Subtract Integers

Model subtraction of integers

We continue to use two-color counters to model subtraction: a blue counter represents a positive $1$ and a red counter represents a negative $1$. Just as before, read $5 - 3$ as “five take away three,” and use the counters the same way.

We’ll model four subtraction facts using $5$ and $3$: $5-3$, $-5-(-3)$, $-5-3$, and $5-(-3)$.

Example. Model $5 - 3$. Start with $5$ positives. Take away $3$ positives. There are $2$ positives left, so $5 - 3 = 2$.

Example. Model $-5 - (-3)$. Start with $5$ negatives. Take away $3$ negatives. There are $2$ negatives left, so $-5 - (-3) = -2$.

Notice that these two examples are alike: in the first, we subtracted $3$ positives from $5$ positives to get $2$ positives; in the second, 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’ll need both positive and negative counters, and sometimes some neutral pairs too — remember, adding a neutral pair doesn’t change the value.

Example. Model $-5 - 3$. Start with $5$ negatives. We need to take away $3$ positives, but there are no positives to take away — so we add $3$ neutral pairs (which doesn’t change the value), giving us $3$ positives to remove. Taking those away leaves the original $5$ negatives plus $3$ more negatives from the neutral pairs: $8$ negatives. So $-5 - 3 = -8$.

Example. Model $5 - (-3)$. Start with $5$ positives. We need to take away $3$ negatives, but there are none — so add $3$ neutral pairs, then remove the $3$ negatives. That leaves the original $5$ positives plus $3$ more positives from the neutral pairs: $8$ positives. So $5 - (-3) = 8$.

Simplify expressions with integers

Do you see a pattern? Let’s think through two more subtractions without actually using counters.

To subtract $-23 - 7$: start with $23$ negative counters. We need to subtract $7$ positives, but there are none, so add $7$ neutral pairs and take away the $7$ positives. What’s left is the original $23$ negatives plus $7$ more negatives from the neutral pair — $30$ negatives: $-23 - 7 = -30$. Notice that to subtract $7$, we added $7$ negatives.

To subtract $30 - (-12)$: start with $30$ positives. We need to subtract $12$ negatives, but there are none, so add $12$ neutral pairs and take away the $12$ negatives. What’s left is the original $30$ positives plus $12$ more positives — $42$ positives: $30 - (-12) = 42$. Notice that to subtract $-12$, we added $12$.

While we may not always use counters, especially with large numbers, practicing with them first gives a concrete way to visualize subtraction without them. Subtraction of signed numbers can also be done by adding the opposite — this is the Subtraction Property:

Compare $6 - 4$ and $6 + (-4)$: both give $2$. When a subtraction problem has only positive numbers, like $6-4$, we just subtract as usual — but knowing that $6-4$ gives the same answer as $6+(-4)$ helps once negative numbers are involved.

Example. Simplify (a) $13 - 8$ and $13 + (-8)$; (b) $-17 - 9$ and $-17 + (-9)$.

(a) $13 - 8 = 5$ and $13 + (-8) = 5$ — subtracting $8$ from $13$ is the same as adding $-8$ to $13$. (b) $-17 - 9 = -26$ and $-17 + (-9) = -26$ — subtracting $9$ from $-17$ is the same as adding $-9$ to $-17$.

Now look at what happens when we subtract a negative: $8 - (-5)$ gives the same result as $8 + 5$ — both are $13$. Subtracting a negative number is like adding a positive.

Example. Simplify (a) $9 - (-15)$ and $9 + 15$; (b) $-7 - (-4)$ and $-7 + 4$.

(a) $9 - (-15) = 24$ and $9 + 15 = 24$ — subtracting $-15$ from $9$ is the same as adding $15$ to $9$. (b) $-7 - (-4) = -3$ and $-7 + 4 = -3$ — subtracting $-4$ from $-7$ is the same as adding $4$ to $-7$.

Here’s a summary of the pattern, based on the results from the earlier examples:

Example. Simplify $-74 - (-58)$. This takes $58$ negatives away from $74$ negatives: $-74 - (-58) = -16$.

These techniques extend to more complicated expressions — follow the order of operations.

Example. Simplify $7 - (-4-3) - 9$. Simplify inside the parentheses first: $7 - (-7) - 9$. Subtract from left to right: $14 - 9$. Subtract: $5$.

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

Evaluate variable expressions with integers

Now let’s practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

Example. Evaluate $x - 4$ when (a) $x=3$, (b) $x=-6$.

(a) Substitute $3$ for $x$: $3-4 = -1$. (b) Substitute $-6$ for $x$: $-6-4 = -10$.

Example. Evaluate $20 - z$ when (a) $z=12$, (b) $z=-12$.

(a) Substitute $12$ for $z$: $20-12 = 8$. (b) Substitute $-12$ for $z$: $20-(-12) = 32$.

Translate word phrases to algebraic expressions

The expression $a - b$ can be read several ways: “$a$ minus $b$,” “the difference of $a$ and $b$,” “subtract $b$ from $a$,” “$b$ subtracted from $a$,” or “$b$ less than $a$.” Be careful to get $a$ and $b$ in the right order — the number subtracted always comes second.

Example. Translate and simplify: (a) the difference of $13$ and $-21$; (b) subtract $24$ from $-19$.

(a) Difference means subtraction, in the order given: $13-(-21) = 34$. (b) Subtract 24 from -19 means take $24$ away from $-19$: $-19-24 = -43$.

Subtract integers in applications

To solve an application problem: identify what you’re asked to find, write a phrase for it, translate the phrase into an expression, simplify, and answer with a complete sentence.

Example. 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?

The difference of $11$ and $-9$: $11 - (-9) = 20$. The difference in temperature was $20$ degrees Fahrenheit.

Geography gives another application, comparing elevations above and below sea level.

Example. Dinesh hiked from Mt. Whitney, the highest point in California at $14{,}497$ feet above sea level, to Death Valley, the lowest point at $282$ feet below sea level. What is the difference in elevation between them?

Elevation of Mt. Whitney minus elevation of Death Valley: $14{,}497 - (-282) = 14{,}779$. The difference in elevation is $14{,}779$ feet.

Checking accounts with overdraft protection combine both positive and negative numbers naturally.

Example. Leslie has $\text{\textdollar}25$ in her checking account and writes a check for $\text{\textdollar}8$. (a) What is the balance after she writes the check? (b) She writes a second check for $\text{\textdollar}20$ — what is the new balance? (c) Leslie’s friend told her that she had lost a check for $\text{\textdollar}10$ that Leslie had given her — what is the balance now?

(a) $\text{\textdollar}25 - \text{\textdollar}8 = \text{\textdollar}17$. (b) $\text{\textdollar}17 - \text{\textdollar}20 = -\text{\textdollar}3$, so she is overdrawn by $\text{\textdollar}3$. (c) The lost check means that $\text{\textdollar}10$ was never actually withdrawn, so it’s added back: $-\text{\textdollar}3 + \text{\textdollar}10 = \text{\textdollar}7$. The balance is now $\text{\textdollar}7$.

Key terms

Subtraction Property — $a - b = a + (-b)$: subtracting a number is the same as adding its opposite.

Practice

Model subtraction of integers

Simplify expressions with integers

Evaluate variable expressions with integers

Translate word phrases to algebraic expressions

Subtract integers in applications

This section is adapted from Prealgebra 2e, Section 3.3: Subtract Integers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the two-color-counter models as accessible inline graphics and the subtraction-pattern summary as a table; condensed prose; omitted the Be Prepared quiz, Manipulative Mathematics and Links to Literacy callouts, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, expanding multipart items into one exercise per part and redrawing the counter models as accessible inline graphics that ask for the value the model leaves.