1. Solving Inequalities

2. The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?

3. The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?
• Think about how to solve this problem.

4. The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?
• Think about how to solve this problem.
• What number did you come up with?

5. The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. Recall Solving Equations
2x − 7 = 11

9. Recall Solving Equations
• Use the properties of equality to
2x − 7 = 11
isolate the variable.

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. 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. 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. 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. 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. 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. 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. 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. 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. Solving Inequality - Example 1
2x − 7 > 11

20. Solving Inequality - Example 1
• Use the properties of equality to isolate the
variable.
2x − 7 > 11

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Solving Inequality Recap

46. Solving Inequality Recap
1) Solve the inequality as you would an
equation.

47. Solving Inequality Recap
1) Solve the inequality as you would an
equation.
2) Graph the solution on a number line.

48. Solving Inequality Recap
1) 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. Solving Inequality Recap
1) 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Equation vs Inequality

76. Equation vs Inequality
• What are the similarities between solving an
Equation and solving an Inequality?

77. Equation vs Inequality
• What are the similarities between solving an
Equation and solving an Inequality?
✓ Solve using the same process.

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. Equation vs Inequality

80. Equation vs Inequality
• What are the differences between solving an Equation and
solving an Inequality?

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. 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. 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. 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. 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. 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. Example 3 - You try...
• Solve, graph, check, and write your solution to
8x − 11 ≤ 13

88. Example 3 - You try...
• Solve, graph, check, and write your solution to
8x − 11 ≤ 13
+11 +11

89. Example 3 - You try...
• Solve, graph, check, and write your solution to
8x − 11 ≤ 13
+11 +11
8x ≤ 24

90. Example 3 - You try...
• Solve, graph, check, and write your solution to
8x − 11 ≤ 13
+11 +11
8x ≤ 24
8 8

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. 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. 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. 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. 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. 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. 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. 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. Example 4 - You try...
• Solve, graph, check, and write your solution to
−3x + 10 > −5

100. Example 4 - You try...
• Solve, graph, check, and write your solution to
−3x + 10 > −5
−10 −10

101. Example 4 - You try...
• Solve, graph, check, and write your solution to
−3x + 10 > −5
−10 −10
−3x > −15

102. Example 4 - You try...
• Solve, graph, check, and write your solution to
−3x + 10 > −5
−10 −10
−3x > −15
−3 −3

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. 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. 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. 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. 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. 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. 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. 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. Graphing Caution!
• Did you notice that the shading always went in the
direction of the inequality?

112. Graphing Caution!
• Did you notice that the shading always went in the
direction of the inequality?
• Graph 1 > x

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. 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. 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. What if parenthesis and/or
variables on both sides of the
inequality?

117. What if parenthesis and/or
variables on both sides of the
inequality?
• Solve the same as you would if it was an equation.

118. What if parenthesis and/or
variables on both sides of the
inequality?
• Solve the same as you would if it was an equation.
• First, remove parenthesis.

119. What if parenthesis and/or
variables on both sides of the
inequality?
• 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. What if parenthesis and/or
variables on both sides of the
inequality?
• 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. What if parenthesis and/or
variables on both sides of the
inequality?
• 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. Example 4 -
1− ( 4 + 3x ) < −3 − 5x

123. Example 4 -
• Remove parenthesis.
1− ( 4 + 3x ) < −3 − 5x

124. Example 4 -
• Remove parenthesis.
1− ( 4 + 3x ) < −3 − 5x
1− 4 − 3x < −3 − 5x

125. Example 4 -
• Remove parenthesis.
1− ( 4 + 3x ) < −3 − 5x
• Combine like terms on the 1− 4 − 3x < −3 − 5x
left.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 5 - Solve
1
3(x − 5) ≥ ( 6x + 8 )
2

145. Example 5 - Solve
• Remove parenthesis.
1
3(x − 5) ≥ ( 6x + 8 )
2

146. Example 5 - Solve
• Remove parenthesis.
1
3(x − 5) ≥ ( 6x + 8 )
2
3x − 15 ≥ 3x + 4

147. Example 5 - Solve
• Remove parenthesis.
1
3(x − 5) ≥ ( 6x + 8 )
• Move variables to one side. 2
3x − 15 ≥ 3x + 4

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. 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. 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. 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. 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. 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. Example 6 - Solve
2(5x + 4) ≥ 10x − 3

155. Example 6 - Solve
• Remove parenthesis.
2(5x + 4) ≥ 10x − 3

156. Example 6 - Solve
• Remove parenthesis.
2(5x + 4) ≥ 10x − 3
10x + 8 ≥ 10x − 3

157. Example 6 - Solve
• Remove parenthesis.
2(5x + 4) ≥ 10x − 3
• Move variables to one side.
10x + 8 ≥ 10x − 3

158. Example 6 - Solve
• Remove parenthesis.
2(5x + 4) ≥ 10x − 3
• Move variables to one side.
10x + 8 ≥ 10x − 3
−10x −10x

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. 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. 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. 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. Example 7 - The local bank in Mathville requires a
minimum balance of $1000. If Josh knows he must write
checks for $525 for rent and $185 for utilities, how much
should he have in his account before writing the checks?

164. Example 7 - The local bank in Mathville requires a
minimum balance of $1000. If Josh knows he must write
checks for $525 for rent and $185 for utilities, how much
should he have in his account before writing the checks?
• Remember this problem?

165. Example 7 - The local bank in Mathville requires a
minimum balance of $1000. If Josh knows he must write
checks for $525 for rent and $185 for utilities, how much
should he have in his account before writing the checks?
• Remember this problem?
• Would $2000 be enough?

166. Example 7 - The local bank in Mathville requires a
minimum balance of $1000. If Josh knows he must write
checks for $525 for rent and $185 for utilities, how much
should he have in his account before writing the checks?
• Remember this problem?
• Would $2000 be enough?
• How about $710?

167. Example 7 - The local bank in Mathville requires a
minimum balance of $1000. If Josh knows he must write
checks for $525 for rent and $185 for utilities, how much
should he have in his account before writing the checks?
• Remember this problem?
• Would $2000 be enough?
• How about $710?
• What about $1710?

168. Example 7 - The local bank in Mathville requires a
minimum balance of $1000. If Josh knows he must write
checks for $525 for rent and $185 for utilities, how much
should 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?
• Let A = Josh’s account balance before writing the checks.

170. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?
Account balance - (sum of checks written) ≥ Min Balance

175. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?
Account balance - (sum of checks written) ≥ Min Balance
A - (525 + 185) ≥ 1000

176. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in his account before writing the checks?
Account balance - (sum of checks written) ≥ Min Balance
A - (525 + 185) ≥ 1000
A - 710 ≥ 1000

177. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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. (Example 7 - continued)
The local bank in Mathville requires a minimum balance
of $1000. If Josh knows he must write checks for $525
for rent and $185 for utilities, how much should he have
in 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
Josh must have at least $1710 in his checking
account before writing the checks.

180. 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.

181. 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

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. 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. 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. 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. 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. (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. 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.
Maximum Perimeter ≥ 2l + 2w

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. 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.
Maximum Perimeter ≥ 2l + 2w
96 ≥ 2 * 2w + 2 * w

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. 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.
Maximum Perimeter ≥ 2l + 2w
96 ≥ 2 * 2w + 2 * w
96 ≥ 4w + 2w

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. 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.
Maximum Perimeter ≥ 2l + 2w
96 ≥ 2 * 2w + 2 * w
96 ≥ 4w + 2w
96 ≥ 6w

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. 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.
Maximum Perimeter ≥ 2l + 2w
96 ≥ 2 * 2w + 2 * w
96 ≥ 4w + 2w
96 ≥ 6w
96 ÷ 6 ≥ 6w ÷ 6

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. 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.
Maximum Perimeter ≥ 2l + 2w
96 ≥ 2 * 2w + 2 * w
96 ≥ 4w + 2w
96 ≥ 6w
96 ÷ 6 ≥ 6w ÷ 6
16 ≥ w

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. 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.
Maximum Perimeter ≥ 2l + 2w
96 ≥ 2 * 2w + 2 * w
96 ≥ 4w + 2w
96 ≥ 6w
96 ÷ 6 ≥ 6w ÷ 6
16 ≥ w
Jim can fence a space with a width less than or equal
to 16 feet.