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Solving Inequalities
Solving Inequalities  Vocabulary 1) set-builder notation 2) interval notation <ul><li>Solve inequalities. </li></ul>
For any two real numbers,  a   and  b ,  exactly one  of the following statements is true . Solving Inequalities
For any two real numbers,  a   and  b ,  exactly one  of the following statements is true . Solving Inequalities
For any two real numbers,  a   and  b ,  exactly one  of the following statements is true . Solving Inequalities
For any two real numbers,  a   and  b ,  exactly one  of the following statements is true . Solving Inequalities
For any two real numbers,  a   and  b ,  exactly one  of the following statements is true . This is known as the  Trichoto...
For any two real numbers,  a   and  b ,  exactly one  of the following statements is true . This is known as the  Trichoto...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of th...
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities  a b
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities  a-c b-c a b c
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities  a-c b-c a b c
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities  a-c b-c a b
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities  a-c b-c a b
Subtraction  Property of Inequality  For any real numbers  a ,  b , and  c : Solving Inequalities  a-c b-c a b
Solving Inequalities  a
Solving Inequalities  a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. a a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. a a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. We use a closed...
Solving Inequalities  a
Solving Inequalities  a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. a a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. a a
Solving Inequalities  We use an open circle (dot) to indicate that  a   is  NOT  part of the solution set. We use a closed...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Solvi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Howev...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multi...
Division  Property of Inequality  Most books run us through the “rules” for division.  Why is this  not  necessary? Solvin...
Division  Property of Inequality  Most books run us through the “rules” for division.  Why is this  not  necessary? Solvin...
Division  Property of Inequality  Most books run us through the “rules” for division.  Why is this  not  necessary? Solvin...
Division  Property of Inequality  Most books run us through the “rules” for division.  Why is this  not  necessary? So,  s...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  4
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  4 set-build...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  Read:  {   ...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  Read:  {   ...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  Read:  {   ...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  -7
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  -7 set-buil...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  Read:  {   ...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  Read:  {   ...
The solution set of an inequality can also be described by using  set-builder notation . Solving Inequalities  Read:  {   ...
The solution set of an inequality can also be described by using  interval notation . Solving Inequalities
The solution set of an inequality can also be described by using  interval notation . Solving Inequalities  The infinity s...
The solution set of an inequality can also be described by using  interval notation . To indicate that an endpoint is  not...
The solution set of an inequality can also be described by using  interval notation . To indicate that an endpoint is  not...
The solution set of an inequality can also be described by using  interval notation . To indicate that an endpoint is  not...
The solution set of an inequality can also be described by using  interval notation . To indicate that an endpoint is  not...
The solution set of an inequality can also be described by using  interval notation . To indicate that an endpoint is  not...
The solution set of an inequality can also be described by using  interval notation . To indicate that an endpoint is  not...
End  of  Lesson Solving Inequalities
Credits  PowerPoint created by Using Glencoe’s Algebra 2 text, © 2005 Robert Fant http://robertfant.com
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Solving Inequalities (Algebra 2)

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Students learn to solve Linear Inequalities, and learn to use set-builder and interval notation.

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Transcript of "Solving Inequalities (Algebra 2)"

  1. 1. Solving Inequalities
  2. 2. Solving Inequalities Vocabulary 1) set-builder notation 2) interval notation <ul><li>Solve inequalities. </li></ul>
  3. 3. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities
  4. 4. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities
  5. 5. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities
  6. 6. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities
  7. 7. For any two real numbers, a and b , exactly one of the following statements is true . This is known as the Trichotomy Property Solving Inequalities
  8. 8. For any two real numbers, a and b , exactly one of the following statements is true . This is known as the Trichotomy Property or the property of order . Solving Inequalities
  9. 9. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Solving Inequalities
  10. 10. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities
  11. 11. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b
  12. 12. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c c
  13. 13. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c c
  14. 14. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c
  15. 15. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c
  16. 16. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c
  17. 17. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities
  18. 18. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b
  19. 19. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b c
  20. 20. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b c
  21. 21. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b
  22. 22. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b
  23. 23. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b
  24. 24. Solving Inequalities a
  25. 25. Solving Inequalities a
  26. 26. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a
  27. 27. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a
  28. 28. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a
  29. 29. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. We use a closed circle (dot) to indicate that a IS part of the solution set. a a
  30. 30. Solving Inequalities a
  31. 31. Solving Inequalities a
  32. 32. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a
  33. 33. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a
  34. 34. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a
  35. 35. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. We use a closed circle (dot) to indicate that a IS part of the solution set. a a
  36. 36. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Solving Inequalities
  37. 37. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities
  38. 38. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities
  39. 39. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:
  40. 40. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:
  41. 41. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:
  42. 42. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:
  43. 43. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:
  44. 44. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:
  45. 45. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities
  46. 46. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities HINT: is the same as
  47. 47. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities HINT: is the same as
  48. 48. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? So, see rules for multiplication! Solving Inequalities HINT: is the same as
  49. 49. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities 4
  50. 50. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities 4 set-builder notation
  51. 51. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation
  52. 52. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation Identify the variable used
  53. 53. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation Identify the variable used Describe the limitations or boundary of the variable
  54. 54. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities -7
  55. 55. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities -7 set-builder notation
  56. 56. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } -7 set-builder notation
  57. 57. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } Identify the variable used -7 set-builder notation
  58. 58. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } Identify the variable used Describe the limitations or boundary of the variable -7 set-builder notation
  59. 59. The solution set of an inequality can also be described by using interval notation . Solving Inequalities
  60. 60. The solution set of an inequality can also be described by using interval notation . Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively.
  61. 61. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively.
  62. 62. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4
  63. 63. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation
  64. 64. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation
  65. 65. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation -7
  66. 66. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation -7 interval notation
  67. 67. End of Lesson Solving Inequalities
  68. 68. Credits PowerPoint created by Using Glencoe’s Algebra 2 text, © 2005 Robert Fant http://robertfant.com
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