The document provides information about integers and inequalities: 1. It defines integers as the set of natural numbers, whole numbers, and their opposites, and illustrates integers on a number line. 2. It explains the concepts of absolute value and inequalities, defining absolute value as the distance from 0 and inequality symbols. 3. It provides examples of marking points on a number line, writing inequalities, and comparing integers using inequality symbols.

3. 1. gain knowledge about the set of
integers
2. compare integers

4. 3. recognize application of integers.
4. define absolute value of a number

6. The consists of all
, all and .
𝐼 = {… , −3, −2, −1, 0, 1, 2, 3, … }

7. that are are called
.
that are are called
.

8. A can be written with a
positive sign (+), but it does not need to have a
sign.
Examples:
1, 3, 5, 7, 99, 100, 1202, …

9. A must be written with a
negative sign (−).
Examples:
−2, −35, −77, −102, −3102,

10. 0
Zero is neither nor .

11. can also be shown by locating them as points on a
.
On a horizontal line number line, are to the
and are to the .
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

12. The number associated with a point on a
number line is the of the point.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

13. The point on the number line assigned to
(0) is called the .
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

14. Numbers that are of the from
zero (origin), but on is
called .
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
The opposite of −4 is 4. The opposite of 4 is −4.

16. Give the opposite of the given integer.
−2 = 2

17. Give the opposite of the given integer.
−3 = 3

18. Give the opposite of the given integer.
15 = −15

19. Give the opposite of the given integer.
120 =−120

20. Give the opposite of the given integer.
77 = −77

21. Give the opposite of the given integer.
0 = 0

22. Got
questions?

23. are also used to describe situations
involving , ,
, , and
.
In all of these examples, integers are used to
count and units.

24. Example 1:
a. 27°C above 0°C
b. 13°C below 0°C
c. 400°C below 0°C
d. 1735°C above 0°C
e. 97°C above zero
a. 27
b. −13
c. −400
d. 1735
e. 97

25. Example 2:
a. A loss of P1
b. A profit of P300
c. A gain of 450m.
d. A gain of P1,503
e. A loss of 300km.
a. −1
b. 300
c. 450
d. 1503
e. −300

26. Example 3:
a. P211 deposit in a bank
b. P2,300 withdraw in a bank
c. P150 deposit in a bank
d. P7,889 withdraw in a bank
e. P1,202 deposit in a bank
a. 211
b. −2300
c. 150
d. -7889
e. 1202

27. Got
questions?

28. E-Math 7
Practice and Application
Test I
Page 27

30. is marking the
on the number line.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

31. Example 1:
Point Z corresponds to a number five units to the right
from the origin.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Z

32. Example 2:
Point M is 7 units to the left from Point Z.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
ZM

33. Example 3:
Point A is the origin.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
ZM A

34. Example 4:
Point P is the additive inverse of Point Z.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
ZM AP

35. Example 5:
Point G is halfway between point M and point A.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
ZM AP G

36. Got
questions?

37. E-Math 7
Practice and Application
Test II
Page 27

39. The of an integer is equal to its
distance from 0.
It’s concept is so important in mathematics that
it has its own symbol.
The absolute value of x is written as | x |.

40. Example 1:
Find each absolute value.
a. | 5 |
Answer:
Since 5 is 5 units from zero, the absolute value of
5 is 5.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

41. Example 2:
Find each absolute value.
a. | −5 |
Answer:
Since −5 is 5 units from zero, the absolute value
of −5 is 5.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

42. Example 3:
Find each absolute value.
a. | 3 |
Answer:
Since 3 is 3 units from zero, the absolute value of
3 is 3.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

43. Example 4:
Find each absolute value.
a. | −3 |
Answer:
Since −3 is 3 units from zero, the absolute value
of −3 is 3.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

44. Example 5:
Find each absolute value.
a. | 0 |
Answer:
The Absolute value of 0 is 0.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

45. Got
questions?

46. Find the value of each expression
Example 1:
9 | − 2|+
= 9 + 2
= 11

47. Find the value of each expression
Example 2:
−7 | 3 |−
= 7 − 3
= 4

48. Find the value of each expression
Example 3:
−17 | − 12 |−
= 17 − 12
= 5

49. Find the value of each expression
Example 4:
−2 | − 4 |∙
= 2 ∙ 4
= 8

50. Find the value of each expression
Example 5:
−10 | − 5 |÷
= 10 ÷ 5
= 2

51. Find the value of each expression
Example 6:
10 − 21
= | − 11 |
= 11

52. Got
questions?

53. E-Math 7
Practice and Application
Test III No. 22
Page 27

55. An is a mathematical statement that
contains the symbols >, ≥, <, ≤, =, 𝑎𝑛𝑑 ≠.
𝑥 > 0
This is read as:
“x is greater than zero”

56. An is a mathematical statement that
contains the symbols >, ≥, <, ≤, =, 𝑎𝑛𝑑 ≠.
𝑥 ≥ 0
This is read as:
“x is greater than or equal zero”

57. An is a mathematical statement that
contains the symbols >, ≥, <, ≤, =, 𝑎𝑛𝑑 ≠.
𝑥 < 0
This is read as:
“x is less than zero”

58. An is a mathematical statement that
contains the symbols >, ≥, <, ≤, =, 𝑎𝑛𝑑 ≠.
𝑥 ≤ 0
This is read as:
“x is less than or equal zero”

59. An is a mathematical statement that
contains the symbols >, ≥, <, ≤, =, 𝑎𝑛𝑑 ≠.
𝑥 = 0
This is read as:
“x is equal to zero”

60. An is a mathematical statement that
contains the symbols >, ≥, <, ≤, =, 𝑎𝑛𝑑 ≠.
𝑥 ≠ 0
This is read as:
“x is not equal to zero”

61. Got
questions?

62. An is used when comparing,
translating different mathematical sentences
including integers.
If integers are graphed on a
, the to the of
is the .

63. Example 1:
Replace each with <, >, 𝑜𝑟 =.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5 4
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

64. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5 4>
“five is four”
Example 1:
Replace each with <, >, 𝑜𝑟 =.
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

65. Example 2:
Replace each with <, >, 𝑜𝑟 =.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
−6 −7
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

66. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
>
“negative six is negative seven”
Example 2:
Replace each with <, >, 𝑜𝑟 =.
−6 −7
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

67. Example 3:
Replace each with <, >, 𝑜𝑟 =.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
−8 4
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

68. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
<
“negative eight is four”
Example 3:
Replace each with <, >, 𝑜𝑟 =.
−8 4
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

69. Example 4:
Replace each with <, >, 𝑜𝑟 =.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
−4 4
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

70. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
<
“negative four is four”
Example 4:
Replace each with <, >, 𝑜𝑟 =.
−4 4
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

71. Example 5:
Replace each with <, >, 𝑜𝑟 =.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
2
10
2
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

72. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
>
“10 over 2 is four”
Example 5:
Replace each with <, >, 𝑜𝑟 =.
10
2
2
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

73. Example 6:
Replace each with <, >, 𝑜𝑟 =.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
8
24
3
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

74. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
=
“24 over 3 is to eight”
Example 6:
Replace each with <, >, 𝑜𝑟 =.
24
3
8
• An is used when comparing, translating
different mathematical sentences including integers.
• If integers are graphed on a ,
the to the of is the
.

75. Got
questions?

76. Two important terms are common in inequality
problems.
1. The phrase “ ” means ≤ (
).
2. The phrase “ ” means ≥ (
).

77. Example 1:
a. The team (t) must have 8
members.
Translation: t 8≥
Two important terms are common in inequality problems.
1. The phrase “ ” means ≤ ( ).
2. The phrase “ ” means ≥ ( ).

78. Example 2:
a. The height (h) must be 6’3”.
Translation: t 8≤
Two important terms are common in inequality problems.
1. The phrase “ ” means ≤ ( ).
2. The phrase “ ” means ≥ ( ).

79. Got
questions?

80. E-Math 7
Practice and Application
Test III
Page 28