Add Integers

Model addition of integers

Most people are comfortable with addition and subtraction facts for positive numbers, but adding or subtracting when negative numbers are involved can feel less automatic. One way to make it concrete is to model addition and subtraction with two-color counters: a blue counter represents a positive $1$, and a red counter represents a negative $1$.

If we have one positive counter and one negative counter, their values add to zero — together they form a neutral pair.

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

Example. Model $5 + 3$.

Start with $5$ positives. Add $3$ more positives. Count the total: $8$ positives. So $5 + 3 = 8$.

Example. Model $-5 + (-3)$.

Start with $5$ negatives. Add $3$ more negatives. Count the total: $8$ negatives. So $-5 + (-3) = -8$.

Example 3.14 and Example 3.15 both add two numbers with the same sign — both positive, or both negative — and in each case the counters are all the same color, so we simply add. Now let’s see what happens when the signs are different.

Example. Model $-5 + 3$.

Start with $5$ negatives. Add $3$ positives. Each positive pairs with a negative to form a neutral pair, which we remove. Two negatives are left over, so $-5 + 3 = -2$. Notice there were more negatives than positives, so the result is negative.

Example. Model $5 + (-3)$.

Start with $5$ positives. Add $3$ negatives. Three neutral pairs form and are removed, leaving $2$ positives. So $5 + (-3) = 2$.

Simplify expressions with integers

Once you can picture the counter model in your mind, you can add any integers without counting out piles of counters. For example, to add $37 + (-53)$: picture $37$ blue counters with $53$ red counters lined up underneath. Since there are more negatives than positives, the sum is negative. Because $53 - 37 = 16$, there are $16$ more negatives left over, so $37 + (-53) = -16$.

For two negatives, such as $-74 + (-27)$: imagine $74$ red counters and $27$ more red counters, for $101$ red counters altogether, so $-74 + (-27) = -101$.

Example. Simplify $19 + (-47)$ and $-32 + 40$.

For $19 + (-47)$: the signs are different, so subtract $19$ from $47$; the answer is negative because there are more negatives than positives: $19 + (-47) = -28$.

For $-32 + 40$: the signs are different, so subtract $32$ from $40$; the answer is positive because there are more positives than negatives: $-32 + 40 = 8$.

Example. Simplify $-14 + (-36)$. The signs are the same, so add; the result is negative because both are negative: $-14 + (-36) = -50$.

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

Example. Simplify $-5 + 3(-2+7)$. Simplify inside the parentheses first: $-5 + 3(5)$. Multiply: $-5 + 15$. Add left to right: $10$.

Evaluate variable expressions with integers

To evaluate an expression, substitute the given number for the variable in the expression, then simplify.

Example. Evaluate $x + 7$ when (a) $x = -2$, (b) $x = -11$.

(a) Substitute $-2$ for $x$: $-2 + 7 = 5$. (b) Substitute $-11$ for $x$: $-11 + 7 = -4$.

Watch the signs carefully when a variable itself carries a minus sign in front of it.

Example. When $n = -5$, evaluate (a) $n+1$, (b) $-n+1$.

(a) Substitute $-5$ for $n$: $-5 + 1 = -4$. (b) Substitute $-5$ for $n$: $-n + 1 = -(-5) + 1 = 5 + 1 = 6$.

Expressions with two variables work the same way — substitute both values, then follow the order of operations.

Example. Evaluate $3a + b$ when $a = 12$ and $b = -30$. Substitute: $3(12) + (-30)$. Multiply: $36 + (-30)$. Add: $6$.

Example. Evaluate $(x+y)^2$ when $x = -18$ and $y = 24$. Substitute: $(-18+24)^2$. Add inside the parentheses: $(6)^2$. Simplify: $36$.

Translate word phrases and applications to expressions with integers

All our earlier work translating word phrases to algebra also applies to expressions with both positive and negative numbers. Remember that the sum and increased by both indicate addition.

Example. Translate and simplify: the sum of $-9$ and $5$. Translate: $-9 + 5$. Simplify: $-4$.

Example. Translate and simplify: the sum of $8$ and $-12$, increased by $3$. Translate: $[8+(-12)]+3$. Simplify: $-4+3$. Add: $-1$.

Positive and negative numbers show up often in everyday situations — temperatures, banking, and sports, for example. Solving these applications is easier with a plan: figure out what you’re looking for, write a phrase for it, translate the phrase into math notation, simplify, and answer in a full sentence.

Example. The temperature in Buffalo, NY, one morning started at $7$ degrees below zero Fahrenheit. By noon, it had warmed up $12$ degrees. What was the temperature at noon?

The temperature warmed up $12$ degrees from $7$ degrees below zero: $-7 + 12 = 5$. The temperature at noon was $5$ degrees Fahrenheit.

Example. A football team took possession of the ball on their $42$-yard line. In the next three plays, they lost $6$ yards, gained $4$ yards, and then lost $8$ yards. On what yard line was the ball at the end of those three plays?

Start at $42$, then lose $6$, gain $4$, lose $8$: $42-6+4-8 = 32$. At the end of the three plays, the ball is on the $32$-yard line.

Key terms

neutral pair — a positive counter and a negative counter together, whose value is $0$. same signs rule — to add two integers with the same sign, add their absolute values and keep the common sign. different signs rule — to add two integers with different signs, subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value.

Practice

Model addition of integers

Simplify expressions with integers

Evaluate variable expressions with integers

Translate word phrases to expressions with integers

Add integers in applications

This section is adapted from Prealgebra 2e, Section 3.2: Add 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 same-signs/different-signs summary as a table; condensed prose; omitted the Be Prepared quiz, Manipulative Mathematics callout, 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, with each multipart exercise expanded into one question per part and the counter-model answers redrawn as accessible inline graphics.