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## Simple operations with 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.

• Like signs:  add and put that sign

• Unlike signs:  subtract and put the sign of the biggest number

### Explanation

4 + 3 = 7

-4 + (-3) = -7

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.

For every a, a + (-a) = 0

Where -a can be read as

1. The additive inverse of  a

2. The opposite of  a

3. The negative of  a

### 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

2. 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.

### 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.

### 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.

### 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:

1. Add all the positive numbers

2. Add all the negative numbers

### 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

-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:

1. Same variable(s)

2. Same exponent(s) on those variable(s)

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

 1 -3 + 5 2 3 - 5 3 7 - (-6) 4 -8 - (-4) 5 8m + 6m 6 -43 + 16 7 3x - 8x + 7x - 10x 8 8y - (-8y) 9 8 + (-4) + 7 10 -6 - 9