Inequalities, comparative Phrases

1. Comparison Statements, Inequalities and Intervals

2. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
Comparison Statements, Inequalities and Intervals

3. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
Comparison Statements, Inequalities and Intervals

4. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
0

5. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
0
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals

6. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
0
+x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals

7. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
and that x is negative means that x < 0.
0
+x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals

8. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
and that x is negative means that x < 0.
0
+– x is negative x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals

9. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
and that x is negative means that x < 0.
0
+– x is negative x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
The phrase “the temperature T is positive” is “0 < T”.

10. “Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals

11. A quantity x is non–positive means x is not positive,
or “x ≤ 0”,
“Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals

12. A quantity x is non–positive means x is not positive,
or “x ≤ 0”,
“Non–Positive” vs. “Non–Negative”
0
+–
x is non–positive
Comparison Statements, Inequalities and Intervals

13. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
“Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals

14. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals

15. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
“More/greater than” vs “Less/smaller than”
Comparison Statements, Inequalities and Intervals

16. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means
“C < x”,
C
x is more than C
Comparison Statements, Inequalities and Intervals

17. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means
“C < x”, and that x is less than C means “x < C”.
Cx is less than C
x is more than C
Comparison Statements, Inequalities and Intervals

18. “No more/greater than” vs “No less/smaller than”
and “At most” vs “At least”
Comparison Statements, Inequalities and Intervals

19. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
and “At most” vs “At least”
Comparison Statements, Inequalities and Intervals

20. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
+–
x is no more than C C
and “At most” vs “At least”
x is at most C
Comparison Statements, Inequalities and Intervals

21. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
A quantity x is “no–less than C” is the same as
“x is at least C” and means “C ≤ x”.
+–
x is no more than C x is no less than CC
and “At most” vs “At least”
x is at most C x is at least C
Comparison Statements, Inequalities and Intervals

22. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
A quantity x is “no–less than C” is the same as
“x is at least C” and means “C ≤ x”.
+–
x is no more than C x is no less than C
“The account balance A is no–less than 500”
is the same as “A is at least 500” or that “500 ≤ A”.
C
and “At most” vs “At least”
x is at most C x is at least C
Comparison Statements, Inequalities and Intervals

23. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
A quantity x is “no–less than C” is the same as
“x is at least C” and means “C ≤ x”.
+–
x is no more than C x is no less than C
“The temperature T is no–more than 250o”
is the same as “T is at most 250o” or that “T ≤ 250o”.
“The account balance A is no–less than 500”
is the same as “A is at least 500” or that “500 ≤ A”.
C
and “At most” vs “At least”
x is at most C x is at least C
Comparison Statements, Inequalities and Intervals

24. We also have the compound statements such as
“x is more than a, but no more than b”.
Comparison Statements, Inequalities and Intervals

25. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
Comparison Statements, Inequalities and Intervals

26. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
Comparison Statements, Inequalities and Intervals

27. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
Comparison Statements, Inequalities and Intervals

28. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
Comparison Statements, Inequalities and Intervals

29. Therefore the statement “the length L of
the stick must be more than 5 feet but no
more than 7 feet” is “5 < L ≤ 7”
We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
Comparison Statements, Inequalities and Intervals

30. Therefore the statement “the length L of
the stick must be more than 5 feet but no
more than 7 feet” is “5 < L ≤ 7”
or that L must be in the interval (5, 7].
We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
75
5 < L ≤ 7
or (5, 7]
L
Comparison Statements, Inequalities and Intervals

31. Therefore the statement “the length L of
the stick must be more than 5 feet but no
more than 7 feet” is “5 < L ≤ 7”
or that L must be in the interval (5, 7].
We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
75
5 < L ≤ 7
or (5, 7]Following is a list of interval notation.
L
Comparison Statements, Inequalities and Intervals

32. Let a, b be two numbers such that a < b, we write
ba
Comparison Statements, Inequalities and Intervals

33. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
Comparison Statements, Inequalities and Intervals

34. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
Comparison Statements, Inequalities and Intervals

35. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
Comparison Statements, Inequalities and Intervals
Note: The notation “(2, 3)”
is to be viewed as an interval or as
a point (x, y) depends on the context.

36. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals

37. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals

38. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
∞a
or a < x, as (a, ∞),
Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals

39. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
–∞ a
or x ≤ a, as (–∞, a],
∞a
or a < x, as (a, ∞),
–∞ a
or x < a, as (–∞, a),
Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals

40. Intersection and Union (∩ & U) of Intervals
Comparison Statements, Inequalities and Intervals

41. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
Comparison Statements, Inequalities and Intervals

42. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
Comparison Statements, Inequalities and Intervals

43. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
2
3
32
Comparison Statements, Inequalities and Intervals

44. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
2
3
3
I ∩ J:
is called the intersection of I and J.
2
Comparison Statements, Inequalities and Intervals

45. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
2
3
3
I ∩ J:
is called the intersection of I and J.
It’s denoted as I ∩ J and this case I ∩ J = (2, 3].
2
Comparison Statements, Inequalities and Intervals

46. The merge of the two intervals I and J shown here
31
I:
42
J:
Comparison Statements, Inequalities and Intervals

47. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
2 41 3
Comparison Statements, Inequalities and Intervals

48. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
Comparison Statements, Inequalities and Intervals

49. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Comparison Statements, Inequalities and Intervals

50. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
Comparison Statements, Inequalities and Intervals

51. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–3 1
0
KWe have
Comparison Statements, Inequalities and Intervals

52. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–2
–3 1
0
K
J
We have
Comparison Statements, Inequalities and Intervals

53. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–2
–3 1
0
K
J
and K U J is
–3
0
We have
Comparison Statements, Inequalities and Intervals
the union:

54. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–2
–3 1
0
so K U J = (–3, ∞).
K
J
and K U J is
–3
0
We have
Comparison Statements, Inequalities and Intervals
the union:

55. b. K ∩ I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
Comparison Statements, Inequalities and Intervals

56. We have 10
b. K ∩ I
–1–4 –3 K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
Comparison Statements, Inequalities and Intervals

57. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
Comparison Statements, Inequalities and Intervals

58. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals

59. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.

60. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:

61. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:

62. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
A: (2, 5 ]

63. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
A: (2, 5 ]

64. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
A: (2, 5 ] B: [4, 7)

65. b. When will there someone working at the shop?
Comparison Statements, Inequalities and Intervals

66. b. When will there someone working at the shop?
Stack the schedules as shown.
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)

67. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
i.e. A U B
A U B:

68. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:

69. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:

70. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
b. When will both be working at the shop?
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)

71. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
b. When will both be working at the shop?
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
A ∩ B:
The time both
persons be working
is the intersection of
their schedule,
i.e. A ∩ B

72. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
b. When will both be working at the shop?
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
A ∩ B:
The time both
persons be working
is the intersection of
their schedule,
i.e. A ∩ B = [4, 5]. So both be there from 4 pm to 5 pm.

73. The interval [a, a] consists of one point {x = a}.
The empty set which contains nothing is denoted as
Φ = { } and interval (a, a) = (a, a] = [a, a) = Φ.
Comparison Statements, Inequalities and Intervals

74. Exercise. A. Draw the following Inequalities. Translate each
inequality into an English phrase. (There might be more than
one way to do it)
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
Exercise. B. Translate each English phrase into an inequality.
Draw the Inequalities.
Let P be the number of people on a bus.
9. There were at least 50 people on the bus.
10. There were no more than 50 people on the bus.
11. There were less than 30 people on the bus.
12. There were no less than 28 people on the bus.
Let T be temperature outside.
13. The temperature is no more than –2o.
14. The temperature is at least than 35o.
15. The temperature is positive.
Inequalities

75. Inequalities
Let M be the amount of money I have.
16. I have at most $25.
17. I have a non–positive amount of money.
18. I have less than $45.
19. I have at least $250.
Let the basement floor number be given as a negative number
and let F be the floor number that we are on.
20. We are below the 7th floor.
21. We are above the first floor.
22. We are not below the 3rd floor basement.
24. We are on at least the 45th floor.
25. We are between the 4th floor basement and the 10th
floor.
26. We are in the basement.