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Subtract Integers

3.3 Subtract Integers

Learning Objectives

By the end of this section, you will be able to:

  • Model subtraction of integers
  • Simplify expressions with integers
  • Evaluate variable expressions with integers
  • Translate words phrases to algebraic expressions
  • Subtract integers in applications

Be Prepared 3.4

Before you get started, take this readiness quiz.

Simplify: 12−(8−1).12−(8−1).
 If you missed this problem, review Example 2.8.

Be Prepared 3.5

Translate the difference of 2020 and −15−15 into an algebraic expression.
 If you missed this problem, review Example 1.36.

Be Prepared 3.6

Add: −18+7.−18+7.
 If you missed this problem, review Example 3.20.

Model Subtraction of Integers

Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.

We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read 5−35−3 as five take away three. When we use counters, we can think of subtraction the same way.

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Subtraction of Signed Numbers" will help you develop a better understanding of subtracting integers.

We will model four subtraction facts using the numbers 55 and 3.3.

5−3−5−(−3)−5−35−(−3)5−3−5−(−3)−5−35−(−3)

Example 3.30

Model: 5−3.5−3.

Solution

Interpret the expression. 5−35−3 means 55 take away 33.
Model the first number. Start with 5 positives. Five light blue circles are arranged in a horizontal line, with the number '5' positioned directly below the middle circle.
Take away the second number. So take away 3 positives. Four light blue circles are enclosed by a pink oval and a left-pointing arrow. The text '2 positives' is written below the grouping, likely denoting a count or selection within the four.
Find the counters that are left. Two light blue circles side-by-side on a white background, resembling abstract bubbles or subtle decorative elements.
5−3=25−3=2.
The difference between 55 and 33 is 22.

Try It 3.59

Model the expression:

6−46−4

Try It 3.60

Model the expression:

7−47−4

Example 3.31

Model: −5−(−3).−5−(−3).

Solution

Interpret the expression. −5−(−3)−5−(−3) means −5−5 take away −3−3.
Model the first number. Start with 5 negatives. Five red circles are displayed in a row, with the number -5 written directly below the middle circles, illustrating a visual representation of the integer negative five.
Take away the second number. So take away 3 negatives. A diagram shows four red circles in a row. The first three are encircled by a purple oval with an arrow. The text '2 negatives' is positioned below the fourth circle.
Find the number of counters that are left. Two light red circles with darker red outlines are shown against a white background.
−5−(−3)=−2−5−(−3)=−2.
The difference between −5−5 and −3−3 is −2−2.

Try It 3.61

Model the expression:

−6−(−4)−6−(−4)

Try It 3.62

Model the expression:

−7−(−4)−7−(−4)

Notice that Example 3.30 and Example 3.31 are very much alike.

  • First, we subtracted 33 positives from 55 positives to get 22 positives.
  • Then we subtracted 33 negatives from 55 negatives to get 22 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.

Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

Example 3.32

Model: −5−3.−5−3.

Solution

Interpret the expression. −5−3−5−3 means −5−5 take away 33.
Model the first number. Start with 5 negatives. Five light red circles with dark red outlines are arranged horizontally, with the number -5 positioned directly below the third circle from the left.
Take away the second number.
So we need to take away 3 positives.
But there are no positives to take away.
Add neutral pairs until you have 3 positives.		 Eight red circles and three light blue circles are shown, with five red circles on the left and a group of three red circles above three blue circles on the right.
Now take away 3 positives. An illustration of subtraction, showing three blue circles highlighted with an oval and an arrow, representing a quantity removed from a larger group of red circles.
Count the number of counters that are left. A diagram shows a row of eight light red circles with a darker red outline. Below the circles, the text '8 negatives' is written in a dark teal color.
−5−3=−8−5−3=−8.
The difference of −5−5 and 33 is −8−8.

Try It 3.63

Model the expression:

−6−4−6−4

Try It 3.64

Model the expression:

−7−4−7−4

Example 3.33

Model: 5−(−3).5−(−3).

Solution

Interpret the expression. 5−(−3)5−(−3) means 55 take away −3−3.
Model the first number. Start with 5 positives. Five light blue circles with a darker blue outline are arranged horizontally on a white background, resembling a progress indicator or a rating system.
Take away the second number, so take away 3 negatives.
But there are no negatives to take away.
Add neutral pairs until you have 3 negatives.		 Eight light blue circles and three red circles are displayed against a white background, arranged in two rows with a gap between groups.
Then take away 3 negatives. Three red circles are grouped by a magenta oval and arrow, signifying their removal or separation from below three of seven light blue circles, likely demonstrating a subtraction or counting concept.
Count the number of counters that are left. Eight light blue circles with a slightly darker blue outline are arranged horizontally on a white background, with the text '8 positives' centered beneath them.
The difference of 55 and −3−3 is 88.
5−(−3)=85−(−3)=8

Try It 3.65

Model the expression:

6−(−4)6−(−4)

Try It 3.66

Model the expression:

7−(−4)7−(−4)

Example 3.34

Model each subtraction.

  1. ⓐ 8 − 2
  2. ⓑ −5 − 4
  3. ⓒ 6 − (−6)
  4. ⓓ −8 − (−3)

Solution

ⓐ
8−28−2
This means 88 take away 22.
Start with 8 positives. Eight light blue circles with a subtle blue outline and shadow are arranged horizontally on a white background.
Take away 2 positives. The first two light blue circles in a sequence are encircled by a magenta oval, with a curving arrow pointing back towards them, signifying a pair or a loop.
How many are left? 66
8−2=68−2=6
This table illustrates the step-by-step process of solving the subtraction problem 8 - 2.
ⓑ
−5−4−5−4
This means −5−5 take away 44.
Start with 5 negatives. Five light red circles, each outlined in a darker red, are arranged horizontally in a row against a plain white background.
You need to take away 4 positives.
Add 4 neutral pairs to get 4 positives.		 Eight light red circles with darker red borders, arranged in two groups of five and three, separated by a gap.
Four light blue circles with blue outlines, arranged horizontally on a white background, potentially indicating a loading state or steps in a process.
Take away 4 positives. Four light blue circles are grouped by a pink oval and arrow, indicating their position beneath four of the seven red circles shown above.
How many are left? A series of nine light pink circles with red outlines are arranged horizontally, separated into a group of five on the left and a group of four on the right by a small gap.
−9−9
−5−4=−9−5−4=−9
Visual demonstration of integer subtraction, showing the steps to calculate -5 - 4 using a chip model.
ⓒ
6−(−6)6−(−6)
This means 66 take away −6−6.
Start with 6 positives. Five light blue circles, outlined in a slightly darker blue, are arranged horizontally across a white background.
Add 6 neutrals to get 6 negatives to take away. Two rows of circles, with 10 light blue circles on top and 6 red circles below, right-aligned, against a white background.
Remove 6 negatives. An illustration showing eleven light blue circles in a row, with five red circles directly below them. A purple oval encircles the red circles, with a curved arrow inside pointing left, suggesting a cyclical process or a moving window.
How many are left? A horizontal row of eleven light blue circles, each with a thin cyan outline, is arranged against a plain white background.
1212
6−(−6)=126−(−6)=12
Step-by-step visual illustration of how to calculate 6 - (-6) using positive and negative integer representations.
ⓓ
−8−(−3)−8−(−3)
This means −8−8 take away −3−3.
Start with 8 negatives. Eight pale red outlined circles in a row.
Take away 3 negatives. A magenta oval groups the first three circles in a row of eight red circles, with a magenta arrow curving from the oval to the left, indicating a grouping or counting action.
How many are left? Five identical light red circles with a darker red outline, arranged horizontally on a plain white background.
−5−5
−8−(−3)=−5−8−(−3)=−5
Illustrates the subtraction of -3 from -8 using a step-by-step visual method with images and mathematical representation.

Try It 3.67

Model each subtraction.

  1. ⓐ 7 - (-8)
  2. ⓑ -7 - (-2)
  3. ⓒ 4 - 1
  4. ⓓ -6 - 8

Try It 3.68

Model each subtraction.

  1. ⓐ 4 - (-6)
  2. ⓑ -8 - (-1)
  3. ⓒ 7 - 3
  4. ⓓ -4 - 2

Example 3.35

Model each subtraction expression:

  1. ⓐ 2−8 2−8
  2. ⓑ −3−(−8)−3−(−8)

Solution

ⓐ
We start with 2 positives.		 Two light blue circles are displayed horizontally, with the numeral '2' printed directly beneath them, indicating a count of two.
We need to take away 8 positives, but we have only 2.
Add neutral pairs until there are 8 positives to take away. An array of 8 blue circles and 6 red circles, illustrating a simple numerical comparison with two blue circles separated from the main group.
Then take away eight positives. Two rows of circles, 8 blue on top with a pink oval and left arrow, and 6 orange on the bottom, likely showing counting or a sequence.
Find the number of counters that are left.
There are 6 negatives.		 Six red circles are aligned horizontally above the text '6 negatives' on a white background, likely illustrating a concept related to negative numbers or values.
2−8=−62−8=−6
ⓑ
We start with 3 negatives.		 Three red circles are arranged horizontally above the number 3, all set against a plain white background.
We need to take away 8 negatives, but we have only 3.
Add neutral pairs until there are 8 negatives to take away. A graphic displaying red and blue circles on a white background. Eight red circles are arranged in two groups on the top row, while five blue circles form a single row below them.
Then take away the 8 negatives. Eight red circles are enclosed by an oval with a leftward arrow, positioned above six blue circles, illustrating two distinct sets of objects.
Find the number of counters that are left.
There are 5 positives.		 Five blue-gray circles are arranged in a horizontal line above the text '5 positives' on a white background, suggesting a count or representation of five positive items.
−3−(−8)=5−3−(−8)=5

Try It 3.69

Model each subtraction expression.

  1. ⓐ 7−97−9
  2. ⓑ −5−(−9)−5−(−9)

Try It 3.70

Model each subtraction expression.

  1. ⓐ 4−74−7
  2. ⓑ −7−(−10)−7−(−10)

Simplify Expressions with Integers

Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.

  • Subtract −23−7.−23−7.
     Think: We start with 2323 negative counters.
     We have to subtract 77 positives, but there are no positives to take away.
     So we add 77 neutral pairs to get the 77 positives. Now we take away the 77 positives.
     So what’s left? We have the original 2323 negatives plus 77 more negatives from the neutral pair. The result is 3030 negatives.
    −23−7=−30−23−7=−30

     Notice, that to subtract 7,7, we added 77 negatives.
  • Subtract 30−(−12).30−(−12).
     Think: We start with 3030 positives.
     We have to subtract 1212 negatives, but there are no negatives to take away.
     So we add 1212 neutral pairs to the 3030 positives. Now we take away the 1212 negatives.
     What’s left? We have the original 3030 positives plus 1212 more positives from the neutral pairs. The result is 4242 positives.
    30−(−12)=4230−(−12)=42

     Notice that to subtract −12,−12, we added 12.12.

While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:

Subtraction Property

a−b=a+(−b)a−b=a+(−b)

Look at these two examples.

This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.

We see that 6−46−4 gives the same answer as 6+(−4).6+(−4).

Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract 6−46−4 long ago. But knowing that 6−46−4 gives the same answer as 6+(−4)6+(−4) helps when we are subtracting negative numbers.

Example 3.36

Simplify:

  1. ⓐ 13−8and13+(−8)13−8and13+(−8)
  2. ⓑ −17−9and−17+(−9)−17−9and−17+(−9)

Solution

ⓐ
13−813−8 and 13+(−8)13+(−8)
Subtract to simplify. 13−8=513−8=5
Add to simplify. 13+(−8)=513+(−8)=5
Subtracting 8 from 13 is the same as adding −8 to 13.
This table demonstrates that subtracting a number is equivalent to adding its opposite, using 13 - 8 and 13 + (-8) as an example.
ⓑ
−17−9−17−9 and −17+(−9)−17+(−9)
Subtract to simplify. −17−9=−26−17−9=−26
Add to simplify. −17+(−9)=−26−17+(−9)=−26
Subtracting 9 from −17 is the same as adding −9 to −17.
This table illustrates that subtracting a number is equivalent to adding its negative counterpart, demonstrated with an example.

Try It 3.71

Simplify each expression:

  1. ⓐ 21−13and21+(−13)21−13and21+(−13)
  2. ⓑ −11−7and−11+(−7)−11−7and−11+(−7)

Try It 3.72

Simplify each expression:

  1. ⓐ 15−7and15+(−7)15−7and15+(−7)
  2. ⓑ −14−8and−14+(−8)−14−8and−14+(−8)

Now look what happens when we subtract a negative.

This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.

We see that 8−(−5)8−(−5) gives the same result as 8+5.8+5. Subtracting a negative number is like adding a positive.

Example 3.37

Simplify:

  1. ⓐ 9−(−15)and9+159−(−15)and9+15
  2. ⓑ −7−(−4)and−7+4−7−(−4)and−7+4

Solution

ⓐ
9−(−15)9−(−15) and 9+159+15
Subtract to simplify. 9−(−15)=249−(−15)=24
Add to simplify. 9+15=249+15=24
Subtracting −15 from 9 is the same as adding 15 to 9.
Demonstrates that subtracting a negative number is equivalent to adding its positive counterpart, using 9 - (-15) = 9 + 15 = 24 as an example.
ⓑ
−7−(−4)−7−(−4) and −7+4−7+4
Subtract to simplify. −7−(−4)=−3−7−(−4)=−3
Add to simplify. −7+4=−3−7+4=−3
Subtracting −4 from −7 is the same as adding 4 to −7
This table illustrates that subtracting a negative number is equivalent to adding a positive number, using the example of -7 - (-4) = -7 + 4.

Try It 3.73

Simplify each expression:

  1. ⓐ 6−(−13)and6+136−(−13)and6+13
  2. ⓑ −5−(−1)and−5+1−5−(−1)and−5+1

Try It 3.74

Simplify each expression:

  1. ⓐ 4−(−19)and4+194−(−19)and4+19
  2. ⓐ −4−(−7)and−4+7−4−(−7)and−4+7

Look again at the results of Example 3.30 - Example 3.33.

5–35–3 –5–(–3)–5–(–3)
22 –2–2
2 positives 2 negatives
When there would be enough counters of the color to take away, subtract.
–5–3–5–3 5–(–3)5–(–3)
–8–8 88
5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives
need neutral pairs need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.
Table 3.4 Subtraction of Integers

Example 3.38

Simplify: −74−(−58).−74−(−58).

Solution

We are taking 58 negatives away from 74 negatives. −74−(−58)−74−(−58)
Subtract. −16−16
This table illustrates the process of subtracting negative numbers, presenting both a verbal description and its corresponding mathematical expression and result.

Try It 3.75

Simplify the expression:

−67−(−38)−67−(−38)

Try It 3.76

Simplify the expression:

−83−(−57)−83−(−57)

Example 3.39

Simplify: 7−(−4−3)−9.7−(−4−3)−9.

Solution

We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.

Simplify inside the parentheses first. A mathematical expression reads '7 minus open parenthesis minus 4 minus 3 close parenthesis minus 9.'
Subtract from left to right. A mathematical expression showing 7 minus negative 7 minus 9.
Subtract. The mathematical expression '14 - 9' is displayed in a clear, dark font against a plain white background, ready for calculation.
The black numeral '5' is prominently displayed against a plain white background.
Illustrates the step-by-step simplification of a mathematical expression using textual descriptions and visual representations.

Try It 3.77

Simplify the expression:

8−(−3−1)−98−(−3−1)−9

Try It 3.78

Simplify the expression:

12−(−9−6)−1412−(−9−6)−14

Example 3.40

Simplify: 3·7−4·7−5·8.3·7−4·7−5·8.

Solution

We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.

Multiply first. The image shows the mathematical expression '3.7 - 4.7 - 5.8' in a horizontal line, rendered in a plain, clear font against a white background.
Subtract from left to right. The numbers 21, 28, and 40 are displayed in a sequence, separated by hyphens, against a white background.
Subtract. The image shows the mathematical expression -7 - 40, centered against a white background.
The number -47 is displayed in black text on a white background.
This table illustrates the sequential steps and their corresponding visual changes when evaluating a mathematical expression, demonstrating the order of operations.

Try It 3.79

Simplify the expression:

6·2−9·1−8·9.6·2−9·1−8·9.

Try It 3.80

Simplify the expression:

2·5−3·7−4·92·5−3·7−4·9

Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

Example 3.41

Evaluate x−4whenx−4when

  1. ⓐ x=3x=3
  2. ⓑ x=−6.x=−6.

Solution

ⓐ To evaluate x−4x−4 when x=3x=3, substitute 33 for xx in the expression.

The image displays the mathematical expression 'x - 4' in black text against a white background. The variable 'x' is followed by a minus sign, and then the numeral '4' completes the simple algebraic expression.
The image displays the text 'Substitute 3 for x.' in a bold, sans-serif font against a white background. The number '3' is highlighted in red, while the rest of the text is a dark, desaturated teal color. The mathematical expression '3 - 4' is displayed, showing a subtraction operation between the numbers three and four.
Subtract. The image displays the number '-1' in black font against a white background.

ⓑ To evaluate x−4x−4 when x=−6,x=−6, substitute −6−6 for xx in the expression.

The mathematical expression 'x - 4' is displayed in black font against a white background.
The text 'Substitute -6 for x.' is displayed in a blue-green gradient font on a white background, with the '-6' highlighted in red. A mathematical expression displays '-6 - 4' in red and black text on a white background, representing the subtraction of 4 from -6.
Subtract. The number negative 10, or minus 10, is displayed in black text on a white background.

Try It 3.81

Evaluate each expression:

y−7wheny−7when

  1. ⓐ y=5y=5
  2. ⓑ y=−8y=−8

Try It 3.82

Evaluate each expression:

m−3whenm−3when

  1. ⓐ m=1m=1
  2. ⓑ m=−4m=−4

Example 3.42

Evaluate 20−zwhen20−zwhen

  1. ⓐ z=12z=12
  2. ⓑ z=−12z=−12

Solution

ⓐ To evaluate 20−zwhenz=12,20−zwhenz=12, substitute 1212 for zz in the expression.

A mathematical expression displays '20 - z' in a simple, clear font against a white background.
The image shows the text 'Substitute 12 for z.' The word 'Substitute' is dark blue-green, the number '12' is red, and 'for z.' is dark blue-green. The text suggests a mathematical or algebraic instruction. A mathematical expression showing '20 - 12', with the number '12' highlighted in red.
Subtract. A single black numeral '8' is displayed against a stark white background.

ⓑ To evaluate 20−zwhenz=−12,substitute−12forzin the expression.20−zwhenz=−12,substitute−12forzin the expression.

The image displays the algebraic expression '20 - z' in a simple, clear font against a white background.
The text reads 'Substitute -12 for z.', with '-12' highlighted in red. A mathematical expression displays '20 - (-12)' in black and red text against a white background.
Subtract. The number 32 is displayed prominently in a gray font on a plain white background, centered within the image frame. The digits appear to be a standard, sans-serif typeface, clean and legible.

Try It 3.83

Evaluate each expression:

17−kwhen17−kwhen

  1. ⓐ k=19k=19
  2. ⓑ k=−19k=−19

Try It 3.84

Evaluate each expression:

−5−bwhen−5−bwhen

  1. ⓐ b=14b=14
  2. ⓑ b=−14b=−14

Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression a−ba−b may be read in several ways as shown below.

This table has six rows. The first row has a - b. The second row states a minus b. The third row states the difference of a and b. The fourth row states subtract b from a. The fifth row states b subtracted from a. The sixth row states b less than a.
Figure 3.18

Be careful to get aa and bb in the right order!

Example 3.43

Translate and then simplify:

  1. ⓐ the difference of 1313 and −21−21
  2. ⓑ subtract 2424 from −19−19

Solution

ⓐ A difference means subtraction. Subtract the numbers in the order they are given.

The text 'the difference of 13 and -21' is displayed, with the word 'difference' highlighted in red.
Translate. A mathematical expression '13 - (-21)' is displayed on a white background, demonstrating the subtraction of a negative number.
Simplify. The number 34 is prominently displayed in the upper right portion of a white background. The digits are in a dark color, providing a clear contrast against the bright, plain surface.

ⓑ Subtract means to take 2424 away from −19.−19.

The text 'subtract 24 from -19' is displayed, instructing a mathematical operation.
Translate. The image displays a mathematical expression '-19 - 24' in a dark gray font on a plain white background, presenting a simple arithmetic problem.
Simplify. The number -43 is displayed in black text on a plain white background.

Try It 3.85

Translate and simplify:

  1. ⓐ the difference of 1414 and −23−23
  2. ⓑ subtract 2121 from −17−17

Try It 3.86

Translate and simplify:

  1. ⓐ the difference of 1111 and −19−19
  2. ⓑ subtract 1818 from −11−11

Subtract Integers in Applications

It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

How To

Solve Application Problems.

  1. Step 1. Identify what you are asked to find.
  2. Step 2. Write a phrase that gives the information to find it.
  3. Step 3. Translate the phrase to an expression.
  4. Step 4. Simplify the expression.
  5. Step 5. Answer the question with a complete sentence.

Example 3.44

In the morning, the temperature in Urbana, Illinois was 1111 degrees Fahrenheit. By mid-afternoon, the temperature had dropped to −9−9 degrees Fahrenheit. What was the difference between the morning and afternoon temperatures?

Solution

Step 1. Identify what we are asked to find. the difference between the morning and afternoon temperatures
Step 2. Write a phrase that gives the information to find it. the difference of 1111 and −9−9
Step 3. Translate the phrase to an expression.
The word difference indicates subtraction.		 11−(−9)11−(−9)
Step 4. Simplify the expression. 2020
Step 5. Write a complete sentence that answers the question. The difference in temperature was 2020 degrees Fahrenheit.
This table illustrates the step-by-step process of solving a temperature difference problem, from identifying the unknown to stating the final answer.

Try It 3.87

In the morning, the temperature in Anchorage, Alaska was 15 degrees Fahrenheit.15 degrees Fahrenheit. By mid-afternoon the temperature had dropped to 30 degrees30 degrees below zero. What was the difference between the morning and afternoon temperatures?

Try It 3.88

The temperature in Denver was −6−6 degrees Fahrenheit at lunchtime. By sunset the temperature had dropped to −15 degree Fahrenheit.−15 degree Fahrenheit. What was the difference between the lunchtime and sunset temperatures?

Geography provides another application of negative numbers with the elevations of places below sea level.

Example 3.45

Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is 14,49714,497 feet above sea level and the elevation of Death Valley is 282282 feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?

Solution

Step 1. What are we asked to find? The difference in elevation between Mt. Whitney and Death Valley
Step 2. Write a phrase. elevation of Mt. Whitney−elevation of Death Valley
Step 3. Translate. 14,497−(−282)14,497−(−282)
Step 4. Simplify. 14,77914,779
Step 5. Write a complete sentence that answers the question. The difference in elevation is 14,77914,779 feet.
Step-by-step solution for calculating the elevation difference between Mt. Whitney and Death Valley.

Try It 3.89

One day, John hiked to the 10,023 foot10,023 foot summit of Haleakala volcano in Hawaii. The next day, while scuba diving, he dove to a cave 80 feet80 feet below sea level. What is the difference between the elevation of the summit of Haleakala and the depth of the cave?

Try It 3.90

The submarine Nautilus is at 340 feet340 feet below the surface of the water and the submarine Explorer is 573 feet573 feet below the surface of the water. What is the difference in the position of the Nautilus and the Explorer?

Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!)

Example 3.46

Leslie has $25$25 in her checking account and she writes a check for $8.$8.

  1. ⓐ What is the balance after she writes the check?
  2. ⓑ She writes a second check for $20.$20. What is the new balance after this check?
  3. ⓒ Leslie’s friend told her that she had lost a check for $10$10 that Leslie had given her with her birthday card. What is the balance in Leslie’s checking account now?

Solution

ⓐ
What are we asked to find? The balance of the account
Write a phrase. $25$25 minus $8$8
Translate The image displays a mathematical subtraction problem, written as '$25 - $8', indicating an operation to find the difference between twenty-five dollars and eight dollars.
Simplify. The number '$17' is displayed in black text on a plain white background, indicating a monetary value of seventeen dollars.
Write a sentence answer. The balance is $17$17.
This table outlines the steps, from question to final answer, for solving a basic mathematical word problem involving an account balance.
ⓑ
What are we asked to find? The new balance
Write a phrase. $17$17 minus $20$20
Translate A numerical text image displays a price range of '$17 - $20' in black font on a white background.
Simplify. The image displays '-$3' in black text against a plain white background, indicating a negative financial value or a cost of three dollars.
Write a sentence answer. She is overdrawn by $3$3.
Step-by-step solution for calculating a bank balance after a withdrawal, illustrating an overdraft scenario.
ⓒ
What are we asked to find? The new balance
Write a phrase. $10$10 more than −$3−$3
Translate The image displays a mathematical expression showing a sum: -$3 + $10. This represents adding a positive $10 to a negative $3, resulting in a net positive value of $7.
Simplify. The image shows the text $7, indicating a price or monetary value of seven dollars.
Write a sentence answer. The balance is now $7$7.
Step-by-step solution demonstrating how to calculate a new financial balance, from identifying the problem to stating the final answer.

Try It 3.91

Araceli has $75$75 in her checking account and writes a check for $27.$27.

  1. ⓐ What is the balance after she writes the check?
  2. ⓑ She writes a second check for $50.$50. What is the new balance?
  3. ⓒ The check for $20$20 that she sent a charity was never cashed. What is the balance in Araceli’s checking account now?

Try It 3.92

Genevieve’s bank account was overdrawn and the balance is −$78.−$78.

  1. ⓐ She deposits a check for $24$24 that she earned babysitting. What is the new balance?
  2. ⓑ She deposits another check for $49.$49. Is she out of debt yet? What is her new balance?

Section 3.3 Exercises

Practice Makes Perfect

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

127.

8 − 2 8 − 2

128.

9 − 3 9 − 3

129.

−5 − ( −1 ) −5 − ( −1 )

130.

−6 − ( −4 ) −6 − ( −4 )

131.

−5 − 4 −5 − 4

132.

−7 − 2 −7 − 2

133.

8 − ( −4 ) 8 − ( −4 )

134.

7 − ( −3 ) 7 − ( −3 )

Simplify Expressions with Integers

In the following exercises, simplify each expression.

135.
  1. ⓐ 15−615−6
  2. ⓑ 15+(−6)15+(−6)
136.
  1. ⓐ 12−912−9
  2. ⓑ 12+(−9)12+(−9)
137.
  1. ⓐ 44−2844−28
  2. ⓑ 44+(−28)44+(−28)
138.
  1. ⓐ 35−1635−16
  2. ⓑ 35+(−16)35+(−16)
139.
  1. ⓐ 8−(−9)8−(−9)
  2. ⓑ 8+98+9
140.
  1. ⓐ 4−(−4)4−(−4)
  2. ⓑ 4+44+4
141.
  1. ⓐ 27−(−18)27−(−18)
  2. ⓑ 27+1827+18
142.
  1. ⓐ 46−(−37)46−(−37)
  2. ⓑ 46+3746+37

In the following exercises, simplify each expression.

143.

15 − ( −12 ) 15 − ( −12 )

144.

14 − ( −11 ) 14 − ( −11 )

145.

10 − ( −19 ) 10 − ( −19 )

146.

11 − ( −18 ) 11 − ( −18 )

147.

48 − 87 48 − 87

148.

45 − 69 45 − 69

149.

31 − 79 31 − 79

150.

39 − 81 39 − 81

151.

−31 − 11 −31 − 11

152.

−32 − 18 −32 − 18

153.

−17 − 42 −17 − 42

154.

−19 − 46 −19 − 46

155.

−103 − ( −52 ) −103 − ( −52 )

156.

−105 − ( −68 ) −105 − ( −68 )

157.

−45 − ( −54 ) −45 − ( −54 )

158.

−58 − ( −67 ) −58 − ( −67 )

159.

8 − 3 − 7 8 − 3 − 7

160.

9 − 6 − 5 9 − 6 − 5

161.

−5 − 4 + 7 −5 − 4 + 7

162.

−3 − 8 + 4 −3 − 8 + 4

163.

−14 − ( −27 ) + 9 −14 − ( −27 ) + 9

164.

−15 − ( −28 ) + 5 −15 − ( −28 ) + 5

165.

71 + ( −10 ) − 8 71 + ( −10 ) − 8

166.

64 + ( −17 ) − 9 64 + ( −17 ) − 9

167.

−16 − ( −4 + 1 ) − 7 −16 − ( −4 + 1 ) − 7

168.

−15 − ( −6 + 4 ) − 3 −15 − ( −6 + 4 ) − 3

169.

( 2 − 7 ) − ( 3 − 8 ) ( 2 − 7 ) − ( 3 − 8 )

170.

( 1 − 8 ) − ( 2 − 9 ) ( 1 − 8 ) − ( 2 − 9 )

171.

− ( 6 − 8 ) − ( 2 − 4 ) − ( 6 − 8 ) − ( 2 − 4 )

172.

− ( 4 − 5 ) − ( 7 − 8 ) − ( 4 − 5 ) − ( 7 − 8 )

173.

25 − [ 10 − ( 3 − 12 ) ] 25 − [ 10 − ( 3 − 12 ) ]

174.

32 − [ 5 − ( 15 − 20 ) ] 32 − [ 5 − ( 15 − 20 ) ]

175.

6 ⋅ 3 − 4 ⋅ 3 − 7 ⋅ 2 6 ⋅ 3 − 4 ⋅ 3 − 7 ⋅ 2

176.

5 ⋅ 7 − 8 ⋅ 2 − 4 ⋅ 9 5 ⋅ 7 − 8 ⋅ 2 − 4 ⋅ 9

177.

5 2 − 6 2 5 2 − 6 2

178.

6 2 − 7 2 6 2 − 7 2

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression for the given values.

179.

x−6whenx−6when

  1. ⓐ x=3x=3
  2. ⓑ x=−3x=−3
180.

x−4whenx−4when

  1. ⓐ x=5x=5
  2. ⓑ x=−5x=−5
181.

5−ywhen5−ywhen

  1. ⓐ y=2y=2
  2. ⓑ y=−2y=−2
182.

8−ywhen8−ywhen

  1. ⓐ y=3y=3
  2. ⓑ y=−3y=−3
183.

4 x 2 − 15 x + 1 when x = 3 4 x 2 − 15 x + 1 when x = 3

184.

5 x 2 − 14 x + 7 when x = 2 5 x 2 − 14 x + 7 when x = 2

185.

−12 − 5 x 2 when x = 6 −12 − 5 x 2 when x = 6

186.

−19 − 4 x 2 when x = 5 −19 − 4 x 2 when x = 5

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

187.
  1. ⓐ The difference of 33 and −10−10
  2. ⓑ Subtract −20−20 from 4545
188.
  1. ⓐ The difference of 88 and −12−12
  2. ⓑ Subtract −13−13 from 5050
189.
  1. ⓐ The difference of −6−6 and 99
  2. ⓑ Subtract −12−12 from −16−16
190.
  1. ⓐ The difference of −8−8 and 99
  2. ⓑ Subtract −15−15 from −19−19
191.
  1. ⓐ 88 less than −17−17
  2. ⓑ −24−24 minus 3737
192.
  1. ⓐ 55 less than −14−14
  2. ⓑ −13−13 minus 4242
193.
  1. ⓐ 2121 less than66
  2. ⓑ 3131 subtracted from −19−19
194.
  1. ⓐ 3434 less than77
  2. ⓑ 2929 subtracted from −50−50

Subtract Integers in Applications

In the following exercises, solve the following applications.

195.

Temperature One morning, the temperature in Urbana, Illinois, was 28° Fahrenheit.28° Fahrenheit. By evening, the temperature had dropped 38° Fahrenheit.38° Fahrenheit. What was the temperature that evening?

196.

Temperature On Thursday, the temperature in Spincich Lake, Michigan, was 22° Fahrenheit.22° Fahrenheit. By Friday, the temperature had dropped 35° Fahrenheit.35° Fahrenheit. What was the temperature on Friday?

197.

Temperature On January 15, the high temperature in Anaheim, California, was 84° Fahrenheit.84° Fahrenheit. That same day, the high temperature in Embarrass, Minnesota was −12° Fahrenheit.−12° Fahrenheit. What was the difference between the temperature in Anaheim and the temperature in Embarrass?

198.

Temperature On January 21, the high temperature in Palm Springs, California, was 89°,89°, and the high temperature in Whitefield, New Hampshire was −31°.−31°. What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

199.

Football At the first down, the Warriors football team had the ball on their 30-yard line.30-yard line. On the next three downs, they gained 2 yards,2 yards, lost 7 yards,7 yards, and lost 4 yards.4 yards. What was the yard line at the end of the third down?

200.

Football At the first down, the Barons football team had the ball on their 20-yard line.20-yard line. On the next three downs, they lost 8 yards,8 yards, gained 5 yards,5 yards, and lost 6 yards.6 yards. What was the yard line at the end of the third down?

201.

Checking Account John has $148$148 in his checking account. He writes a check for $83.$83. What is the new balance in his checking account?

202.

Checking Account Ellie has $426$426 in her checking account. She writes a check for $152.$152. What is the new balance in her checking account?

203.

Checking Account Gina has $210$210 in her checking account. She writes a check for $250.$250. What is the new balance in her checking account?

204.

Checking Account Frank has $94$94 in his checking account. He writes a check for $110.$110. What is the new balance in his checking account?

205.

Checking Account Bill has a balance of −$14−$14 in his checking account. He deposits $40$40 to the account. What is the new balance?

206.

Checking Account Patty has a balance of −$23−$23 in her checking account. She deposits $80$80 to the account. What is the new balance?

Everyday Math

207.

Camping Rene is on an Alpine hike. The temperature is−7°.−7°. Rene’s sleeping bag is rated “comfortable to −20°”.−20°”. How much can the temperature change before it is too cold for Rene’s sleeping bag?

208.

Scuba Diving Shelly’s scuba watch is guaranteed to be watertight to −100feet.−100feet. She is diving at −45feet−45feet on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed?

Writing Exercises

209.

Explain why the difference of 99 and −6−6 is 15.15.

210.

Why is the result of subtracting 3−(−4)3−(−4) the same as the result of adding 3+4?3+4?

Self Check

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

A self-assessment checklist for mathematics skills related to integers. The table lists five 'I can...' statements (e.g., model subtraction of integers) and three columns for assessment: Confidently, With some help, and No-I don't get it!

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?