The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on the real number line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the positions of two numbers on the real number line, with the number farther to the right said to be greater than the number to its left. The document provides examples of inequalities and how to represent sets of numbers using inequalities, such as all numbers between two values a and b. It also outlines steps for solving inequalities algebraically.

1. Inequalities

2. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
Inequalities

3. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
Inequalities

4. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3
Inequalities

5. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½
Inequalities

6. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
Inequalities
–π  –3.14..

7. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
–π  –3.14..

8. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
–π  –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.

9. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
+–
RL
–π  –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.

10. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
+–
R
We write this as L < R and called this the natural form because
it corresponds to their respective positions on the real line.
L
<
–π  –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.

11. We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a real number is
called the real (number) line.
Inequalities
+–
R
We write this as L < R and called this the natural form because
it corresponds to their respective positions on the real line.
This relation may also be written as R > L (less preferable).
L
<
–π  –3.14..
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.

12. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
Inequalities

13. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities

14. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x".

15. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a).

16. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–
a
open dot
a < x

17. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x.
a < x

18. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x

19. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.

20. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
+–
a a < x < b b

21. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b. A line segment as such is
called an interval.
+–
a a < x < b b

22. To solve inequalities:
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.

23. To solve inequalities:
1. Simplify both sides of the inequalities
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.

24. To solve inequalities:
1. Simplify both sides of the inequalities
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side

25. To solve inequalities:
1. Simplify both sides of the inequalities
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x

26. To solve inequalities:
1. Simplify both sides of the inequalities
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18

27. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18

28. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18 move 18 and –3x (change sign)

29. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18 move 18 and –3x (change sign)
6 – 18 > 3x

30. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18 move 18 and –3x (change sign)
6 – 18 > 3x
– 12 > 3x

31. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18 move 18 and –3x (change sign)
6 – 18 > 3x
– 12 > 3x

32. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18 move 18 and –3x (change sign)
6 – 18 > 3x
– 12 > 3x
–12/3 3x/3>
div. by 3 (no need to switch >)

33. To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Inequalities
Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.
3(2 – x) > 2(x + 9) – 2x simplify each side
6 – 3x > 2x + 18 – 2x
6 – 3x > 18 move 18 and –3x (change sign)
6 – 18 > 3x
– 12 > 3x
–12/3 3x/3>
–4 > x or x < –4
div. by 3 (no need to switch >)
+
–4

34. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw
Inequalities
We also have inequalities in the form of intervals.

35. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.

36. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.

37. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6
–6 –6 –6
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.

38. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6
–6 –6 –6
6 > –2x > –10
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.

39. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6
–6 –6 –6
6 > –2x > –10 div. by -2, switch inequality sign
6
-2
-2x
-2<
-10
-2
<
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.

40. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6
–6 –6 –6
6 > –2x > –10
-3 < x < 5
div. by -2, switch inequality sign
6
-2
-2x
-2<
-10
-2
<
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.

41. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6
–6 –6 –6
6 > –2x > –10
-3 < x < 5
div. by -2, switch inequality sign
6
-2
-2x
-2<
-10
-2
<
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x. The answer is an interval of
numbers.

42. Example C. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6
–6 –6 –6
6 > –2x > –10
0
+
-3 < x < 5
5
div. by -2, switch inequality sign
6
-2
-2x
-2<
-10
-2
<
-3
Inequalities
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x. The answer is an interval of
numbers.

43. Comparison Statements, Inequalities and Intervals

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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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”.

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

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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

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

61. “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

62. “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

63. “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

64. “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

65. “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

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

67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

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

75. 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

76. 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

77. 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.

78. 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

79. 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

80. 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

81. 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

82. Inequalities
Exercise. A. Draw the following Inequalities. Indicate clearly
whether the end points are included or not.
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
B. Write in the natural form then draw them.
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
C. Draw the following intervals, state so if it is impossible.
9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 2
13. 6 > x ≥ 8 14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9
D. Solve the following Inequalities and draw the solution.
17. x + 5 < 3 18. –5 ≤ 2x + 3 19. 3x + 1 < –8
20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x
22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 9
24. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)
26. x + 2(x – 3) < 2(x – 1) – 2
27. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13

83. Inequalities
F. Solve the following interval inequalities.
28. –4 ≤
2
x 29. 7 >
3
–x 30. < –4–x
E. Clear the denominator first then solve and draw the solution.
5
x2 3
1 2
3
2
+ ≥ x31. x4
–3
3
–4
– 1> x32.
x
2 8
3 3
4
5– ≤33. x
8 12
–5 7
1+ >34.
x
2 3
–3 2
3
4
4
1–+ x35. x4 6
5 5
3
–1
– 2+ < x36.
x
12 2
7 3
6
1
4
3–– ≥ x37.
≤
40. – 2 < x + 2 < 5 41. –1 ≥ 2x – 3 ≥ –11
42. –5 ≤ 1 – 3x < 10 43. 11 > –1 – 3x > –7
38. –6 ≤ 3x < 12 39. 8 > –2x > –4