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Prealgebra

3.1 Introduction to Integers

Prealgebra3.1 Introduction to Integers

Learning Objectives

By the end of this section, you will be able to:
  • Locate positive and negative numbers on the number line
  • Order positive and negative numbers
  • Find opposites
  • Simplify expressions with absolute value
  • Translate word phrases to expressions with integers

Be Prepared 3.1

Before you get started, take this readiness quiz.

  1. Plot 0,1,and30,1,and3 on a number line.
     If you missed this problem, review Example 1.1.
  2. Fill in the appropriate symbol: (=, <, or >):2___4(=, <, or >):2___4
     If you missed this problem, review Example 2.3.

Locate Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than 0.0. Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example, −1°F−1°F (read as “negative one degree Fahrenheit”) is 1degree1degree below 0.0. A minus sign is shown before a number to indicate that it is negative. Figure 3.2 shows −20°F,−20°F, which is 20degrees20degrees below 0.0.

This figure is a thermometer scaled in degrees Fahrenheit. The thermometer has a reading of 20 degrees.
Figure 3.2 Temperatures below zero are described by negative numbers.

Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.

Elevations can also be represented by negative numbers. The elevation at sea level is 0 feet.0 feet. Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about 1,302feet1,302feet below sea level, so the elevation of the Dead Sea can be represented as −1,302feet.−1,302feet. See Figure 3.3.

This figure is a drawing of a side view of the coast of Israel, showing different elevations. The Mediterranean Sea is labeled 0 feet elevation and the Dead Sea is labeled negative 1302 feet elevation. The country of Jordan is also labeled in the figure.
Figure 3.3 The surface of the Mediterranean Sea has an elevation of 0ft.0ft. The diagram shows that nearby mountains have higher (positive) elevations whereas the Dead Sea has a lower (negative) elevation.

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of 500feet.500feet. Its position would then be −500feet−500feet as labeled in Figure 3.4.

This figure is a drawing of a submarine underwater. In the water is also a vertical number line, scaled in feet. The number line has 0 feet at the surface and negative 500 feet below the water where the submarine is located.
Figure 3.4 Depths below sea level are described by negative numbers. A submarine 500ft500ft below sea level is at −500ft.−500ft.

Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at 00 and showed the counting numbers increasing to the right as shown in Figure 3.5. The counting numbers (1, 2, 3, …)(1, 2, 3, …) on the number line are all positive. We could write a plus sign, +,+, before a positive number such as +2+2 or +3,+3, but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

This figure is a number line scaled from 0 to 6.
Figure 3.5

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with −1−1 at the first mark to the left of 0,−20,−2 at the next mark, and so on. See Figure 3.6.

This figure is a number line with 0 in the middle. Then, the scaling has positive numbers 1 to 4 to the right of 0 and negative numbers, negative 1 to negative 4 to the left of 0.
Figure 3.6 On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Number Line-part 2" will help you develop a better understanding of integers.

Example 3.1

Plot the numbers on a number line:

  1. ⓐ33
  2. ⓑ−3−3
  3. ⓒ−2−2

Try It 3.1

Plot the numbers on a number line.

  1. ⓐ11
  2. ⓑ−1−1
  3. ⓒ−4−4

Try It 3.2

Plot the numbers on a number line.

  1. ⓐ−4−4
  2. ⓑ4 4
  3. ⓐ−1−1

Order Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See Figure 3.10.

This figure is a number line. Above the number line there is an arrow pointing to the right labeled increasing. Below the number line there is an arrow pointing to the left labeled decreasing.
Figure 3.10

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation a<ba<b (read aa is less than bb) when aa is to the left of bb on the number line. We write a>ba>b (read aa is greater than bb) when aa is to the right of bb on the number line. This is shown for the numbers 33 and 55 in Figure 3.11.

This figure is a number line with points 3 and 5 labeled with dots. Below the number line is the statements 3 is less than 5 and 5 is greater than 3.
Figure 3.11 The number 33 is to the left of 55 on the number line. So 33 is less than 5,5, and 55 is greater than 3.3.

The numbers lines to follow show a few more examples.

ⓐ

This figure is a number line with points 1 and 4 labeled with dots.

44 is to the right of 11 on the number line, so 4>1.4>1.

11 is to the left of 44 on the number line, so 1<4.1<4.

ⓑ

This figure is a number line with points negative 2 and 1 labeled with dots.

−2−2 is to the left of 11 on the number line, so −2<1.−2<1.

11 is to the right of −2−2 on the number line, so 1>−2.1>−2.

ⓒ

This figure is a number line with points negative 3 and negative 1 labeled with dots.

−1−1 is to the right of −3−3 on the number line, so −1>−3.−1>−3.

−3−3 is to the left of −1−1 on the number line, so −3<−1.−3<−1.

Example 3.2

Order each of the following pairs of numbers using << or >:>:

  1. ⓐ14___614___6
  2. ⓑ−1___9−1___9
  3. ⓒ−1___−4−1___−4
  4. ⓓ2___−202___−20

Try It 3.3

Order each of the following pairs of numbers using << or >.>.

  1. ⓐ15___715___7
  2. ⓑ −2___5−2___5
  3. ⓒ −3___−7−3___−7
  4. ⓓ5___−175___−17

Try It 3.4

Order each of the following pairs of numbers using << or >.>.

  1. ⓐ8___138___13
  2. ⓑ3___−43___−4
  3. ⓒ−5___−2−5___−2
  4. ⓓ9___−219___−21

Find Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers 22 and −2−2 are the same distance from zero, they are called opposites. The opposite of 22 is −2,−2, and the opposite of −2−2 is 22 as shown in Figure 3.13(a). Similarly, 33 and −3−3 are opposites as shown in Figure 3.13(b).

This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites.
Figure 3.13

Opposite

The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero.

Example 3.3

Find the opposite of each number:

  1. ⓐ77
  2. ⓑ−10−10

Try It 3.5

Find the opposite of each number:

  1. ⓐ4 4
  2. ⓑ−3−3

Try It 3.6

Find the opposite of each number:

  1. ⓐ88
  2. ⓑ−5−5

Opposite Notation

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol “−”,“−”, in three different ways.

10−410−4 Between two numbers, the symbol indicates the operation of subtraction.
 We read 10−410−4 as 10 minus 44.
−8−8 In front of a number, the symbol indicates a negative number.
 We read −8−8 as negative eight.
−x−x In front of a variable or a number, it indicates the opposite.
We read−x−x as the opposite of xx.
−(−2)−(−2) Here we have two signs. The sign in the parentheses indicates that the number is negative 2.
 The sign outside the parentheses indicates the opposite. We read −(−2)−(−2) as the opposite of −2.−2.

Opposite Notation

−a−a means the opposite of the number aa

The notation −a−a is read the opposite of a.a.

Example 3.4

Simplify: −(−6).−(−6).

Try It 3.7

Simplify:

−(−1)−(−1)

Try It 3.8

Simplify:

−(−5)−(−5)

Integers

The set of counting numbers, their opposites, and 00 is the set of integers.

Integers

Integers are counting numbers, their opposites, and zero.

…−3,−2,−1,0,1,2,3……−3,−2,−1,0,1,2,3…

We must be very careful with the signs when evaluating the opposite of a variable.

Example 3.5

Evaluate −x:−x:

  1. ⓐ when x=8x=8
  2. ⓑ when x=−8.x=−8.

Try It 3.9

Evaluate −n:−n:

  1. ⓐ whenn=4whenn=4
  2. ⓑ whenn=−4whenn=−4

Try It 3.10

Evaluate: −m:−m:

  1. ⓐ whenm=11whenm=11
  2. ⓑ whenm=−11whenm=−11

Simplify Expressions with Absolute Value

We saw that numbers such as 55 and −5−5 are opposites because they are the same distance from 00 on the number line. They are both five units from 0.0. The distance between 00 and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of 55 is written as |5|,|5|, and the absolute value of −5−5 is written as |−5||−5| as shown in Figure 3.16.

This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.
Figure 3.16

Absolute Value

The absolute value of a number is its distance from 00 on the number line.

The absolute value of a number nn is written as |n|.|n|.

|n|≥0for all numbers|n|≥0for all numbers

Example 3.6

Simplify:

  1. ⓐ |3||3|
  2. ⓑ |−44||−44|
  3. ⓒ |0||0|

Try It 3.11

Simplify:

  1. ⓐ|12||12|
  2. ⓑ−|−28|−|−28|

Try It 3.12

Simplify:

  1. ⓐ |9||9|
  2. ⓑ (b)−|37|(b)−|37|

We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.

Example 3.7

Evaluate:

  1. ⓐ|x|whenx=−35|x|whenx=−35
  2. ⓑ|−y|wheny=−20|−y|wheny=−20
  3. ⓒ −|u|whenu=12−|u|whenu=12
  4. ⓓ −|p|whenp=−14−|p|whenp=−14

Try It 3.13

Evaluate:

  1. ⓐ|x|whenx=−17|x|whenx=−17
  2. ⓑ|−y|wheny=−39|−y|wheny=−39
  3. ⓒ−|m|whenm=22−|m|whenm=22
  4. ⓓ−|p|whenp=−11−|p|whenp=−11

Try It 3.14

  1. ⓐ|y|wheny=−23|y|wheny=−23
  2. ⓑ|−y|wheny=−21|−y|wheny=−21
  3. ⓒ−|n|whenn=37−|n|whenn=37
  4. ⓓ−|q|whenq=−49−|q|whenq=−49

Example 3.8

Fill in <,>,or=<,>,or= for each of the following:

  1. ⓐ|−5|___−|−5||−5|___−|−5|
  2. ⓑ8___−|−8|8___−|−8|
  3. ⓒ−9___−|−9|−9___−|−9|
  4. ⓓ−|−7|___−7−|−7|___−7

Try It 3.15

Fill in <,>,or=for each of the following:<,>,or=for each of the following:

  1. ⓐ|−9|___−|−9||−9|___−|−9|
  2. ⓑ2___−|−2|2___−|−2|
  3. ⓒ−8___|−8|−8___|−8|
  4. ⓓ−|−5|___−5−|−5|___−5

Try It 3.16

Fill in <,>,or=<,>,or= for each of the following:

  1. ⓐ7___−|−7|7___−|−7|
  2. ⓑ−|−11|___−11−|−11|___−11
  3. ⓒ|−4|___−|−4||−4|___−|−4|
  4. ⓓ−1___|−1|−1___|−1|

Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

Example 3.9

Simplify:

  1. ⓐ|9−3||9−3|
  2. ⓑ4|−2|4|−2|

Try It 3.17

Simplify:

  1. ⓐ|12−9||12−9|
  2. ⓑ3|−6|3|−6|

Try It 3.18

Simplify:

  1. ⓐ|27−16||27−16|
  2. ⓑ9|−7|9|−7|

Example 3.10

Simplify: |8+7|−|5+6|.|8+7|−|5+6|.

Try It 3.19

Simplify: |1+8|−|2+5||1+8|−|2+5|

Try It 3.20

Simplify: |9−5|−|7−6||9−5|−|7−6|

Example 3.11

Simplify: 24−|19−3(6−2)|.24−|19−3(6−2)|.

Try It 3.21

Simplify: 19−|11−4(3−1)|19−|11−4(3−1)|

Try It 3.22

Simplify: 9−|8−4(7−5)|9−|8−4(7−5)|

Translate Word Phrases into Expressions with Integers

Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates −20.−20. So does the word opposite in “the opposite of 20.”20.”

Example 3.12

Translate each phrase into an expression with integers:

  1. ⓐ the opposite of positive fourteen
  2. ⓑ the opposite of −11−11
  3. ⓒ negative sixteen
  4. ⓓ two minus negative seven

Try It 3.23

Translate each phrase into an expression with integers:

  1. ⓐ the opposite of positive nine
  2. ⓑ the opposite of −15−15
  3. ⓒ negative twenty
  4. ⓓ eleven minus negative four

Try It 3.24

Translate each phrase into an expression with integers:

  1. ⓐ the opposite of negative nineteen
  2. ⓑ the opposite of twenty-two
  3. ⓒ negative nine
  4. ⓓ negative eight minus negative five

As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example.

Example 3.13

Translate into an expression with integers:

  1. ⓐ The temperature is 12degrees Fahrenheit12degrees Fahrenheit below zero.
  2. ⓑ The football team had a gain of 3yards.3yards.
  3. ⓒ The elevation of the Dead Sea is 1,302feet1,302feet below sea level.
  4. ⓓ A checking account is overdrawn by $40.$40.

Try It 3.25

Translate into an expression with integers:

The football team had a gain of 5yards.5yards.

Try It 3.26

Translate into an expression with integers:

The scuba diver was 30feet30feet below the surface of the water.

Section 3.1 Exercises

Practice Makes Perfect

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the given points on a number line.

1.
  1. ⓐ22
  2. ⓑ−2−2
  3. ⓒ−5−5
2.
  1. ⓐ55
  2. ⓑ−5−5
  3. ⓒ−2−2
3.
  1. ⓐ−8−8
  2. ⓑ88
  3. ⓒ−6−6
4.
  1. ⓐ−7 −7
  2. ⓑ77
  3. ⓒ−1−1

Order Positive and Negative Numbers on the Number Line

In the following exercises, order each of the following pairs of numbers, using << or >.>.

5.
  1. ⓐ9__49__4
  2. ⓑ−3__6−3__6
  3. ⓒ−8__−2−8__−2
  4. ⓓ1__−101__−10
6.
  1. ⓐ6__2;6__2;
  2. ⓑ−7__4;−7__4;
  3. ⓒ−9__−1;−9__−1;
  4. ⓓ9__−39__−3
7.
  1. ⓐ−5__1;−5__1;
  2. ⓑ−4__−9;−4__−9;
  3. ⓒ6__10;6__10;
  4. ⓓ3__−83__−8
8.
  1. ⓐ−7__3;−7__3;
  2. ⓑ−10__−5;−10__−5;
  3. ⓒ2__−6;2__−6;
  4. ⓓ8__98__9

Find Opposites

In the following exercises, find the opposite of each number.

9.
  1. ⓐ22
  2. ⓑ−6−6
10.
  1. ⓐ99
  2. ⓑ−4−4
11.
  1. ⓐ−8−8
  2. ⓑ11
12.
  1. ⓐ−2−2
  2. ⓑ66

In the following exercises, simplify.

13.

− ( −4 ) − ( −4 )

14.

− ( −8 ) − ( −8 )

15.

− ( −15 ) − ( −15 )

16.

− ( −11 ) − ( −11 )

In the following exercises, evaluate.

17.

−mwhen−mwhen

  1. ⓐm=3m=3
  2. ⓑm=−3m=−3
18.

−pwhen−pwhen

  1. ⓐp=6p=6
  2. ⓑp=−6p=−6
19.

−cwhen−cwhen

  1. ⓐc=12c=12
  2. ⓑc=−12c=−12
20.

−dwhen−dwhen

  1. ⓐd=21d=21
  2. ⓑd=−21d=−21

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

21.
  1. ⓐ|7||7|
  2. ⓑ|−25||−25|
  3. ⓒ|0||0|
22.
  1. ⓐ|5||5|
  2. ⓑ|20||20|
  3. ⓒ|−19||−19|
23.
  1. ⓐ|−32||−32|
  2. ⓑ|−18||−18|
  3. ⓒ|16||16|
24.
  1. ⓐ|−41||−41|
  2. ⓑ|−40||−40|
  3. ⓒ|22||22|

In the following exercises, evaluate each absolute value expression.

25.
  1. ⓐ|x|whenx=−28|x|whenx=−28
  2. ⓑ|−u|whenu=−15|−u|whenu=−15
26.
  1. ⓐ|y|wheny=−37|y|wheny=−37
  2. ⓑ|−z|whenz=−24|−z|whenz=−24
27.
  1. ⓐ−|p|whenp=19−|p|whenp=19
  2. ⓑ−|q|whenq=−33−|q|whenq=−33
28.
  1. ⓐ−|a|whena=60−|a|whena=60
  2. ⓑ−|b|whenb=−12−|b|whenb=−12

In the following exercises, fill in <,>,or=<,>,or= to compare each expression.

29.
  1. ⓐ−6__|−6|−6__|−6|
  2. ⓑ−|−3|__−3−|−3|__−3
30.
  1. ⓐ−8__|−8|−8__|−8|
  2. ⓑ−|−2|__−2−|−2|__−2
31.
  1. ⓐ|−3|__−|−3||−3|__−|−3|
  2. ⓑ4__−|−4|4__−|−4|
32.
  1. ⓐ|−5|__−|−5||−5|__−|−5|
  2. ⓑ9__−|−9|9__−|−9|

In the following exercises, simplify each expression.

33.

| 8 − 4 | | 8 − 4 |

34.

| 9 − 6 | | 9 − 6 |

35.

8 | −7 | 8 | −7 |

36.

5 | −5 | 5 | −5 |

37.

| 15 − 7 | − | 14 − 6 | | 15 − 7 | − | 14 − 6 |

38.

| 17 − 8 | − | 13 − 4 | | 17 − 8 | − | 13 − 4 |

39.

18 − | 2 ( 8 − 3 ) | 18 − | 2 ( 8 − 3 ) |

40.

15 − | 3 ( 8 − 5 ) | 15 − | 3 ( 8 − 5 ) |

41.

8 ( 14 − 2 | −2 | ) 8 ( 14 − 2 | −2 | )

42.

6 ( 13 − 4 | −2 | ) 6 ( 13 − 4 | −2 | )

Translate Word Phrases into Expressions with Integers

Translate each phrase into an expression with integers. Do not simplify.

43.
  1. ⓐ the opposite of 88
  2. ⓑ the opposite of −6−6
  3. ⓒ negative three
  4. ⓓ 44 minus negative 33
44.
  1. ⓐ the opposite of 1111
  2. ⓑ the opposite of −4−4
  3. ⓒ negative nine
  4. ⓓ 88 minus negative 22
45.
  1. ⓐ the opposite of 2020
  2. ⓑ the opposite of −5−5
  3. ⓒ negative twelve
  4. ⓓ 1818 minus negative 77
46.
  1. ⓐ the opposite of 1515
  2. ⓑ the opposite of −9−9
  3. ⓒ negative sixty
  4. ⓓ1212 minus 55
47.

a temperature of 6degrees6degrees below zero

48.

a temperature of 14degrees14degrees below zero

49.

an elevation of 40feet40feet below sea level

50.

an elevation of 65feet65feet below sea level

51.

a football play loss of 12yards12yards

52.

a football play gain of 4yards4yards

53.

a stock gain of $3$3

54.

a stock loss of $5$5

55.

a golf score one above par

56.

a golf score of 33 below par

Everyday Math

57.

Elevation The highest elevation in the United States is Mount McKinley, Alaska, at 20,320feet20,320feet above sea level. The lowest elevation is Death Valley, California, at 282feet282feet below sea level. Use integers to write the elevation of:

  1. ⓐ Mount McKinley
  2. ⓑ Death Valley
58.

Extreme temperatures The highest recorded temperature on Earth is 58° Celsius,58° Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature is 90°90° below 0° Celsius,0° Celsius, recorded in Antarctica in 1983. Use integers to write the:

  1. ⓐ highest recorded temperature
  2. ⓑ lowest recorded temperature
59.

State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of $540 million.$540 million. That same month, Texas estimated it would have a budget deficit of $27 billion.$27 billion. Use integers to write the budget:

  1. ⓐ surplus
  2. ⓑ deficit
60.

College enrollments Across the United States, community college enrollment grew by 1,400,0001,400,000 students from 20072007 to 2010.2010. In California, community college enrollment declined by 110,171110,171 students from 20092009 to 2010.2010. Use integers to write the change in enrollment:

  1. ⓐ growth
  2. ⓑ decline

Writing Exercises

61.

Give an example of a negative number from your life experience.

62.

What are the three uses of the “−” sign in algebra? Explain how they differ.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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