.
- 17. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. 2D Linear Inequalities Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. x y 0 0
- 18. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. 2D Linear Inequalities Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. x y 0 4 6 0
- 19. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. 2D Linear Inequalities Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. x y 0 4 6 0 (0, 4) (6, 0)
- 20. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. 2D Linear Inequalities Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. Draw a dotted line because 2x + 3y = 12 is not part of the solution. (Use a solid line for ≤ or ≥ ) x y 0 4 6 0 2x + 3y = 12 (0, 4) (6, 0)
- 21. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. 2D Linear Inequalities Trick: Sample a point on the axes. Use (0, 0) if it’s possible. Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. Draw a dotted line because 2x + 3y = 12 is not part of the solution. (Use a solid line for ≤ or ≥ ) x y 0 4 6 0 2x + 3y = 12 (0, 4) (6, 0)
- 22. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. If the selected point fits inequality then the half-plane containing the test point is the solution. 2D Linear Inequalities Trick: Sample a point on the axes. Use (0, 0) if it’s possible. Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. Draw a dotted line because 2x + 3y = 12 is not part of the solution. (Use a solid line for ≤ or ≥ ) x y 0 4 6 0 2x + 3y = 12 (0, 4) (6, 0)
- 23. In general, to solve a linear inequalities Ax + By > C or Ax + By < C means to identify which side of the line Ax + By = C is the half-plane that is the solution of the inequality in question. To accomplish this, after graphing the line Ax + By = C, sample any point not on the line. If the selected point fits inequality then the half-plane containing the test point is the solution. Otherwise, the other side is the solution the inequality. 2D Linear Inequalities Trick: Sample a point on the axes. Use (0, 0) if it’s possible. Example B. Shade 2x + 3y > 12. Use the intercepts method to graph 2x + 3y = 12. Draw a dotted line because 2x + 3y = 12 is not part of the solution. (Use a solid line for ≤ or ≥ ) x y 0 4 6 0 2x + 3y = 12 (0, 4) (6, 0)
- 24. 2D Linear Inequalities To match the region that fits 2x + 3y > 12, sample (0, 0). 2x + 3y = 12 (0, 4) (6, 0)
- 25. 2D 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 (0, 4) (6, 0)
- 26. 2D 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 (0, 4) (6, 0) 2x + 3y > 12
- 27. 2D 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. If the inequality is > or < , then the boundary is also part of the solution and we draw a solid line to express that. 2x + 3y = 12 (0, 4) (6, 0) 2x + 3y > 12
- 28. 2D 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. To find the region that fits a system of two x&y 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. 2x + 3y = 12 (0, 4) (6, 0) 2x + 3y > 12
- 29. 2D 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. To find the region that fits a system of two x&y 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. 2x + 3y = 12 (0, 4) (6, 0) 2x + 3y > 12
- 30. 2D 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. To find the region that fits a system of two x&y 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. 2x + 3y = 12 (0, 4) (6, 0) 2x + 3y > 12
- 31. 2D 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. To find the region that fits a system of two x&y 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. 2x + 3y = 12 (0, 4) (6, 0) 2x + 3y > 12
- 32. 2D 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. To find the region that fits a system of two x&y 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. 2x + 3y = 12 (0, 4) (6, 0) 2x + 3y > 12
- 33. Example C. Shade the system 2x – y < –2 x + y < 5 Find and label the tip of the region. { 2D Linear Inequalities
- 34. 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. 2D Linear Inequalities
- 35. 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. 2D Linear Inequalities
- 36. 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 2D Linear Inequalities Find the intercepts.
- 37. 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 2D Linear Inequalities Find the intercepts.
- 38. 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 2D Linear Inequalities Find the intercepts.
- 39. 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. 2D Linear Inequalities Find the intercepts.
- 40. 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. 2D Linear Inequalities Find the intercepts.
- 41. 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), 2D Linear Inequalities Find the intercepts.
- 42. 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. 2D Linear Inequalities Find the intercepts.
- 43. 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. 2D Linear Inequalities Find the intercepts.
- 44. 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. 2D Linear Inequalities Find the intercepts.
- 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. 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. 2D Linear Inequalities Find the intercepts.
- 46. For x + y < 5, graph x = y = 5 2D Linear Inequalities
- 47. For x + y < 5, graph x = y = 5 x y 0 0 2D Linear Inequalities
- 48. For x + y < 5, graph x = y = 5 x y 0 5 5 0 2D Linear Inequalities
- 49. For x + y < 5, graph x = y = 5 x y 0 5 5 0 2D Linear Inequalities
- 50. 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. 2D Linear Inequalities
- 51. 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. 2D Linear Inequalities
- 52. 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), 2D Linear Inequalities
- 53. 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. 2D Linear Inequalities
- 54. 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. 2D Linear Inequalities
- 55. 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 them. 2D Linear Inequalities
- 56. 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 them. 2D Linear Inequalities For x + y < 5, graph x = y = 5
- 57. 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 them. The region that fits the system is the region has both shading. 2D Linear Inequalities For x + y < 5, graph x = y = 5
- 58. 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 them. 2D Linear Inequalities For x + y < 5, graph x = y = 5
- 59. 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 them. The region that fits the system is the region has both shading. 2D Linear Inequalities For x + y < 5, graph x = y = 5
- 60. 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 them. The region that fits the system is the region has both shading. 2D 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 their point of intersection.
- 61. 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 them. The region that fits the system is the region has both shading. 2D 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 their point of intersection.
- 62. 2D Linear Inequalities 2x – y = –2 x + y = 5+)
- 63. 2D Linear Inequalities 2x – y = –2 x + y = 5+) 3x = 3
- 64. 2D Linear Inequalities 2x – y = –2 x + y = 5+) 3x = 3 x = 1
- 65. 2D Linear Inequalities Set x = 1 in x + y = 5, 2x – y = –2 x + y = 5+) 3x = 3 x = 1
- 66. 2D 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
- 67. 2D 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) 2x – y < –2 x + y < 5{
- 68. 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 2D 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.