3.3 Subtract Integers

Learning objectives

Learning objectives

• Model subtraction of integers

• Simplify expressions with integers

• Evaluate variable expressions with integers

• Translate words phrases to algebraic expressions

• Subtract integers in applications

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 $3.$

Example 3.30

Solution

1. Model the minuend. Draw five blue counters to represent the $5$ in $5-3$, matching the first figure.
2. Identify what to remove. Three counters must be taken away from the group.
3. Remove the counters. Circle three of the five blue counters and point to them with an arrow, as shown in the second figure.
4. Count the remainder. Two blue counters remain, matching the final figure.
5. State the result. The model shows that $5-3=2$.

Takeaway: To model subtraction, remove a counter for each unit in the subtracted amount and count what remains.

Example 3.31

Solution

1. Model the starting value. Draw five red counters to represent $-5$, matching the first figure.
2. Interpret the negative subtrahend. Subtracting $-3$ means removing three negative counters.
3. Remove the counters. Take away three of the five red counters, as indicated by the arrow in the second figure.
4. Count the remainder. Two red counters remain, as shown in the final figure.
5. State the result. The model shows that $-5-(-3)=-2$.

Takeaway: Subtracting a negative integer means removing negative counters, so both the count and the sign of the remainder remain negative.

Notice that Example 3.30 and Example 3.31 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.

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

Solution

1. Start with the negative value. Draw five red counters to represent $-5$, as shown in the first figure.
2. Add neutral pairs for the subtrahend. Subtracting the positive $3$ means removing three blue counters, so add three neutral pairs — three blue and three red counters — giving eight red and three blue counters.
3. Remove the positives. Take away the three blue counters, leaving the eight red counters untouched.
4. Read the remainder. The remaining eight red counters represent $-8$, so the model gives $-5-3=-8$.

Takeaway: Subtracting a positive from a negative deepens the negative, so the result keeps the negative sign and grows in magnitude.

Example 3.33

Solution

1. Start with the positive value. Draw five blue counters to represent $5$, as shown in the first figure.
2. Add neutral pairs for the subtrahend. Subtracting the negative $-3$ means removing three red counters, so add three neutral pairs — three red and three blue counters — giving eight blue and three red counters.
3. Remove the negatives. Take away the three red counters, leaving the eight blue counters untouched.
4. Read the remainder. The remaining eight blue counters represent $8$, so the model gives $5-(-3)=8$.

Takeaway: Subtracting a negative is the same as adding its opposite, so the result moves away from zero in the positive direction.

Example 3.34

Solution

1. Model $8-2$. Draw eight blue counters and remove two of them, leaving six blue counters, so $8-2=6$.
2. Model $-5-4$. Draw five red counters for $-5$. Subtracting the positive $4$ needs four blue counters to remove, so add four neutral pairs, giving nine red and four blue counters; removing the four blue counters leaves nine red counters, so $-5-4=-9$.
3. Model $6-(-6)$. Draw six blue counters for $6$. Subtracting the negative $-6$ needs six red counters to remove, so add six neutral pairs, giving twelve blue and six red counters; removing the six red counters leaves twelve blue counters, so $6-(-6)=12$.
4. Model $-8-(-3)$. Draw eight red counters for $-8$ and remove three of them, because the $-3$ being subtracted is already present; five red counters remain, so $-8-(-3)=-5$.
5. State the results. The four models give $6$, $-9$, $12$, and $-5$, with the figures showing the neutral pairs added and the counters removed in each case.

Takeaway: Adding neutral pairs creates the counters a subtraction must remove; subtracting a negative adds its magnitude, while subtracting a positive from a negative deepens it.

Example 3.35

Solution

1. Model $2-8$. Start with two blue counters for $2$. Subtracting the positive $8$ needs eight blue counters, so add six neutral pairs, giving eight blue and six red counters; removing the eight blue counters leaves six red counters, so $2-8=-6$.
2. Model $-3-(-8)$. Start with three red counters for $-3$. Subtracting the negative $-8$ needs eight red counters, so add five neutral pairs, giving eight red and five blue counters; removing the eight red counters leaves five blue counters, so $-3-(-8)=5$.
3. Compare the remainders. The first model leaves six negative counters, and the second leaves five positive counters.
4. State the results. The expressions evaluate to $-6$ and $5$, and the figures show the neutral pairs added and the counters removed in each case.

Takeaway: When the subtracted quantity is larger in magnitude than the starting value, the result takes the opposite sign of that subtracted quantity.

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.

Notice, that to subtract $\mathrm{7,}$ 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.

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:

Look at these two examples.

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 3.36

Solution

1. Use the subtraction rule. Subtracting a positive integer is equivalent to adding its opposite.
2. Simplify $13-8$. The calculation gives $13-8=5$, since eight units are removed from thirteen.
3. Compare with $13+(-8)$. Adding the opposite $-8$ to $13$ also gives $13+(-8)=5$.
4. Simplify $-17-9$. Removing nine from a value of $-17$ gives $-17-9=-26$.
5. Compare with $-17+(-9)$. Adding the opposite $-9$ gives the same result, $-17+(-9)=-26$.

Takeaway: Subtracting a positive number matches adding its opposite: $13-8=13+(-8)=5$ and $-17-9=-17+(-9)=-26$.

Now look what happens when we subtract a negative.

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

Example 3.37

Solution

1. Restate the rule. Subtracting a negative integer is equivalent to adding the positive opposite of that integer.
2. Simplify $9-(-15)$. The negative being subtracted has opposite $15$, so this becomes $9+15=24$.
3. Check the equivalent sum. Evaluating $9+15$ directly also gives $24$, confirming the first result.
4. Simplify $-7-(-4)$. The opposite of $-4$ is $4$, so this becomes $-7+4=-3$.
5. Check the equivalent sum. Adding $4$ to $-7$ gives $-3$, so both pairs have matching values.

Takeaway: Subtracting a negative number matches adding its positive opposite, so $9-(-15)=9+15=24$ and $-7-(-4)=-7+4=-3$.

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

Table

3.4

Subtraction of Integers

Example 3.38

Solution

1. Rewrite the subtraction. The opposite of $-58$ is $58$, so $-74-(-58)$ can be rewritten as $-74+58$.
2. Compare the magnitudes. The magnitudes are $74$ and $58$, and their difference is $74-58=16$.
3. Apply the sign rule. The larger magnitude is negative, so the sum of the unlike signs is $-16$.
4. Check by adding. Combining $-74$ with $58$ leaves $16$ negative units, so the result is $-16$.

Takeaway: Subtracting $-58$ from $-74$ adds the opposite of $-58$, giving $-74-(-58)=-16$.

Example 3.39

Solution

1. Simplify the parentheses. Inside $-4-3$, compute $-4-3=-7$, so the expression becomes $7-(-7)-9$, matching the second figure.
2. Subtract the negative value. Since subtracting $-7$ means adding $7$, the expression becomes $7+7-9$.
3. Add from left to right. Compute $7+7=14$, as shown in the figure displaying $14-9$.
4. Finish the subtraction. Compute $14-9=5$, matching the final figure.
5. State the result. The expression $7-(-4-3)-9$ equals $5$.

Takeaway: Evaluate the innermost parentheses before applying the remaining subtraction signs from left to right.

Example 3.40

Solution

1. Identify the products. The expression contains $3\cdot7$, $4\cdot7$, and $5\cdot8$, as shown in the first figure.
2. Compute the first product. Multiply $3\cdot7=21$.
3. Compute the second product. Multiply $4\cdot7=28$, so the expression becomes $21-28-5\cdot8$, matching the sequence of figures.
4. Compute the third product. Multiply $5\cdot8=40$, giving $21-28-40$.
5. Subtract and state the result. Compute $21-28=-7$ and then $-7-40=-47$, matching the final figure. The expression equals $-47$.

Takeaway: Complete every multiplication before performing subtraction, even when the products are separated by minus signs.

Evaluate Variable Expressions with Integers

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

Example 3.41

Solution

1. Substitute the first value. Replace $x$ with $3$ in $x-4$, giving $3-4$.
2. Evaluate the first expression. Subtracting $4$ from $3$ gives $-1$, matching the first calculation figure.
3. Substitute the second value. Replace $x$ with $-6$ in the original expression, giving $-6-4$.
4. Evaluate the second expression. Subtracting $4$ from $-6$ gives $-10$, matching the later calculation figure.
5. State both values. The expression equals $-1$ when $x=3$ and equals $-10$ when $x=-6$.

Takeaway: To evaluate $x-4$, substitute the given value for $x$, yielding $-1$ when $x=3$ and $-10$ when $x=-6$.

Example 3.42

Solution

1. Substitute the first value. Replace $z$ with $12$ in $20-z$, giving $20-12$.
2. Evaluate the first expression. Subtracting $12$ from $20$ gives $8$, as shown in the first set of figures.
3. Substitute the second value. Replace $z$ with $-12$, giving the expression $20-(-12)$.
4. Evaluate the second expression. Subtracting a negative adds $12$, so $20+12=32$.
5. State both values. The expression equals $8$ when $z=12$ and equals $32$ when $z=-12$.

Takeaway: To evaluate $20-z$, substitute the given value for $z$, yielding $8$ when $z=12$ and $32$ when $z=-12$.

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.

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

Example 3.43

Solution

1. Identify the operation. Both phrases request subtraction, with the quantities written in the order stated.
2. Translate ⓐ. The difference of $13$ and $-21$ becomes $13-(-21)$.
3. Simplify ⓐ. Subtracting $-21$ adds $21$, so $13+21=34$.
4. Translate ⓑ. Subtracting $24$ from $-19$ places $-19$ first and gives $-19-24$.
5. Simplify ⓑ. The unlike signs have magnitudes $19$ and $24$, so the result is $-(24-19)=-43$.

Takeaway: Subtract in the order stated, so $13-(-21)=34$ and subtracting $24$ from $-19$ gives $-19-24=-43$.

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.

Example 3.44

Solution

1. Identify the requested difference. The question asks how far apart the morning value $11$ and afternoon value $-9$ are.
2. Write a subtraction phrase. Use the morning temperature minus the afternoon temperature: the difference of $11$ and $-9$.
3. Translate the phrase. Because difference means subtraction in the stated order, write $11-(-9)$.
4. Simplify the expression. Subtracting $-9$ adds $9$, so $11+9=20$.
5. Answer with the unit. The difference between the temperatures was $20$ degrees Fahrenheit.

Takeaway: The temperature difference is the morning value minus the afternoon value, so $11-(-9)=20$ degrees Fahrenheit.

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

Example 3.45

Solution

1. Assign signed elevations. Mt. Whitney is $14,497$ feet above sea level, while Death Valley is $-282$ feet relative to sea level.
2. Write the difference. Subtract the Death Valley elevation from the Mt. Whitney elevation to get $14,497-(-282)$.
3. Interpret the negative subtraction. Removing $-282$ is the same as adding $282$, so the expression becomes $14,497+282$.
4. Add the values. Compute $14,497+282=14,779$.
5. Answer with the unit. The difference in elevation between Mt. Whitney and Death Valley is $14,779$ feet.

Takeaway: Because Death Valley is below sea level, subtract $-282$ from $14,497$, giving $14,497-(-282)=14,779$ 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 3.46

Solution

1. Subtract the first check. Begin with $\$25$ and subtract $\$8$ to obtain a balance of $\$17$, as shown in the first figures.
2. Subtract the second check. Write the new balance as $\$17-\$20=-\$3$, matching the figure displaying the negative balance.
3. Add the returned check. Leslie receives the missing $\$10$ check, so add $\$10$ to the balance: $-\$3+\$10=\$7$.
4. State all three results. The balances after the first check, second check, and returned check are $\$17$, $-\$3$, and $\$7$, respectively, as shown in the final figures.

Takeaway: Track a balance by subtracting each withdrawal and adding each amount returned to the account.

Section 3.3 Exercises

Section 3.3 Exercises

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

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

Translate Word Phrases to Algebraic Expressions

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

Subtract Integers in Applications

In the following exercises, solve the following applications.

Everyday Math

Writing Exercises

Self Check

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

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

1. Model the expression:

$-6-(-4)$

2. Model the expression:

$-7-(-4)$

3. Model the expression:

$-6-4$

4. Model the expression:

$-7-4$

1. Model each subtraction.

1. ⓐ 7 - (-8)

2. ⓑ -7 - (-2)

3. ⓒ 4 - 1

4. ⓓ -6 - 8

2. Model each subtraction.

1. ⓐ 4 - (-6)

2. ⓑ -8 - (-1)

3. ⓒ 7 - 3

4. ⓓ -4 - 2

3. Model each subtraction expression.

1. ⓐ $\,7-9$

2. ⓑ $\,-5-(-9)$

4. Model each subtraction expression.

1. ⓐ $\,4-7$

2. ⓑ $\,-7-(-10)$

1. Simplify each expression:

1. ⓐ $\,21-13\,\mathrm{and}\,21+(-13)\,$

2. ⓑ $\,-11-7\,\mathrm{and}\,-11+(-7)$

2. Simplify each expression:

1. ⓐ $\,15-7\,\mathrm{and}\,15+(-7)\,$

2. ⓑ $\,-14-8\,\mathrm{and}\,-14+(-8)$

3. Simplify each expression:

1. ⓐ $\,6-(-13)\,\mathrm{and}\,6+13\,$

2. ⓑ $\,-5-(-1)\,\mathrm{and}\,-5+1$

4. Simplify each expression:

1. ⓐ $\,4-(-19)\,\mathrm{and}\,4+19\,$

2. ⓐ $-4-(-7)\,\mathrm{and}\,-4+7$

1. Simplify the expression:

$-67-(-38)$

2. Simplify the expression:

$-83-(-57)$

3. Simplify the expression:

$6\cdot{}2-9\cdot{}1-8\cdot{}9.$

4. Simplify the expression:

$2\cdot{}5-3\cdot{}7-4\cdot{}9$

1. Evaluate each expression:

$y-7\,\mathrm{when}\,$

1. ⓐ $\,y=5\,\,$

2. ⓑ $\,y=-8$

2. Evaluate each expression:

$m-3\,\mathrm{when}\,$

1. ⓐ $\,m=1\,$

2. ⓑ $\,m=-4$

3. Evaluate each expression:

$17-k\,\mathrm{when}\,$

1. ⓐ $\,k=19\,$

2. ⓑ $\,k=-19$

4. Evaluate each expression:

$-5-b\,\mathrm{when}\,$

1. ⓐ $\,b=14\,$

2. ⓑ $\,b=-14$

1. Translate and simplify:

1. ⓐ the difference of $14$ and $-23$

2. ⓑ subtract $21$ from $-17$

2. Translate and simplify:

1. ⓐ the difference of $11$ and $-19$

2. ⓑ subtract $18$ from $-11$

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

4. The temperature in Denver was $-6$ degrees Fahrenheit at lunchtime. By sunset the temperature had dropped to $\mathrm{-15 degree Fahrenheit.}$ What was the difference between the lunchtime and sunset temperatures?

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

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

3. Araceli has $\text{\textdollar 75}$ in her checking account and writes a check for $\text{\textdollar 27}.$

1. ⓐ What is the balance after she writes the check?

2. ⓑ She writes a second check for $\text{\textdollar 50}.$ What is the new balance?

3. ⓒ The check for $\text{\textdollar 20}$ that she sent a charity was never cashed. What is the balance in Araceli’s checking account now?

4. Genevieve’s bank account was overdrawn and the balance is $-\text{\textdollar 78}.$

1. ⓐ She deposits a check for $\text{\textdollar 24}$ that she earned babysitting. What is the new balance?

2. ⓑ She deposits another check for $\text{\textdollar 49}.$ Is she out of debt yet? What is her new balance?

1. $-5-(-1)$

2. $-5-4$

3. 1. ⓐ $\,15-6\,$

2. ⓑ $\,15+(-6)$

4. 1. ⓐ $\,44-28\,$

2. ⓑ $\,44+(-28)$

1. 1. ⓐ $\,8-(-9)\,$

2. ⓑ $8+9$

2. 1. ⓐ $\,27-(-18)\,$

2. ⓑ $\,27+18$

3. $48-87$

4. $31-79$

1. $-31-11$

2. $-17-42$

3. $-103-(-52)$

4. $8-3-7$

1. $-5-4+7$

2. $-16-(-4+1)-7$

3. $6\cdot{}3-4\cdot{}3-7\cdot{}2$

4. ${5}^{2}-{6}^{2}$

1. $x-6\,\mathrm{when}\,$

1. ⓐ $\,x=3\,$

2. ⓑ $\,x=-3$

2. $5-y\,\mathrm{when}\,$

1. ⓐ $\,y=2\,$

2. ⓑ $\,y=-2$

3. $4{x}^{2}-15x+1\,\mathrm{when}\,x=3$

4. $-12-5{x}^{2}\,\mathrm{when}\,x=6$

1. 1. ⓐ The difference of $3$ and $-10$

2. ⓑ Subtract $-20$ from $45$

2. 1. ⓐ The difference of $-6$ and $9$

2. ⓑ Subtract $-12$ from $-16$

3. 1. ⓐ $\,8$ less than $-17$

2. ⓑ $\,-24$ minus $37$

4. 1. ⓐ $\,21$ less than$\,6$

2. ⓑ $\,31$ subtracted from $-19$

1. Temperature One morning, the temperature in Urbana, Illinois, was $\mathrm{28^{\circ} Fahrenheit.}$ By evening, the temperature had dropped $\mathrm{38^{\circ} Fahrenheit.}$ What was the temperature that evening?

2. Temperature On January 15, the high temperature in Anaheim, California, was $\mathrm{84^{\circ} Fahrenheit.}$ That same day, the high temperature in Embarrass, Minnesota was $\mathrm{-12^{\circ} Fahrenheit.}$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

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

4. Checking Account John has $\text{\textdollar 148}$ in his checking account. He writes a check for $\text{\textdollar 83.}$ What is the new balance in his checking account?

1. Checking Account Gina has $\text{\textdollar 210}$ in her checking account. She writes a check for $\text{\textdollar 250.}$ What is the new balance in her checking account?

2. Checking Account Bill has a balance of $\mathrm{-\$14}$ in his checking account. He deposits $\text{\textdollar 40}$ to the account. What is the new balance?

3. Camping Rene is on an Alpine hike. The temperature is$-7^{\circ}.$ Rene’s sleeping bag is rated “comfortable to $-\mathrm{20}\mathrm{^{\circ}".}$ How much can the temperature change before it is too cold for Rene’s sleeping bag?

4. Explain why the difference of $9$ and $-6$ is $15.$