Simple operations with rational numbers
Adding rational numbers
When doing any mathematical operation in Algebra, use your rules to find the sign of your answer first, then add,
subtract, multiply, divide or raise a number to a power.
Remember our earlier suggestion that you write down your rules and definitions to use for future reference.
Rules for Addition

Like signs: add and put that sign

Unlike signs: subtract and put the sign of the biggest number
Examples 
Explanation 
4 + 3 = 7 
Think like signs; add; answer positive 
4 + (3) = 7 
Think like signs; add; answer negative 
4 + (3) = 1 
Think unlike signs; subtract; answer positive because biggest number is positive 
4 + 3 = 1 
Think unlike signs; subtract; answer negative because the biggest number is negative 
Subtracting Rational Numbers
When doing any mathematical operation in Algebra, use your rules to find the sign of your answer first then add,
subtract, multiply, divide or raise a number to a power.
Remember our earlier suggestion that you write down your rules and definitions to use for future reference.
To develop the rule for subtraction, we need to develop another property of numbers.
Additive Inverse Property:
For every a, a + (a) = 0
Where a can be read as

The additive inverse of a

The opposite of a

The negative of a
Examples 
Explanation 
5 and 5 
Because 5 + (5) = 0 remember
unlike sign; subtract
where a = 5 and a
= 5 
7 and 7 
Because 7 + 7 = 0
where a = 7 and a
= 7 
As you can see, a is not always negative. It has the opposite sign of a. So if a is positive, then a is negative; but, if a is negative then a is positive.
Rules for Subtraction

To subtract a number, add its additive inverse

Change the sign and add
We will use "change the sign and add" as our way of remembering the rule for subtraction.
Again, a minus sign in front of a grouping symbol tells us to change the sign.
Examples 
Explanation 
3  (5) 
Change the 5 to a plus 5 then add
unlike signs; subtract and put the sign of the biggest number 
3 + 5
2 
If there are no grouping symbols in a problem, then you add.
Examples

Explanation 
3  5 

8 
Like sign; add and put that sign 
If there is a plus sign in front of a grouping symbol, then rewrite as is and add.
Examples

Explanation 
3 + (5) 
Remember: order of operation tells us to do
grouping symbols first. 
3  5 
Because there is a plus in front of ( ) rewrite as 5 
8 
Like signs; add and put that sign 
The Commutative and Associative properties allow us to add numbers in any order we choose. If you are adding
more than two numbers, I suggest that you:

Add all the positive numbers

Add all the negative numbers

Subtract the two answers
Examples
1.

5  3 + 7 + 4  8  4

Explanation 

16  15 
Rename 5 + 7+ 4 as 16
Rename 3  8  4 as 15 

1 
Unlike signs; subtract and put the sign of the biggest
number 
2.

5  (3)  7  (8)



5 + 3  7 + 8 
( ) tells us to change the sign
3 to 3, 8 to 8 

11  12 
Add 3 + 8 = 11, add 5 7 = 12 

1 
Subtract and put the sign of the biggest number 
Adding and subtracting like terms
We can also use our rules for addition and subtraction to combine like terms.
Remember that like terms are:

Same variable(s)

Same exponent(s) on those variable(s)
Examples: Simplify
1.

3x + 5x

Explanation 

2x 
Unlike signs; subtract and put the sign of the biggest number 
2.

a + 2a  4a

Remember: a = 1a 

5a + 2a 
Add the a  4a = 5a 

3a 
Subtract 5a + 2a = 3a 
Try these exercises
Add or subtract

3 + 5

3  5

7  (6)

8  (4)

8m + 6m

43 + 16

3x  8x + 7x  10x

8y  (8y)

8 + (4) + 7

6  9
Answers to questions

3 + 5 = 2

3  5 = 2

7  (6)
7 + 6
13

8  (4)
8 + 4
4

8m + 6m
14m

43 + 16
27

3x  8x + 7x 10x
10x 18x
8x

8y  (8y)
8y + 8y
16y

8 + (4) + 7
15  4
11

6  9
15