Math Core

Lesson 1.2 · Integers and Order of Operations

Adding and subtracting integers

Adding and subtracting with negative numbers is the single skill you will use most in Algebra 1. Every time you combine like terms or solve an equation, you are adding signed numbers. The good news: a number line and two short rules handle every case.

Adding on a number line

Start at the first number. Then move by the second number: right if it is positive, left if it is negative.

To find −4+7-4 + 7, start at −4-4 and move 77 units right. You land on 33.

−6−5−4−3−2−10123456+7
−4 + 7 = 3

To find 2+(−5)2 + (-5), start at 22 and move 55 units left. You land on −3-3.

−6−5−4−3−2−10123456−5
2 + (−5) = −3

The rules for adding

Drawing a number line every time gets slow. The pictures above lead to two rules based on absolute value.

Adding integers

  • Same signs: add the absolute values and keep the common sign. −6+(−3)=−9-6 + (-3) = -9
  • Different signs: subtract the smaller absolute value from the larger one, and take the sign of the number with the larger absolute value. −9+4=−5-9 + 4 = -5, because 9−4=59 - 4 = 5 and −9-9 has the larger absolute value.

A number plus its opposite is always 00: a+(−a)=0a + (-a) = 0.

Subtracting means adding the opposite

Subtraction can look tricky with negatives, but it turns into addition with one change.

Subtracting integers

To subtract a number, add its opposite:

a−b=a+(−b)a - b = a + (-b)

So 3−8=3+(−8)=−53 - 8 = 3 + (-8) = -5, and −2−(−6)=−2+6=4-2 - (-6) = -2 + 6 = 4.

Why does subtracting a negative move you right? Subtraction undoes addition. Adding −6-6 moves you 66 units left, so subtracting −6-6 must move you 66 units right. Here is −2−(−6)-2 - (-6):

−6−5−4−3−2−10123456+6
−2 − (−6) = −2 + 6 = 4

Common mistake

Change only the operation and the sign of the number being subtracted. The first number stays exactly as it is. In −5−3-5 - 3, the −5-5 does not change: −5−3=−5+(−3)=−8-5 - 3 = -5 + (-3) = -8, not 5+(−3)5 + (-3).

Worked example: Using the addition rules

Find each sum.

  1. −7+(−8)-7 + (-8)
  2. −12+5-12 + 5
  3. 15+(−6)15 + (-6)

Solutions.

  1. Same signs: 7+8=157 + 8 = 15, keep the negative sign. The sum is −15-15.
  2. Different signs: 12−5=712 - 5 = 7. The number with the larger absolute value is −12-12, so the sum is −7-7.
  3. Different signs: 15−6=915 - 6 = 9. The number with the larger absolute value is 1515, so the sum is 99.

Worked example: Rewriting subtraction

Find each difference.

  1. 4−114 - 11
  2. −3−(−10)-3 - (-10)
  3. −6−9-6 - 9

Solutions.

  1. 4−11=4+(−11)=−74 - 11 = 4 + (-11) = -7
  2. −3−(−10)=−3+10=7-3 - (-10) = -3 + 10 = 7
  3. −6−9=−6+(−9)=−15-6 - 9 = -6 + (-9) = -15

Worked example: A longer string

Evaluate −8+13−(−4)−10-8 + 13 - (-4) - 10.

First rewrite every subtraction as adding the opposite:

−8+13+4+(−10)-8 + 13 + 4 + (-10)

Now you can add in any order. Group the positives and the negatives:

(13+4)+(−8+(−10))=17+(−18)=−1(13 + 4) + (-8 + (-10)) = 17 + (-18) = -1

Tip

To check a subtraction, add back. If −3−(−10)=7-3 - (-10) = 7, then 7+(−10)7 + (-10) should give −3-3. It does.

Practice

Practice 1

Find −4+(−7)-4 + (-7).

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find −9+15-9 + 15.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Find 5−145 - 14.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Find −7−(−12)-7 - (-12).

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Which expression has the same value as −6−2-6 - 2?

Practice 6

Evaluate −3−8+(−2)-3 - 8 + (-2).

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Evaluate 10−(−6)+(−15)−(−3)10 - (-6) + (-15) - (-3).

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A submarine is at −140-140 meters. It rises 4545 meters, then dives 6868 meters, then rises 140140 meters. Where does it end up, in meters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.