Solving Inequalities Notes

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  • Solving Inequalities Notes

    1. 1. Solving Inequalities
    2. 2. The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?
    3. 3. The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Think about how to solve this problem.
    4. 4. The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Think about how to solve this problem.• What number did you come up with?
    5. 5. The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Think about how to solve this problem.• What number did you come up with?• Ask yourself if the number makes sense in the context of this problem.
    6. 6. The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Think about how to solve this problem.• What number did you come up with?• Ask yourself if the number makes sense in the context of this problem.• How did you solve this problem?
    7. 7. The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Think about how to solve this problem.• What number did you come up with?• Ask yourself if the number makes sense in the context of this problem.• How did you solve this problem?• Keep your answers and we’ll return to this problem later.
    8. 8. Recall Solving Equations 2x − 7 = 11
    9. 9. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable.
    10. 10. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable.• Undo subtraction (or addition) first using the opposite operation.
    11. 11. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation.
    12. 12. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18
    13. 13. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18• Then undo multiplication (or division) using the opposite operation.
    14. 14. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18• Then undo multiplication (or division) 2 2 using the opposite operation.
    15. 15. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18• Then undo multiplication (or division) 2 2 using the opposite operation. x=9
    16. 16. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18• Then undo multiplication (or division) 2 2 using the opposite operation. x=9• The coefficient of x is 1. Check your solution in the original problem.
    17. 17. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18• Then undo multiplication (or division) 2 2 using the opposite operation. x=9• The coefficient of x is 1. Check your solution in the original problem. Check: ? 2 ⋅ 9 − 7 = 11 11 = 11
    18. 18. Recall Solving Equations• Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7• Undo subtraction (or addition) first using the opposite operation. 2x = 18• Then undo multiplication (or division) 2 2 using the opposite operation. x=9• The coefficient of x is 1. Check your solution in the original problem. Check: ?• It works so your solution is correct. 2 ⋅ 9 − 7 = 11 11 = 11
    19. 19. Solving Inequality - Example 1 2x − 7 > 11
    20. 20. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11
    21. 21. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the opposite operation.
    22. 22. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation.
    23. 23. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18
    24. 24. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18• Then undo multiplication (or division) using the opposite operation.
    25. 25. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18• Then undo multiplication (or division) using 2 2 the opposite operation.
    26. 26. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18• Then undo multiplication (or division) using 2 2 the opposite operation. x>9
    27. 27. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18• Then undo multiplication (or division) using 2 2 the opposite operation.• The coefficient of x is 1. x>9
    28. 28. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18• Then undo multiplication (or division) using 2 2 the opposite operation.• The coefficient of x is 1. x>9• Steps were the same!
    29. 29. Solving Inequality - Example 1• Use the properties of equality to isolate the variable. 2x − 7 > 11• Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18• Then undo multiplication (or division) using 2 2 the opposite operation.• The coefficient of x is 1. x>9• Steps were the same!• Final step is to check the solution.
    30. 30. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10
    31. 31. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.
    32. 32. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality.
    33. 33. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality.• Shade to the right because 10 > 9.
    34. 34. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality.• Shade to the right because 10 > 9.• Check the solution in the ORIGINAL problem.
    35. 35. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality. Original Problem:• Shade to the right because 10 > 9. 2x − 7 > 11• Check the solution in the ORIGINAL problem.
    36. 36. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality. Original Problem:• Shade to the right because 10 > 9. 2x − 7 > 11• Check the solution in the ORIGINAL problem.• Check the solution 10.
    37. 37. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality. Original Problem:• Shade to the right because 10 > 9. 2x − 7 > 11• Check the solution in the ORIGINAL Check: problem. ? 2 ⋅10 − 7 > 11• Check the solution 10. 13 > 11
    38. 38. Example 1 (con’t)• Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10• Open circle at 9 because >, not ≥.• Always shade in the direction of a number that works in the inequality. Original Problem:• Shade to the right because 10 > 9. 2x − 7 > 11• Check the solution in the ORIGINAL Check: problem. ? 2 ⋅10 − 7 > 11• Check the solution 10. 13 > 11• This is true and therefore a solution.
    39. 39. Example 1 (con’t)• It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 Original Problem: 2x − 7 > 11 Check:
    40. 40. Example 1 (con’t)• It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10• Pick a number that is not a solution and the number where the circle is placed. Original Problem: 2x − 7 > 11 Check:
    41. 41. Example 1 (con’t)• It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10• Pick a number that is not a solution and the number where the circle is placed. Original Problem:• 0 is not a solution. 2x − 7 > 11 Check:
    42. 42. Example 1 (con’t)• It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10• Pick a number that is not a solution and the number where the circle is placed. Original Problem:• 0 is not a solution. 2x − 7 > 11• The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? 2 ⋅ 0 − 7 > 11 −7 > 11 x
    43. 43. Example 1 (con’t)• It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10• Pick a number that is not a solution and the number where the circle is placed. Original Problem:• 0 is not a solution. 2x − 7 > 11• The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ?• 9 is not a solution because of the open 2 ⋅ 0 − 7 > 11 circle. −7 > 11 x
    44. 44. Example 1 (con’t)• It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10• Pick a number that is not a solution and the number where the circle is placed. Original Problem:• 0 is not a solution. 2x − 7 > 11• The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ?• 9 is not a solution because of the open 2 ⋅ 0 − 7 > 11 circle. −7 > 11 x ?• 11 is not greater than 11. Again this is 2 ⋅ 9 − 7 > 11 good because 9 is not a solution. 11 > 11 x
    45. 45. Solving Inequality Recap
    46. 46. Solving Inequality Recap1) Solve the inequality as you would an equation.
    47. 47. Solving Inequality Recap1) Solve the inequality as you would an equation.2) Graph the solution on a number line.
    48. 48. Solving Inequality Recap1) Solve the inequality as you would an equation.2) Graph the solution on a number line.3) Check at least 3 numbers in the original inequality.
    49. 49. Solving Inequality Recap1) Solve the inequality as you would an equation.2) Graph the solution on a number line.3) Check at least 3 numbers in the original inequality.4) Write the solution.
    50. 50. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph.• Check solution in original inequality.• -5 is NOT greater than -3!• Why didn’t it work?• Negative means reverse. When dividing by a negative, don’t we reverse all the signs?• The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
    51. 51. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3!• Why didn’t it work?• Negative means reverse. When dividing by a negative, don’t we reverse all the signs?• The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
    52. 52. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4• Why didn’t it work?• Negative means reverse. When dividing by a negative, don’t we reverse all the signs?• The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
    53. 53. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4• Why didn’t it work? -2 -2• Negative means reverse. When dividing by a negative, don’t we reverse all the signs?• The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
    54. 54. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4• Why didn’t it work? -2 -2• Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the signs?• The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
    55. 55. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1 −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
    56. 56. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1 −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
    57. 57. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality. −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
    58. 58. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality. −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3
    59. 59. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
    60. 60. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4• Why didn’t it work? -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
    61. 61. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4• Why didn’t it work? -2 -2• Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the -5 -4 -3 -2 -1 0 1 2 3 4 5 signs? Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
    62. 62. Example 2 - What’s wrong here...• Solve for x. −2x + 1 > −3• Graph. −1 −1• Check solution in original inequality.• -5 is NOT greater than -3! −2x > −4• Why didn’t it work? -2 -2• Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the -5 -4 -3 -2 -1 0 1 2 3 4 5 signs? Check: ?• The same goes for the inequality. When −2 ⋅ 3 + 1>− 3 dividing (or multiplying) by a negative, the inequality must reverse. −5 > −3 x
    63. 63. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2.• Graph.• Check solution in the original inequality.• 1 is greater than -3!• Write the solution.
    64. 64. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph.• Check solution in the original inequality.• 1 is greater than -3!• Write the solution.
    65. 65. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4• Check solution in the original inequality.• 1 is greater than -3!• Write the solution.
    66. 66. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original inequality.• 1 is greater than -3!• Write the solution.
    67. 67. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original x<2 inequality.• 1 is greater than -3!• Write the solution.
    68. 68. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2 x<2
    69. 69. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2 x<2 -5 -4 -3 -2 -1 0 1 2 3 4 5
    70. 70. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5
    71. 71. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 0 + 1>− 3 1 > −3
    72. 72. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5• 1 is greater than -3! Check: ? −2 ⋅ 0 + 1>− 3 1 > −3
    73. 73. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5• 1 is greater than -3! Check: ?• Write the solution. −2 ⋅ 0 + 1>− 3 1 > −3
    74. 74. Example 2 (con’t)• Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1• Graph. −2x > −4 -2 -2• Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5• 1 is greater than -3! Check: ?• Write the solution. −2 ⋅ 0 + 1>− 3 {x | x < 2} 1 > −3
    75. 75. Equation vs Inequality
    76. 76. Equation vs Inequality• What are the similarities between solving an Equation and solving an Inequality?
    77. 77. Equation vs Inequality• What are the similarities between solving an Equation and solving an Inequality? ✓ Solve using the same process.
    78. 78. Equation vs Inequality• What are the similarities between solving an Equation and solving an Inequality? ✓ Solve using the same process. ✓ Always check solution.
    79. 79. Equation vs Inequality
    80. 80. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality?
    81. 81. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses.
    82. 82. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to >
    83. 83. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to <
    84. 84. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥
    85. 85. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥ ‣ ≥ changes to ≤
    86. 86. Equation vs Inequality• What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥ ‣ ≥ changes to ≤ ✓ Check multiple numbers to include solutions and non- solutions.
    87. 87. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13
    88. 88. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11
    89. 89. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24
    90. 90. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24 8 8
    91. 91. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24 8 8 x≤3
    92. 92. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 8 8 x≤3
    93. 93. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 8 8 x≤3
    94. 94. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: 8 8 x≤3
    95. 95. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? 8 8 8 ⋅ 0 − 11≤ 13 −11 ≤ 13 x≤3
    96. 96. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 29 ≤ 13 x
    97. 97. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 ? 29 ≤ 13 x 8 ⋅ 3 − 11≤ 13 13 ≤ 13
    98. 98. Example 3 - You try...• Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 ? 29 ≤ 13 x 8 ⋅ 3 − 11≤ 13 13 ≤ 13 {x | x ≤ 3}
    99. 99. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5
    100. 100. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10
    101. 101. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15
    102. 102. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15 −3 −3
    103. 103. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15 −3 −3 x<5
    104. 104. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 −3 −3 x<5
    105. 105. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 −3 −3 x<5
    106. 106. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: −3 −3 x<5
    107. 107. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? −3 −3 −3 ⋅ 0 + 10 >− 5 10 > −5 x<5
    108. 108. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x
    109. 109. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x ? −3 ⋅ 5 + 10 >− 5 −5 > −5 x
    110. 110. Example 4 - You try...• Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x ? −3 ⋅ 5 + 10 >− 5 {x | x < 5} −5 > −5 x
    111. 111. Graphing Caution!• Did you notice that the shading always went in the direction of the inequality?
    112. 112. Graphing Caution!• Did you notice that the shading always went in the direction of the inequality?• Graph 1 > x
    113. 113. Graphing Caution!• Did you notice that the shading always went in the direction of the inequality?• Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5
    114. 114. Graphing Caution!• Did you notice that the shading always went in the direction of the inequality?• Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5• Notice the shading is opposite the direction of the inequality. This is because x is on the right of the symbol instead of the left.
    115. 115. Graphing Caution!• Did you notice that the shading always went in the direction of the inequality?• Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5• Notice the shading is opposite the direction of the inequality. This is because x is on the right of the symbol instead of the left.• Always check numbers when determining shading!
    116. 116. What if parenthesis and/orvariables on both sides of theinequality?
    117. 117. What if parenthesis and/orvariables on both sides of theinequality?• Solve the same as you would if it was an equation.
    118. 118. What if parenthesis and/orvariables on both sides of theinequality?• Solve the same as you would if it was an equation.• First, remove parenthesis.
    119. 119. What if parenthesis and/orvariables on both sides of theinequality?• Solve the same as you would if it was an equation.• First, remove parenthesis.• Second, combine like terms on the same side of the inequality (if any).
    120. 120. What if parenthesis and/orvariables on both sides of theinequality?• Solve the same as you would if it was an equation.• First, remove parenthesis.• Second, combine like terms on the same side of the inequality (if any).• Third, get variables on the same side of inequality.
    121. 121. What if parenthesis and/orvariables on both sides of theinequality?• Solve the same as you would if it was an equation.• First, remove parenthesis.• Second, combine like terms on the same side of the inequality (if any).• Third, get variables on the same side of inequality.• Finally, solve for the variable.
    122. 122. Example 4 - 1− ( 4 + 3x ) < −3 − 5x
    123. 123. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x
    124. 124. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x 1− 4 − 3x < −3 − 5x
    125. 125. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left.
    126. 126. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x
    127. 127. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side.
    128. 128. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x
    129. 129. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x.
    130. 130. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3
    131. 131. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3 +3 +3
    132. 132. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3 +3 +3 2x < 0
    133. 133. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3 +3 +3 2x < 0 2 2
    134. 134. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3 +3 +3 2x < 0 2 2 x<0
    135. 135. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3 +3 +3• Graph solution. 2x < 0 2 2 x<0
    136. 136. Example 4 -• Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x• Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x• Move variables on one side. +5x +5x• Solve for x. −3 + 2x < −3 +3 +3• Graph solution. 2x < 0 2 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x<0
    137. 137. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5• The solution checks out.
    138. 138. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 Check:• The solution checks out.
    139. 139. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check:1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− 1 < −3 + 5 0<2• The solution checks out.
    140. 140. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ?1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 −6 < −8 x• The solution checks out.
    141. 141. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ?1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x• The solution checks out.
    142. 142. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ?1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x• The solution checks out.
    143. 143. Example 4 Continued...• Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ?1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x• The solution checks out. {x | x < 0}
    144. 144. Example 5 - Solve 1 3(x − 5) ≥ ( 6x + 8 ) 2
    145. 145. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) 2
    146. 146. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) 2 3x − 15 ≥ 3x + 4
    147. 147. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2 3x − 15 ≥ 3x + 4
    148. 148. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2 3x − 15 ≥ 3x + 4 −3x −3x
    149. 149. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2 3x − 15 ≥ 3x + 4 −3x −3x −15 ≥ 4
    150. 150. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2• What happen to x? 3x − 15 ≥ 3x + 4 −3x −3x −15 ≥ 4
    151. 151. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2• What happen to x? 3x − 15 ≥ 3x + 4• Is the resulting inequality −3x −3x true? −15 ≥ 4
    152. 152. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2• What happen to x? 3x − 15 ≥ 3x + 4• Is the resulting inequality −3x −3x true? −15 ≥ 4• Because the result is a false statement, there is no solution to this inequality.
    153. 153. Example 5 - Solve• Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 )• Move variables to one side. 2• What happen to x? 3x − 15 ≥ 3x + 4• Is the resulting inequality −3x −3x true? −15 ≥ 4• Because the result is a false statement, there is no No Solution solution to this inequality.
    154. 154. Example 6 - Solve 2(5x + 4) ≥ 10x − 3
    155. 155. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3
    156. 156. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3 10x + 8 ≥ 10x − 3
    157. 157. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3• Move variables to one side. 10x + 8 ≥ 10x − 3
    158. 158. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3• Move variables to one side. 10x + 8 ≥ 10x − 3 −10x −10x
    159. 159. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3• Move variables to one side. 10x + 8 ≥ 10x − 3 −10x −10x 8 ≥ −3
    160. 160. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3• Move variables to one side.• The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x 8 ≥ −3
    161. 161. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3• Move variables to one side.• The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x• Yes! Because the result is a true statement, any number can be 8 ≥ −3 substituted into the original inequality.
    162. 162. Example 6 - Solve• Remove parenthesis. 2(5x + 4) ≥ 10x − 3• Move variables to one side.• The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x• Yes! Because the result is a true statement, any number can be 8 ≥ −3 substituted into the original inequality. All Real Numbers• The graph is the entire number line so the solution is All Real -5 -4 -3 -2 -1 0 1 2 3 4 5 numbers.
    163. 163. Example 7 - The local bank in Mathville requires aminimum balance of $1000. If Josh knows he must writechecks for $525 for rent and $185 for utilities, how muchshould he have in his account before writing the checks?
    164. 164. Example 7 - The local bank in Mathville requires aminimum balance of $1000. If Josh knows he must writechecks for $525 for rent and $185 for utilities, how muchshould he have in his account before writing the checks?• Remember this problem?
    165. 165. Example 7 - The local bank in Mathville requires aminimum balance of $1000. If Josh knows he must writechecks for $525 for rent and $185 for utilities, how muchshould he have in his account before writing the checks?• Remember this problem?• Would $2000 be enough?
    166. 166. Example 7 - The local bank in Mathville requires aminimum balance of $1000. If Josh knows he must writechecks for $525 for rent and $185 for utilities, how muchshould he have in his account before writing the checks?• Remember this problem?• Would $2000 be enough?• How about $710?
    167. 167. Example 7 - The local bank in Mathville requires aminimum balance of $1000. If Josh knows he must writechecks for $525 for rent and $185 for utilities, how muchshould he have in his account before writing the checks?• Remember this problem?• Would $2000 be enough?• How about $710?• What about $1710?
    168. 168. Example 7 - The local bank in Mathville requires aminimum balance of $1000. If Josh knows he must writechecks for $525 for rent and $185 for utilities, how muchshould he have in his account before writing the checks?• Remember this problem?• Would $2000 be enough?• How about $710?• What about $1710?• This situation can be modeled with an Inequality. We would use an Inequality instead of an Equation because an exact amount is not needed. There is a range of acceptable values.
    169. 169. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Let A = Josh’s account balance before writing the checks.
    170. 170. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Let A = Josh’s account balance before writing the checks.• The minimum balance is $1000 so this is the least amount allowed in the account.
    171. 171. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Let A = Josh’s account balance before writing the checks.• The minimum balance is $1000 so this is the least amount allowed in the account.• Checks are deducted from the account balance. What operation shows a deduction?
    172. 172. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Let A = Josh’s account balance before writing the checks.• The minimum balance is $1000 so this is the least amount allowed in the account.• Checks are deducted from the account balance. What operation shows a deduction? • Subtraction!
    173. 173. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?• Let A = Josh’s account balance before writing the checks.• The minimum balance is $1000 so this is the least amount allowed in the account.• Checks are deducted from the account balance. What operation shows a deduction? • Subtraction! • $525 and $185 need subtracted from the account balance.
    174. 174. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?Account balance - (sum of checks written) ≥ Min Balance
    175. 175. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000
    176. 176. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000
    177. 177. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710
    178. 178. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710 A ≥ 1710
    179. 179. (Example 7 - continued)The local bank in Mathville requires a minimum balanceof $1000. If Josh knows he must write checks for $525for rent and $185 for utilities, how much should he havein his account before writing the checks?Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710 A ≥ 1710Josh must have at least $1710 in his checkingaccount before writing the checks.
    180. 180. Example 8 - Jim has several rectangular spaces that needenclosed by a fence. He has 96 feet of fencing but not surewhich space to enclose. All the spaces have lengths twiceas long as the width. Write an inequality and solve so Jimcan determine the dimensions the fencing will cover.
    181. 181. Example 8 - Jim has several rectangular spaces that needenclosed by a fence. He has 96 feet of fencing but not surewhich space to enclose. All the spaces have lengths twiceas long as the width. Write an inequality and solve so Jimcan determine the dimensions the fencing will cover.• Let w = width
    182. 182. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.• Let w = width• length = twice as long as the width = 2w.
    183. 183. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.• Let w = width• length = twice as long as the width = 2w.• Enclosing a space refers to perimeter, the distance around the space.
    184. 184. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.• Let w = width• length = twice as long as the width = 2w.• Enclosing a space refers to perimeter, the distance around the space.• This space is rectangular. Opposite sides have the same measure.
    185. 185. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.• Let w = width• length = twice as long as the width = 2w.• Enclosing a space refers to perimeter, the distance around the space.• This space is rectangular. Opposite sides have the same measure.• Perimeter, P, of a rectangle is P = 2l + 2w, where l is length and w is width.
    186. 186. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.• Let w = width• length = twice as long as the width = 2w.• Enclosing a space refers to perimeter, the distance around the space.• This space is rectangular. Opposite sides have the same measure.• Perimeter, P, of a rectangle is P = 2l + 2w, where l is length and w is width.• Because Jim has 96 feet of fence, this is the maximum perimeter.
    187. 187. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w
    188. 188. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w
    189. 189. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w
    190. 190. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w
    191. 191. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6
    192. 192. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6 16 ≥ w
    193. 193. (Example 8 - continued)Jim has several rectangular spaces that need enclosed by a fence.He has 96 feet of fencing but not sure which space to enclose. Allthe spaces have lengths twice as long as the width. Write aninequality and solve so Jim can determine the dimensions thefencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6 16 ≥ wJim can fence a space with a width less than or equalto 16 feet.

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