UNIT TWO: Prealgebra in a Technical World
2.1 The Integers
SWBAT 1. Read, write and say integers.
2. Compare integers.
3. Find opposites.
4. Find absolute values.
Read, Write and Say Integers
“Got VISA?” In 1970, college students were not eligible for credit cards. Today, special
high-rate credit cards are available to people who may have trouble paying their bills.
When we borrow or take on debt, we can record this amount using a negative number.
We record our income as a positive number. Together these are called signed numbers.
In this section we begin our study of signed numbers by
studying the set of numbers called integers.
Definition: The integers are the counting numbers, their opposites, and zero.
Zero is neither positive nor negative. We can write the integers using
braces { } to denote a set:
{. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . }
The diagram at right gives the relationship
of groups within the integers. The counting
numbers { 1, 2, 3, 4, . . . } were the first numbers
humans used, and these are still the numbers
children learn first. The whole numbers are the set
of counting numbers and zero { 0, 1, 2, 3, . . . } .
The integers do not include fractions or decimals.
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SECTION 2.1 The Integers
The integers can all be put on a number line. An integer number line stretches to
infinity in both directions and lets us see which numbers are smaller and which are larger:
Positive integers are all greater than zero, so they are on the right of zero. Negative
integers are all less than zero, so they are on the left of zero.
Positive numbers do not need a " + " sign in front of them; for instance, we assume
that 3 = +3. Sometimes we may choose to use the + sign for clarity, but it is never necessary.
Negative integers, like all negative numbers, always have a dash (−) in front of them to show
that they are negative.
Subtraction is also shown by a dash. We also use a dash to mean “opposite." This can
be confusing. As you read this section and study this unit, you will want to refer to the table
below. It is a good idea to mark this page in your book!
Expression
3−7
−7 or (−7)
−𝑥
−(8)
−(6 − 3)
−4 − (−3)
How to Say It
“3 subtract 7” “take 7 from 3”
“3 take away 7”
“3 minus 7”
“negative seven”
“the opposite of 7”
“the opposite of 𝑥”
“the opposite of 8”
“the opposite of the difference of 6 and 3”
“negative four minus negative three"
Integers are used in business, technology, science, the trades, medicine, games, and
household finances.
Example 1: Fill in the correct negative integers.
a. Joshua pays $382 for car insurance. In his check book he writes ______.
ANSWER: −$𝟑𝟖𝟐
b. The coldest natural temperature ever recorded on Earth was 128° below zero Fahrenheit ,
________, recorded in Antarctica on July 21, 1983.
ANSWER: −𝟏𝟐𝟖° F
UNIT TWO: Prealgebra in a Technical World
c. When Oregon State University scientists test for oxygen in the water off Cape Perpetua, the
scientists take samples at 150 feet below sea level, that is ________.
ANSWER: −𝟏𝟓𝟎 ft
 Check Point 1
Translate into negative integers.
a. Jonathan lost 38 points in this round. Jonathan’s score is ________ in this round.
b. The shore of the Dead Sea has an elevation of 1,371 feet below sea level. The shore of the
Dead Sea has an elevation of ____________ft.
c. The Seahawks lost 4 yards on their last play. The Seahawk's yardage was recorded as
__________ yards on their last play.
Compare Integers
We compare integers two at a time. The first integer is either greater than (>) the
second integer, less than (<) the second integer, or equal to (=) the second integer. The
number line shows how two numbers are related. The farther left on the number line, the
smaller the number. The farther right on the number line, the larger the number.
Example 2: Compare integers using < or > and use the number line to check your results.
a.
3 ____ − 4
b. −7 ___ − 9
− 3 ____ − 4
4 ______3
− 3 ____4
− 9 ____ 07
− 7____ 9
7____9
− 3 > −4
4>3
−3<4
−7<9
7 <9
ANSWERS:
a.
𝟑 > −4
b. −𝟕 > −9
−9< 7
 Check Point 2
Compare integers using < or > and use the number line to check your results here.
a.
5 ____ 6
b. 11 ____ − 7
− 5 _____ (−6)
− 11 _____ 7
− 5 _____6
6 _______ − 5
− 7 _____ − 11
7_______11
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SECTION 2.1 The Integers
Find Opposites
DEFINITION: Opposites are numbers that are the same distance from zero and
are located in opposite directions from zero on the number line.
We can find opposites on the number line. Notice that the opposite of a
positive number is negative, and the opposite of a negative is positive. Zero is its own opposite.
On the number line we see that 6 is the same distance from zero as -6, but since they
are located on opposite sides of zero, they are opposites. Similarly, -2 and 2 are opposites:
Often the opposite of a negative number shows up in technical reading. For example, in
working a medical calculation, a nursing student comes to a point where she has – (−8) on her
paper. She needs to know what this means.
We say this number is “the opposite of −8.” We also simplify this number by thinking
about the number line. The opposite of negative 8 is positive 8.
PROPERTY: The Op-Op Property: For any number 𝑥, −(−𝑥) = 𝑥
We can take the opposite of the opposite of the opposite of a number, and this will
simplify too. In Example 3 we have simplified each number and hope that you will find the
pattern for taking repeated opposites. Use this pattern to complete the Check Point problems.
Example 3: Simplify.
a. −(−19) =______
b. −(−(−67)) = ______
c. −(−(−(−2))) = ______
d. −(−(−4)) = ______
e. −(−55) = ______
f.
ANSWERS: a. 19
b. −67
c. 2
d. −4
e. 55
f. −1
−(−(−(−(−1)))) =______
UNIT TWO: Prealgebra in a Technical World
 Check Point 3
Simplify.
a. −(−87) = _________________________
b. −(−(−(−(−82)))) = __________________
c. −(−(−5) = _______________________
d. −(−(−(−(82)))) = ____________________
e. −(−63) = _________________________
Write the pattern for finding repeated opposites below and be prepared to discuss in class:
______________________________________________________________________________
Find Absolute Values
When we add, subtract, multiply, and divide integers, the distance the integer is from
zero is very important.
When we travel, distance is never negative. If we travel 20 miles north, this is the same
distance as traveling 20 miles south. Both numbers are positive even though the directions are
opposite.
We use absolute value to find the distance a number is from zero no matter which
direction it is from zero.
DEFINITION: The absolute value of any number is its distance from zero.
Distance is always positive. We use straight brackets to represent the absolute
value of the number |𝑥|.
The absolute value of any number is positive, |𝒙| > 0.
The opposite of the absolute value of any number is negative, −|𝒙| < 0 .
Example 4: Simplify.
a. |3| = _____
b. |−54| = _____
c. |73| = _____
d. |−4,563| = _____
e. −|3| = _____
f. −|−54| = _____
g. −|73| = _____
h. −|−4,563| = _____
ANSWERS: a. 3
b. 54
c. 73
d. 4,563
e. -3
f. -54
g. -73
h. -4,563
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SECTION 2.1 The Integers
 Check Point 4
a. |−34| = ______
b. |34| = _______
c. |2| = _____
d. −|34| = ______
e. −|2| = ______
f. −|900| = _____
g. −|−34| = ______
h. −|−2| = _____

Check Point 5
Carefully determine if parentheses or absolute value bars are being used, and then simplify
accordingly.
a. (−45) = ______
b. −(−45) = _______
c. −|28| = _____
d. | − 45| = ______
e. −|−2| = ______
f. −(28) = _____
g. −|−45| = ______
h. −(−2) = _____
Read Integers on a Graph and Solve Applications
Example 5: By reading the graph below, record the approximate average monthly low
temperature for the months of January, February, and March for Barrow, Alaska.
January: ______
February: ______
March: ______
Average Monthly Low for Barrow, Alaska
40
30
degrees Fahrenheit
90
20
10
0
-10
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
-20
-30
ANSWERS: January ≈ -18°F , February ≈ -23°F, March ≈ -21°F
Sep
Oct
Nov
Dec
UNIT TWO: Prealgebra in a Technical World
 Check Point 6
In the table provided,
write the difference
in stock price for the
given days. The first
difference has been
placed in the table
for you.
Example 6: Cheryl recorded all of her VISA transactions in a ledger. She is up to date through
the month of September.
Date
Transaction
In the first week of October, she kept these
9/26
RCC bookstore textbooks
receipts to record: October 1: Gas $42,
9/27
Groceries
October 1: prescription $26, October 7:
9/30
VISA payment
Amount
−183
−43
−400
payment to VISA $100.
Record these payments in her ledger to the
right.
 Check Point 7
Jeffrey is keeping track of his
Date
Transaction
Amount
transactions. On October 1 he
deposits $1,890, writes a rent
check for $695, and writes the
$297 check for his car payment.
On October 3, Jeffrey writes a
$252 check for groceries. On
October 6, Jeffrey deposits a
$40 refund check and then writes a check for $122 for cable. Fill in his check ledger.
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SECTION 2.1 The Integers
UNIT TWO: Prealgebra in a Technical World
2.1 Exercise Set
Name _______________________________
Skills
Find the absolute value.
1. |20|
2.
|−11|
3. |−21|
4. |0|
5.
|43|
6. |−14|
7.
|9|
8. |1|
9. |−1|
10. |−49|
Write the opposite of the following:
11. 2
12. −7
13. −5
14. −6
15. 9
16. −15
17. 29
18. 11
19. −1
20. 0
Find the integer that makes the following equations true.
21.
5 + ______ = 0
22. −3 + ______ = 0
23. −22 + _____ = 0 24.
14 + ______ = 0
For each pair of numbers below, determine which number is larger. Place the appropriate
< or > symbols in between the pair.
25.
5______6
26.
−2______ − 1
27.
0_______ − 1
28.
−9_______5
29.
−11______ − 1
30.
−11_____ − 9
31.
−101______ − 100
32.
−233_____4
33.
−225_____ − 226
34.
−289_____290
35.
0_____ − 12
36.
−131_____0
Find −𝑥 when given the following values for x:
37.
𝑥 = −1
38.
−𝑥 = __________
40.
𝑥 = 488
−𝑥 = __________
𝑥 =0
39.
−𝑥 = __________
41.
𝑥 =1
−𝑥 = __________
𝑥 = 17
−𝑥 = __________
42.
𝑥 = −131
−𝑥 = __________
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SECTION 2.1 The Integers
43.
a. Circle the following numbers that are integers:
b. Circle the following numbers that are NOT integers:
−40
−40
1/2
−1.5
3/4
1/2
−1.5
3/4
0
0
44. Place the following numbers on the number line: −1 −4 −11 2 −6 5
−10
10
0
Use one of the symbols, =, < or > to compare the two numbers with grouping symbols. Simplify
first.
45. |−6|______0
46. −|4|______0
47. −(4)______|−4|
48. −(−1)_______ − |−2|
49. −|9|_____|−9| 50. −(−6)___ − |6| 51. −(−2)_____ − 2 52.
−2______ − (−9)
Applications
Translate the words in italics in the following sentences using inequality symbols, < or >.
For example: Mary makes more money per hour than Alan. The answer is: >
53.
The number of cactus plants is less than other plants. ______
54.
Jonny complains that his dish of ice cream is smaller than his sister’s. ______
55.
I was taller than my sister at age 4. ______
56.
The studs used in the living room are shorter than those in the bedroom. ______
57.
Anita's spelling test score always exceeds that of her brother. ______
58.
The water level behind the dam is lower than it was last year. ______
UNIT TWO: Prealgebra in a Technical World
59.
Simplify −|−4|. (Remember that absolute value symbols are grouping symbols.)
60.
a. Simplify –(–4).
b. Why is this result different from that for number 59?
61.
You were over (spent more than) your
62. You were over your budget last month
budget last month by $50. The month
by $32. The month before you were
before you were under your budget by
under your budget by $41. Express each
$27. Express each of these numbers as
of these numbers as integers.
integers.
63.
65.
67.
The coldest temperature recorded in
64. The coldest temperature ever recorded
Fairbanks, Alaska, was 66° F degrees
on Earth on July 21, 1983, was reported
below zero. Express this temperature as
as 129° F below zero. Express this
an integer.
temperature as an integer.
Death Valley, California, has an elevation
66. The Salton Trough runs in the
of 86 meters (283 feet) below sea level.
southwestern U.S. and into Mexico, and
Express this elevation, in both meters
has an elevation of 69 meters below sea
and feet, using integers.
level. Express this elevation as an integer.
The Dead Sea Depression, located along
68. The San Julian's Great Depression,
the borders of Israel, Syria, and Jordan,
located in Argentina, has an elevation of
has an elevation of about 413 meters
103 meters (344 feet) below sea level.
(1,355 feet) below sea level. Express this
Express this elevation, in both meters
elevation, in both meters and feet, using
and feet, using integers.
integers.
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SECTION 2.1 The Integers
69.
Your credit card bill for one month was
70. Your credit card bill for one month was
$223. Express this debt as an integer.
71.
$491. Express this debt as an integer.
Your golf score was 4 below par. Express 72. (See number 71) You are 2 above par in
this score as an integer.
golf. Is this a positive or negative score?
Review
For problems 73 and 74, simplify using correct order of operations and show your work below.
73.
4 ∙ 32 − 6(3 + 20 ) ÷ 8 − (6 − 2)
75. Find the area of the following shape:
18 cm
74.
5 + 92 − 2 ∙ 32 ÷ [3(2 − 40 )]
76. Find the area of the following shape:
1616cm
cm
18 cm
6 cm
5 cm
6 cm
12 cm
12 cm
77. What is the average of 76, 89, 80, and 75?
78. What is the average of 15, 27, 25, and 5?
79. Add 189 + 54 mentally and then explain
80. Subtract 189 – 94 mentally and then
how you did this mentally.
81. Divide 365 by 5 mentally and explain how
you did this mentally.
explain how you did this mentally.
82. Multiply 1,200 by 5 mentally and explain
how you did this mentally.