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# 2 6 inequalities

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• 1. Inequalities
• 2. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
• 3. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 -2 -1 0 1 2 3 +
• 4. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 -2 -1 0 1 2/3 2 3 +
• 5. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 -2 -1 0 1 2/3 2 3 2½ +
• 6. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 –π -2 –3.14.. -1 0 1 2/3 2 3 2½ π + 3.14..
• 7. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 –π -2 –3.14.. -1 0 1 2/3 2 + 3 2½ π 3.14.. This line with each position addressed by a real number is called the real (number) line.
• 8. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 –π -2 –3.14.. -1 0 1 2/3 2 3 2½ π + 3.14.. This line with each position addressed by a real number is called the real (number) line. 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. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 –π -2 -1 –3.14.. 0 1 2 3 2½ π 2/3 + 3.14.. This line with each position addressed by a real number is called the real (number) line. 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. – L R +
• 10. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 –π -2 -1 0 –3.14.. 1 2 3 2½ π 2/3 + 3.14.. This line with each position addressed by a real number is called the real (number) line. 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. L < R + – We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line.
• 11. Inequalities We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left. – -3 –π -2 -1 0 –3.14.. 1 2 3 2½ π 2/3 + 3.14.. This line with each position addressed by a real number is called the real (number) line. 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. L < 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).
• 12. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements
• 13. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements.
• 14. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x".
• 15. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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 a<x + – open dot
• 17. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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 a<x + – open dot If we want all the numbers x greater than or equal to a (including a), we write it as a < x.
• 18. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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 a<x + – 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 a<x – solid dot +
• 19. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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 a<x + – 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 a<x – + solid dot The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.
• 20. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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 a<x + – 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 a<x – + solid dot 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. Inequalities Example A. 2 < 4, –3< –2, 0 > –1 are true statements and –2 < –5 , 5 < 3 are false statements. 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 a<x + – 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 a<x – + solid dot 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<x<b b a + –
• 22. Example B. a. Draw –1 < x < 3. Inequalities
• 23. Example B. a. Draw –1 < x < 3. It’s in the natural form. Inequalities
• 24. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly.
• 25. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 3
• 26. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 b. Draw 0 > x > –3 0 3
• 27. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. 3
• 28. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. 3
• 29. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. x – -3 0 3 +
• 30. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 3 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. x – -3 0 Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. +
• 31. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 3 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. x – -3 0 Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Adding or subtracting the same quantity to both retains the inequality sign, +
• 32. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 3 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. x – -3 0 Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c. +
• 33. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 3 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. x – -3 0 Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c. For example 6 < 12, then 6 + 3 < 12 + 3. +
• 34. Inequalities Example B. a. Draw –1 < x < 3. It’s in the natural form. Mark the numbers and x on the line in order accordingly. x + – -1 0 3 b. Draw 0 > x > –3 Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly. x – -3 0 Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution. Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c. For example 6 < 12, then 6 + 3 < 12 + 3. We use the this fact to solve inequalities. +
• 35. Inequalities Example C. Solve x – 3 < 12 and draw the solution.
• 36. Inequalities Example C. Solve x – 3 < 12 and draw the solution. x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3
• 37. Inequalities 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
• 38. Inequalities 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 x – 0 15 +
• 39. Inequalities 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 x – 0 A number c is positive means that 0 < c. 15 +
• 40. Inequalities 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 x – 0 15 + 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. Inequalities 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 x – 0 15 + 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. Inequalities 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 x – 0 15 + 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. Inequalities 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 x – 0 15 + 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. Inequalities 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 x – 0 15 + 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. Inequalities 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 x – 0 15 + 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. Inequalities 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 x – 0 15 + 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. Example D. Solve 3x > 12 and draw the solution.
• 47. Inequalities 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 x – 0 15 + 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. Example D. Solve 3x > 12 and draw the solution. 3x > 12
• 48. Inequalities 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 x – 0 15 + 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. Example D. Solve 3x > 12 and draw the solution. 3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3
• 49. Inequalities 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 x – 0 15 + 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. 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
• 50. Inequalities 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 x – 0 15 + 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. 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 x – 0 4 +
• 51. Inequalities A number c is negative means c < 0.
• 52. Inequalities A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign,
• 53. Inequalities 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 .
• 54. Inequalities 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 . For example 6 < 12 is true.
• 55. Inequalities 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 . For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true.
• 56. Inequalities 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 . 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.
• 57. Inequalities 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 . 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. – 0 6 < 12 +
• 58. Inequalities 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 . 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. – –6 0 6 < 12 +
• 59. Inequalities 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 . 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. – –12 < –6 0 6 < 12 +
• 60. Inequalities 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 . 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. – 0 < –6 < 12 6 Example E. Solve –x + 2 < 5 and draw the solution. –12 +
• 61. Inequalities 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 . 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. – 0 < –6 < 12 6 Example E. Solve –x + 2 < 5 and draw the solution. –12 –x + 2 < 5 +
• 62. Inequalities 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 . 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. – 0 < –6 < 12 6 Example E. Solve –x + 2 < 5 and draw the solution. –12 –x + 2 < 5 –x < 3 subtract 2 from both sides +
• 63. Inequalities 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 . 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. – 0 < –6 < 12 6 Example E. Solve –x + 2 < 5 and draw the solution. –12 –x + 2 < 5 –x < 3 –(–x) > –3 x > –3 + subtract 2 from both sides multiply by –1 to get x, reverse the inequality
• 64. Inequalities 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 . 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. – 0 < –6 < 12 6 Example E. Solve –x + 2 < 5 and draw the solution. –12 + –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
• 65. Inequalities 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 . 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. – 0 < –6 < 12 6 Example E. Solve –x + 2 < 5 and draw the solution. –12 + –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
• 66. Inequalities To solve inequalities: 1. Simplify both sides of the inequalities
• 67. Inequalities 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
• 68. Inequalities 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).
• 69. Inequalities 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.
• 70. Inequalities 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.
• 71. Inequalities 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.
• 72. Inequalities 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. Example F. Solve 3x + 5 > x + 9
• 73. Inequalities 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. Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 move the x and 5, change side-change sign
• 74. Inequalities 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. Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 3x – x > 9 – 5 move the x and 5, change side-change sign
• 75. Inequalities 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. Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 3x – x > 9 – 5 2x > 4 move the x and 5, change side-change sign
• 76. Inequalities 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. Example F. Solve 3x + 5 > x + 9 3x + 5 > x + 9 3x – x > 9 – 5 2x > 4 2x 4 2 > 2 move the x and 5, change side-change sign div. 2
• 77. Inequalities 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. 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 4 2 > 2 x > 2 or 2 < x
• 78. Inequalities 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. 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 4 2 > 2 x > 2 or 2 < x + – 0 2
• 79. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x
• 80. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x simplify each side
• 81. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x simplify each side
• 82. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 simplify each side
• 83. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x simplify each side move 18 and –3x (change sign)
• 84. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign)
• 85. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x –12 > 3x 3 3 simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >)
• 86. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x –12 > 3x 3 3 –4 > x or x < –4 simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >)
• 87. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >) –12 > 3x 3 3 –4 > x or x < –4 -4 0 +
• 88. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >) –12 > 3x 3 3 –4 > x or x < –4 -4 0 + We also have inequalities in the form of intervals.
• 89. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >) –12 > 3x 3 3 –4 > x or x < –4 -4 0 + We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities.
• 90. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >) –12 > 3x 3 3 –4 > x or x < –4 -4 0 + 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,
• 91. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >) –12 > 3x 3 3 –4 > x or x < –4 -4 0 + 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.
• 92. Inequalities Example G. Solve 3(2 – x) > 2(x + 9) – 2x 3(2 – x) > 2(x + 9) – 2x 6 – 3x > 2x + 18 – 2x 6 – 3x > 18 6 – 18 > 3x – 12 > 3x simplify each side move 18 and –3x (change sign) div. by 3 (no need to switch >) –12 > 3x 3 3 –4 > x or x < –4 -4 0 + 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.
• 93. Inequalities Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw
• 94. Inequalities Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 –6 –6 –6 subtract 6
• 95. Inequalities Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 –6 –6 –6 6 > –2x > –10 subtract 6
• 96. Inequalities Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 –6 –6 –6 6 > –2x > –10 -2x -10 6 < < -2 -2 -2 subtract 6 div. by -2, switch inequality sign
• 97. Inequalities Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 –6 –6 –6 6 > –2x > –10 -2x -10 6 < < -2 -2 -2 -3 < x < 5 subtract 6 div. by -2, switch inequality sign
• 98. Inequalities Example H. (Interval Inequality) Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 –6 –6 –6 6 > –2x > –10 -2x -10 6 < < -2 -2 -2 -3 < x < 5 -3 0 subtract 6 div. by -2, switch inequality sign 5 +
• 99. 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
• 100. Inequalities E. Clear the denominator first then solve and draw the solution. 28. –4 ≤ x 29. 7 > –x 30. –x < –4 2 3 5 2 2 31. 1 x + 3 ≥ 3 x 32. –3 x > –4 x – 1 2 4 3 7 33. 3 x – 3 ≤ 5 34. –5 x + 12 > 1 2 8 4 8 2 4 5 35. –3 x + 3 ≤ 3 x – 1 36. 4 x + 5 < –1 x – 2 2 4 6 3 7 37. 12 x – 3 ≥ 1 x – 3 2 6 4 F. Solve the following interval inequalities. 38. –6 ≤ 3x < 12 39. 8 > –2x > –4 41. –1 ≥ 2x – 3 ≥ –11 40. – 2 < x + 2 < 5 43. 11 > –1 – 3x > –7 42. –5 ≤ 1 – 3x < 10