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3. Lesson 6-1 Solving Inequalities by Addition and Subtraction
Lesson 6-2 Solving Inequalities by Multiplication and Division
Lesson 6-3 Solving Multi-Step Inequalities
Lesson 6-4 Solving Compound Inequalities
Lesson 6-5 Solving Open Sentences Involving Absolute Value
Contents

4. Example 1 Solve by Adding Example 2 Graph the Solution
Example 3 Solve by Subtracting
Example 4 Variables on Both Sides
Example 5 Write and Solve an Inequality
Example 6 Write an Inequality to Solve a Problem
Lesson 1 Contents

5. Solve Then check your solution.
Original inequality
Add 12 to each side.
This means all numbers greater than 77.
Check Substitute 77, a number less than 77, and a number greater than 77.
Answer: The solution is the set {all numbers greater than 77}.
Example 1-1a

6. Solve Then check your solution.
Answer: or {all numbers less than 14}
Example 1-1b

7. Solve Then graph it on a number line.
Original inequality
Add 9 to each side.
Simplify.
Answer: Since is the same as y  21, the solution set is
The heavy arrow pointing to the left shows that the inequality includes all the numbers less than 21.
The dot at 21 shows that 21 is included in the inequality.
Example 1-2a

8. Solve Then graph it on a number line.
Answer:
Example 1-2b

9. Solve Then graph the solution.
Original inequality
Subtract 23 from each side.
Simplify.
Answer: The solution set is
Example 1-3a

10. Solve Then graph the solution.
Answer:
Example 1-3b

11. Then graph the solution.
Original inequality
Subtract 12n from each side.
Simplify.
Answer: Since is the same as the solution set is
Example 1-4a

12. Then graph the solution.
Answer:
Example 1-4b

13. Write an inequality for the sentence below. Then solve the inequality.
Seven times a number is greater than 6 times that number minus two.
Seven times a number
is greater than
six times that number
minus
two. 7n 6n 2
> –
Original inequality
Subtract 6n from each side.
Simplify.
Answer: The solution set is
Example 1-5a

14. Write an inequality for the sentence below. Then solve the inequality.
Three times a number is less than two times that number plus 5.
Answer:
Example 1-5b

15. Entertainment Alicia wants to buy season passes to two theme parks
Entertainment Alicia wants to buy season passes to two theme parks. If one season pass cost $54.99, and Alicia has $100 to spend on passes, the second season pass must cost no more than what amount?
Words The total cost of the two passes must be less than or equal to $100.
Variable Let the cost of the second pass.
Inequality
100
The total cost
is less than or equal to
$100.
Example 1-6a

16. Subtract 54.99 from each side.
Solve the inequality.
Original inequality
Subtract from each side.
Simplify.
Answer: The second pass must cost no more than $45.01.
Example 1-6a

17. Michael scored 30 points in the four rounds of the free throw contest
Michael scored 30 points in the four rounds of the free throw contest. Randy scored 11 points in the first round, 6 points in the second round, and 8 in the third round. How many points must he score in the final round to surpass Michael’s score?
Answer: 6 points
Example 1-6b

18. End of Lesson 1

19. Example 1 Multiply by a Positive Number
Example 2 Multiply by a Negative Number
Example 3 Write and Solve an Inequality
Example 4 Divide by a Positive Number
Example 5 Divide by a Negative Number
Example 6 The Word “not”
Lesson 2 Contents

20. Then check your solution.
Original inequality
Multiply each side by 12. Since we multiplied by a positive number, the inequality symbol stays the same.
Simplify.
Example 2-1a

21. Answer: The solution set is
Check To check this solution, substitute 36, a number less that 36 and a number greater than 36 into the inequality.
Answer: The solution set is
Example 2-1a

22. Then check your solution.
Answer:
Example 2-1b

23. Multiply each side by and change
Original inequality
Multiply each side by and change
Simplify.
Answer: The solution set is
Example 2-2a

24. Answer:
Example 2-2b

25. Write an inequality for the sentence below. Then solve the inequality.
Four-fifths of a number is at most twenty.
Four-fifths of
is at most
twenty.
a number r  20
Example 2-3a

26. Multiple each side by and do not change the inequality’s direction.
Original inequality
Multiple each side by and do not
change the inequality’s direction.
Simplify.
Answer: The solution set is .
Example 2-3a

27. Write an inequality for the sentence below. Then solve the inequality.
Two-thirds of a number is less than 12.
Answer:
Example 2-3b

28. Answer: The solution set is
Original inequality
Divide each side by 12 and do not change the direction of the inequality sign.
Simplify.
Check
Answer: The solution set is
Example 2-4a

29. Answer:
Example 2-4b

30. Divide each side by –8 and change < to >.
using two methods.
Method 1 Divide.
Original inequality
Divide each side by –8 and change < to >.
Simplify.
Example 2-5a

31. Method 2 Multiply by the multiplicative inverse.
Original inequality
Multiply each side by and change < to >.
Simplify.
Answer: The solution set is
Example 2-5a

32. using two methods.
Answer:
Example 2-5b

33. Multiple-Choice Test Item Which inequality does not have the solution
A B C D
Read the Test Item
You want to find the inequality that does not have the solution set
Solve the Test Item
Consider each possible choice.
Example 2-6a

34. A. B. C. D.
Answer: B
Example 2-6a

35. Multiple-Choice Test Item
Which inequality does not have the solution ?
A B C D
Answer: C
Example 2-6b

36. End of Lesson 2

37. Example 1 Solve a Real-World Problem
Example 2 Inequality Involving a Negative Coefficient
Example 3 Write and Solve an Inequality
Example 4 Distributive Property
Example 5 Empty Set
Lesson 3 Contents

38. Science The inequality F > 212 represents the
temperatures in degrees Fahrenheit for which water is
a gas (steam). Similarly, the inequality
represents the temperatures in degrees Celsius for
which water is a gas. Find the temperature in degrees
Celsius for which water is a gas.
Example 3-1a

39. Subtract 32 from each side.
Original inequality
Subtract 32 from each side.
Simplify.
Multiply each side by
Simplify.
Answer: Water will be a gas for all temperatures greater than 100°C.
Example 3-1a

40. Science The boiling point of helium is –452°F. Solve
the inequality to find the temperatures
in degrees Celsius for which helium is a gas.
Answer: Helium will be a gas for all temperatures greater than –268.9°C.
Example 3-1b

41. Then check your solution.
Original inequality
Subtract 13 from each side.
Simplify.
Divide each side by –11 and change
Simplify.
Example 3-2a

42. Answer: The solution set is
Check To check the solution, substitute –6, a number less than –6, and a number greater than –6.
Answer: The solution set is
Example 3-2a

43. Then check your solution.
Answer:
Example 3-2b

44. Write an inequality for the sentence below. Then solve the inequality.
Four times a number plus twelve is less than a number minus three.
Four times a number
plus
is less than
a number minus three.
twelve 4n +
< 12
Example 3-3a

45. Subtract n from each side.
Original inequality
Subtract n from each side.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution set is
Example 3-3a

46. Write an inequality for the sentence below. Then solve the inequality.
6 times a number is greater than 4 times the number minus 2.
Answer:
Example 3-3b

47. Distributive Property Combine like terms. Add c to each side.
Original inequality
Distributive Property
Combine like terms.
Add c to each side.
Simplify.
Subtract 6 from each side.
Simplify.
Divide each side by 4.
Simplify.
Example 3-4a

48. Answer: Since is the same as the solution set is
Example 3-4a

49. Answer:
Example 3-4b

50. Distributive Property
Original inequality
Distributive Property
Combine like terms.
Subtract 4s from each side.
This statement is false.
Answer: Since the inequality results in a false statement, the solution set is the empty set Ø.
Example 3-5a

51. Answer: Ø
Example 3-5b

52. End of Lesson 3

53. Example 1 Graph an Intersection
Example 2 Solve and Graph an Intersection
Example 3 Write and Graph a Compound Inequality
Example 4 Solve and Graph a Union
Lesson 4 Contents

54. Graph the solution set of
Find the intersection.
Example 4-1a

55. Answer:. The solution set is. Note that the. graph of
Answer: The solution set is Note that the graph of includes the point 5. The graph of does not include 12.
Example 4-1a

56. Graph the solution set of and
Example 4-1b

57. Then graph the solution set.
First express using and. Then solve each inequality.
and
Example 4-2a

58. The solution set is the intersection of the two graphs.
Find the intersection.
Example 4-2a

59. Answer: The solution set is
Example 4-2a

60. Then graph the solution set.
Answer:
Example 4-2b

61. Variables Let c be the cost of staying at the resort per night.
Travel A ski resort has several types of hotel rooms and several types of cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount that a quest would pay per night at the resort.
Words The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night.
Variables Let c be the cost of staying at the resort per night.
Inequality
Cost per night
is at most
$89 or
the cost
is at least
$109. c 89
109 or
Example 4-3a

62. Now graph the solution set.
Find the union.
Example 4-3a

63. Answer:
Example 4-3a

64. Answer: where c is the cost per seat
Ticket Sales A professional hockey arena has seats available in the Lower Bowl level that cost at most $65 per seat. The arena also has seats available at the Club Level and above that cost at least $80 per seat. Write and graph a compound inequality that describes the amount a spectator would pay for a seat at the hockey game.
Answer: where c is the cost per seat
Example 4-3b

65. Then graph the solution set.or
Example 4-4a

66. Graph
Graph
Answer:
Notice that the graph of contains every point in the graph of So, the union is the graph of The solution set is
Example 4-4a

67. Then graph the solution set.
Answer:
Example 4-4b

68. End of Lesson 4

69. Example 1 Solve an Absolute Value Equation
Example 2 Write an Absolute Value Equation
Example 3 Solve an Absolute Value Inequality (<)
Example 4 Solve an Absolute Value Inequality (>)
Lesson 5 Contents

70. The distance from –6 to –11 is 5 units.
Method 1 Graphing
means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction.
The distance from –6 to –11 is 5 units.
The distance from –6 to –1 is 5 units.
Answer: The solution set is
Example 5-1a

71. Method 2 Compound Sentence
Write as or
Case 1
Case 2
Original inequality
Subtract 6 from each side.
Simplify.
Answer: The solution set is
Example 5-1a

72. Answer: {12, –2}
Example 5-1b

73. Write an equation involving the absolute value for the graph.
Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1.
The distance from 1 to –4 is 5 units.
The distance from 1 to 6 is 5 units.
So, an equation is .
Example 5-2a

74. Check Substitute –4 and 6 into
Answer:
Check Substitute –4 and 6 into
Example 5-2a

75. Write an equation involving the absolute value for the graph.
Answer:
Example 5-2b

76. Then graph the solution set.
Write as and
Case 1
Case 2
Original inequality
Add 3 to each side.
Simplify.
Answer: The solution set is
Example 5-3a

77. Then graph the solution set.
Answer:
Example 5-3b

78. Then graph the solution set.
Write as or
Case 1
Case 2
Original inequality
Add 3 to each side.
Simplify.
Divide each side by 3.
Simplify.
Example 5-4a

79. Answer: The solution set is
Example 5-4a

80. Then graph the solution set.
Answer:
Example 5-4b

81. End of Lesson 5

82. End of Slide Show