Subtract Integers

3.3 Subtract Integers

The topics covered in this section are:

  1. Model subtraction of integers
  2. Simplify expressions with integers
  3. Evaluate variable expressions with integers
  4. Translate words phrases to algebraic expressions
  5. Subtract integers in applications

3.3.1 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 $5-3$ as five take away three. When we use counters, we can think of subtraction the same way.

We will model four subtraction facts using the numbers $5$ and $3$.

  • $5-3$
  • $-5-(-3)$
  • $-5-3$
  • $5-(-3)$

Example 1

Model: $5-3$.

Solution

Interpret the expression.$5-3$ means $5$ take away $3$.
Model the first number. Start with $5$ positives..
Take away the second number. So take away $3$ positives..
Find the counters that are left..
$5-3=2$.
The difference between $5$ and $3$ is $2$.

Example 2

Model: $-5-(-3)$.

Solution

Interpret the expression.$-5-(-3)$ means $-5$ take away $-3$.
Model the first number. Start with $5$ negatives..
Take away the second number. So take away $3$ negatives..
Find the number of counters that are left..
$-5-(-3)=-2$.
The difference between $-5$ and $-3$ is $-2$.

Notice that Example 1 and Example 2 are very much alike.

  • First, we subtracted $3$ positives from $5$ positives to get $2$ positives.
  • Then we subtracted $3$ negatives from $5$ negatives to get $2$ 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

Model: $-5-3$.

Solution

Interpret the expression.$-5-3$ means $-5$ take away $3$.
Model the first number. Start with $5$ negatives..
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.
.
Now take away $3$ positives..
Count the number of counters that are left..
$-5-3=-8$.
The difference of $-5$ and $3$ is $-8$.

Example 4

Model: $5-(-3)$.

Solution

Interpret the expression.$5-(-3)$ means $5$ take away $-3$.
Model the first number. Start with $5$ positives..
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.
.
Then take away $3$ negatives..
Count the number of counters that are left..
The difference of $5$ and $-3$ is $8$.
$5-(-3)=8$

Example 5

Model each subtraction.

  1. $8-2$
  2. $-5-4$
  3. $6-(-6)$
  4. $-8-(-3)$
Solution

Part 1.$8-2$
This means $8$ take away 2.
Start with $8$ positives..
Take away $2$ positives..
How many are left?$6$
$8-2=6$
Part 2.$-5-4$
This means $-5$ take away $4$.
Start with $5$ negatives..
You need to take away $4$ positives.
Add $4$ neutral pairs to get $4$ positives.
.
.
Take away $4$ positives..
How many are left?.
$-9$
$-5-4=-9$
Part 3.$6-(-6)$
This means $6$ take away $-6$.
Start with $6$ positives..
Add $6$ neutrals to get $6$ negatives to take away..
Remove $6$ negatives..
How many are left?.
$12$
$6-(-6)=12$
Part 4.$-8-(-3)$
This means $-8$ take away $-3$.
Start with $8$ negatives..
Take away $3$ negatives..
How many are left?.
$-5$
$-8-(-3)=-5$

Example 6

Model each subtraction expression:

  1. $2-8$
  2. $-3-(-8)$
Solution

Part 1. We start with $2$ positives..
We need to take away $8$ positives, but we have only $2$.
Add neutral pairs until there are $8$ positives to take away..
Then take away eight positives..
Find the number of counters that are left.
There are $6$ negatives.
.
$2-8=-6$
Part 2. We start with $3$ negatives..
We need to take away $8$ negatives, but we have only $3$.
Add neutral pairs until there are $8$ negatives to take away..
Then take away the $8$ negatives..
Find the number of counters that are left.
There are $5$ positives.
.
$-3-(-8)=5$

3.3.2 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 $-23-7$
Think: We start with $23$ negative counters.
We have to subtract $7$ positives, but there are no positives to take away.
So we add $7$ neutral pairs to get the $7$ positives. Now we take away the $7$ positives.
So what’s left? We have the original $23$ negatives plus $7$ more negatives from the neutral pair. The result is $30$ negatives.

$-23-7=-30$

Notice, that to subtract $7$, we added $7$ negatives.

Subtract $30-(-12)$.
Think: We start with $30$ positives.
We have to subtract $12$ negatives, but there are no negatives to take away.
So we add $12$ neutral pairs to the $30$ positives. Now we take away the $12$ negatives.
What’s left? We have the original $30$ positives plus $12$ more positives from the neutral pairs. The result is $42$ positives.

$30-(-12)=42$

Notice that to subtract $-12$, we added $12$.

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

$a-b=a+(-b)$

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 $6-4$ gives the same answer as $6+(-4)$.

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 $6-4$ long ago. But knowing that $6-4$ gives the same answer as $6+(-4)$ helps when we are subtracting negative numbers.

Example 7

Simplify:

  1. $13-8$ and $13+(-8)$
  2. $-17-9$ and $-17+(-9)$
Solution

Part 1.$13-8$ and $13+(-8)$
Subtract to simplify.$13-8=5$
Add to simplify.$13+(-8)=5$
Subtracting $8$ from $13$ is the same as adding $-8$ to $13$.
Part 2.$-17-9$ and $-17+(-9)$
Subtract to simplify.$-17-9=-26$
Add to simplify.$-17+(-9)=-26$
Subtracting $9$ from $-17$ is the same as adding $-9$ to $-17$.

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 $8-(-5)$ gives the same result as $8+5$. Subtracting a negative number is like adding a positive.

Example 8

Simplify:

  1. $9-(-15)$ and $9+15$
  2. $-7-(-4)$ and $-7+4$
Solution

Part 1.$9-(-15)$ and $9+15$
Subtract to simplify.$9-(-15)=24$
Add to simplify.$9+15=24$
Subtracting $-15$ from $9$ is the same as adding $15$ to $9$.
Part 2.$-7-(-4)$ and $-7+4$
Subtract to simplify.$-7-(-4)=-3$
Add to simplify.$-7+4=-3$
Subtracting $-4$ from $-7$ is the same as adding $4$ to $-7$

Look again at the results of Example 1 through Example 4.

$5-3$$-5-(-3)$
$2$$-2$
$2$ positives$2$ negatives
When there would be enough counters of the color to take away, subtract.
$-5-3$$5-(-3)$
$-8$$8$
$5$ negatives, want to subtract $3$ positives$5$ positives, want to subtract $3$ negatives
need neutral pairsnee neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.

Example 9

Simplify: $-74-(-58)$.

Solution

We are taking $58$ negatives away from $74$ negatives.$-74-(-58)$
Subtract.$-16$

Example 10

Simplify: $7-(-4-3)-9$.

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.$7-(-4-3)-9$
Subtract from left to right.$7-(-7)-9$
Subtract.$14-9$
$5$

Example 11

Simplify: $3 \cdot 7 – 4 \cdot 7 – 5 \cdot 8$.

Solution

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

Multiply first.$3 \cdot 7 – 4 \cdot 7 – 5 \cdot 8$
Subtract from left to right.$21-28-40$
Subtract.$-7-40$
$-47$

3.3.3 Evaluate Variable Expressions with Integers

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

Example 12

Evaluate $x-4$ when

  1. $x=3$
  2. $x=-6$.
Solution

Part 1. To evaluate $x-4$ when $x=3$, substitute $3$ for $x$ in the expression.$x-4$
Substitute $3$ for $x$.$3-4$
Subtract.$-1$
Part 2. To evaluate $x-4$ when $x=-6$, substitute $-6$ for $x$ in the expression.$x-4$
Substitute $-6$ for $x$.$-6-4$
Subtract.$-10$

Example 13

Evaluate $20-z$ when

  1. $z=12$
  2. $z=-12$
Solution

Part 1. To evaluate $20-z$ when $z=12$, substitute $12$ for $z$ in the expression.$20-z$
Substitute $12$ for $z$.$20-12$
Subtract.$8$
Part 2. To evaluate $20-z$ when $z=-12$, substitute $-12$ for $z$ in the expression.$20-z$
Substitute $-12$ for $z$.$20-(-12)$
Subtract.$32$

3.3.4 Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression $a-b$ 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.

Be careful to get $a$ and $b$ in the right order!

Example 14

Translate and the simplify:

  1. the difference of $13$ and $-12$
  2. subtract $24$ from $-19$
Solution

Part 1.difference means subtraction. Subtract the numbers in the order they are given.the difference of $13$ and $-21$
Translate.$13-(-21)$
Simplify.$34$
Part 2. Subtract means to take $24$ away from $-19$.subtract $24$ from $-19$
Translate.$-19-24$
Simplify.$-43$

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

Example 15

In the morning, the temperature in Urbana, Illinois was $11$ degrees Fahrenheit. By mid-afternoon, the temperature had dropped to $-9$ 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 $11$ and $-9$
Step 3. Translate the phrase to an expression.
The word difference indicates subtraction.
$11-(-9)$
Step 4. Simplify the expression.$20$
Step 5. Write a complete sentence that answers the question.The difference in temperature was $20$ degrees Fahrenheit.

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

Example 16

Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is $14,497$ feet above sea level and the elevation of Death Valley is $282$ 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.$14,497-(-282)$
Step 4. Simplify.$14,779$
Step 5. Write a complete sentence that answers the question.The difference in elevation is $14,779$ feet.

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 17

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

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

Part 1.
What are we asked to find?The balance of the account
Write a phrase.$ \$ 25$ minus $ \$ 8$
Translate$ \$ 25 – \$ 8$
Simplify.$ \$ 17$
Write a sentence answer.The balance is $ \$ 17$.
Part 2.
What are we asked to find?The new balance
Write a phrase.$ \$ 17$ minus $ \$ 20$
Translate$ \$ 17 – \$ 20$
Simplify.$ – \$ 3$
Write a sentence answer.She is overdrawn by $ \$3$.
Part 3.
What are we asked to find?The new balance
Write a phrase.$ \$ 10$ more than $- \$ 3$
Translate$- \$ 3 + \$ 10$
Simplify.$ \$ 7$
Write a sentence answer.The balance is now $ \$ 7$.
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