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# 2 6 inequalities

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### 2 6 inequalities

1. 1. Inequalities
2. 2. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. Inequalities
3. 3. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – Inequalities
4. 4. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 Inequalities
5. 5. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ Inequalities
6. 6. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ π  3.14.. Inequalities –π  –3.14..
7. 7. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ π  3.14.. This line with each position addressed by a real number is called the real (number) line. Inequalities –π  –3.14..
8. 8. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ π  3.14.. This line with each position addressed by a real number is called the real (number) line. Inequalities –π  –3.14.. Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
9. 9. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ π  3.14.. This line with each position addressed by a real number is called the real (number) line. Inequalities +– RL –π  –3.14.. Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
10. 10. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ π  3.14.. This line with each position addressed by a real number is called the real (number) line. Inequalities +– R We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line. L < –π  –3.14.. Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
11. 11. We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. -2 20 1 3 + -1-3 – 2/3 2½ π  3.14.. This line with each position addressed by a real number is called the real (number) line. Inequalities +– R We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line. This relation may also be written as R > L (less preferable). L < –π  –3.14.. Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
12. 12. Example A. 2 < 4, –3< –2, 0 > –1 are true statements Inequalities
13. 13. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities
14. 14. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x".
15. 15. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a).
16. 16. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture, +– a open dot a < x
17. 17. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture, +– a open dot If we want all the numbers x greater than or equal to a (including a), we write it as a < x. a < x
18. 18. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture, +– a open dot If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture +– a solid dot a < x a < x
19. 19. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture, +– a open dot If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture +– a solid dot a < x a < x The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.
20. 20. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture, +– a open dot If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture +– a solid dot a < x a < x The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. +– a a < x < b b
21. 21. Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. Inequalities If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture, +– a open dot If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture +– a solid dot a < x a < x The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. A line segment as such is called an interval. +– a a < x < b b
22. 22. Example B. a. Draw –1 < x < 3. Inequalities
23. 23. Example B. a. Draw –1 < x < 3. Inequalities It’s in the natural form.
24. 24. Example B. a. Draw –1 < x < 3. Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly.
25. 25. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly.
26. 26. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly.
27. 27. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0.
28. 28. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
29. 29. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 0 + -3 – x Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
30. 30. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 0 + -3 – x Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Inequalities It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
31. 31. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 0 + -3 – x Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Inequalities Adding or subtracting the same quantity to both retains the inequality sign, It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
32. 32. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 0 + -3 – x Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Inequalities Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c. It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
33. 33. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 0 + -3 – x Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Inequalities For example 6 < 12, then 6 + 3 < 12 + 3. Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c. It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
34. 34. Example B. a. Draw –1 < x < 3. 0 3 + -1 – x b. Draw 0 > x > –3 0 + -3 – x Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Inequalities For example 6 < 12, then 6 + 3 < 12 + 3. We use the this fact to solve inequalities. Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c. It’s in the natural form. Mark the numbers and x on the line in order accordingly. Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
35. 35. Example C. Solve x – 3 < 12 and draw the solution. Inequalities
36. 36. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 Inequalities
37. 37. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 Inequalities
38. 38. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x
39. 39. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c.
40. 40. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign,
41. 41. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b
42. 42. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
43. 43. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true,
44. 44. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12
45. 45. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
46. 46. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. Example D. Solve 3x > 12 and draw the solution. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
47. 47. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. Example D. Solve 3x > 12 and draw the solution. 3x > 12 For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
48. 48. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. Example D. Solve 3x > 12 and draw the solution. 3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
49. 49. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. Example D. Solve 3x > 12 and draw the solution. 3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 x > 4 or 4 < x For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
50. 50. Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15 0 15 +– Inequalities x A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. Example D. Solve 3x > 12 and draw the solution. 3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 x > 4 or 4 < x 40 +– For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true. x
51. 51. A number c is negative means c < 0. Inequalities
52. 52. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, Inequalities
53. 53. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then Inequalities
54. 54. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities
55. 55. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities For example 6 < 12 is true.
56. 56. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true.
57. 57. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
58. 58. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities 60 +– 12< For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
59. 59. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities 60 +– 12–6 < For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
60. 60. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
61. 61. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Example E. Solve –x + 2 < 5 and draw the solution. Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
62. 62. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Example E. Solve –x + 2 < 5 and draw the solution. –x + 2 < 5 Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
63. 63. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Example E. Solve –x + 2 < 5 and draw the solution. –x + 2 < 5 subtract 2 from both sides –x < 3 Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
64. 64. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Example E. Solve –x + 2 < 5 and draw the solution. –x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
65. 65. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Example E. Solve –x + 2 < 5 and draw the solution. –x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 or –3 < x Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
66. 66. A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb . Example E. Solve –x + 2 < 5 and draw the solution. –x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 or –3 < x 0 + -3 – Inequalities 60 +– 12–6–12 << For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
67. 67. To solve inequalities: 1. Simplify both sides of the inequalities Inequalities
68. 68. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides Inequalities
69. 69. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). Inequalities
70. 70. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. Inequalities
71. 71. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. Inequalities
72. 72. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities
73. 73. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9
74. 74. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign
75. 75. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign 3x – x > 9 – 5
76. 76. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign 3x – x > 9 – 5 2x > 4
77. 77. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign 3x – x > 9 – 5 2x > 4 div. 2 2x 2 4 2>
78. 78. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign 3x – x > 9 – 5 2x > 4 div. 2 2x 2 4 2> x > 2 or 2 < x
79. 79. To solve inequalities: 1. Simplify both sides of the inequalities 2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive. Inequalities Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign 3x – x > 9 – 5 2x > 4 div. 2 20 +– 2x 2 4 2> x > 2 or 2 < x
80. 80. Example G. Solve 3(2 – x) > 2(x + 9) – 2x Inequalities
81. 81. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side Inequalities
82. 82. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x Inequalities
83. 83. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 Inequalities
84. 84. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x Inequalities
85. 85. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x Inequalities
86. 86. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x –12 3 3x 3 > div. by 3 (no need to switch >) Inequalities
87. 87. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x –12 3 3x 3 > –4 > x or x < –4 div. by 3 (no need to switch >) Inequalities
88. 88. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x 0 + –12 3 3x 3 > -4 div. by 3 (no need to switch >) Inequalities –4 > x or x < –4
89. 89. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x 0 + –12 3 3x 3 > -4 div. by 3 (no need to switch >) Inequalities We also have inequalities in the form of intervals. –4 > x or x < –4
90. 90. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x 0 + –12 3 3x 3 > -4 div. by 3 (no need to switch >) Inequalities We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. –4 > x or x < –4
91. 91. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x 0 + –12 3 3x 3 > -4 div. by 3 (no need to switch >) Inequalities We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, –4 > x or x < –4
92. 92. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x 0 + –12 3 3x 3 > -4 div. by 3 (no need to switch >) Inequalities We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x. –4 > x or x < –4
93. 93. Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 move 18 and –3x (change sign) 6 – 18 > 3x – 12 > 3x 0 + –12 3 3x 3 > -4 div. by 3 (no need to switch >) Inequalities We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x. The answer is an interval of numbers. –4 > x or x < –4
94. 94. Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw Inequalities
95. 95. Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 Inequalities
96. 96. Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 Inequalities
97. 97. Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 div. by -2, switch inequality sign 6 -2 -2x -2< -10 -2 < Inequalities
98. 98. Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 -3 < x < 5 div. by -2, switch inequality sign 6 -2 -2x -2< -10 -2 < Inequalities
99. 99. Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 0 + -3 < x < 5 5 div. by -2, switch inequality sign 6 -2 -2x -2< -10 -2 < -3 Inequalities
100. 100. Inequalities Exercise. A. Draw the following Inequalities. Indicate clearly whether the end points are included or not. 1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12 B. Write in the natural form then draw them. 5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12 C. Draw the following intervals, state so if it is impossible. 9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 2 13. 6 > x ≥ 8 14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9 D. Solve the following Inequalities and draw the solution. 17. x + 5 < 3 18. –5 ≤ 2x + 3 19. 3x + 1 < –8 20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x 22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 9 24. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1) 26. x + 2(x – 3) < 2(x – 1) – 2 27. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13
101. 101. Inequalities F. Solve the following interval inequalities. 28. –4 ≤ 2 x 29. 7 > 3 –x 30. < –4–x E. Clear the denominator first then solve and draw the solution. 5 x2 3 1 2 3 2 + ≥ x31. x4 –3 3 –4 – 1> x32. x 2 8 3 3 4 5– ≤33. x 8 12 –5 7 1+ >34. x 2 3 –3 2 3 4 4 1–+ x35. x4 6 5 5 3 –1 – 2+ < x36. x 12 2 7 3 6 1 4 3–– ≥ x37. ≤ 40. – 2 < x + 2 < 5 41. –1 ≥ 2x – 3 ≥ –11 42. –5 ≤ 1 – 3x < 10 43. 11 > –1 – 3x > –7 38. –6 ≤ 3x < 12 39. 8 > –2x > –4