Step by step
The question
Question: work out 1/(x + 3) − 1/(x − 3).
1 Find the common bottom
The bottoms are (x + 3) and (x − 3).
The common bottom must have every factor of both bottoms, each one once.
It is (x + 3)(x − 3).
Careful!
We can't take away the tops and take away the bottoms.
Fractions need the same bottom first.
2 Make the bottoms the same
The first fraction is missing (x − 3).
We multiply its top and bottom by (x − 3).
The second fraction is missing (x + 3).
We multiply its top and bottom by (x + 3).
(x − 3)/(x − 3) is 1, so the value stays the same.
3 Take away the tops
Now one fraction: (x − 3) − (x + 3)/(x + 3)(x − 3).
4 Open the brackets
A bracket with a plus or nothing in front can just be removed.
A minus in front of a bracket changes the sign of every part inside.
Now we have x − 3 − x − 3.
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
x − x is 0.
It is like 1 apple take away 1 apple: that is 0 apples.
−3 − 3 is −6.
So the expression is −6.
6 Cross out the same
Is a whole factor on top the same as one on the bottom?
No.
So nothing crosses out.
We leave the bottom as factors.
So the answer is −6/(x + 3)(x − 3).
The bottoms can't be 0.
So x can't be −3 or 3.
7 Check
Let's check with a number: x = 1.
First in the question.
1/4 take away 1 over −2 is 3/4.
Now in our answer.
We get 3/4 too.
Both are 3/4.
So −6/(x + 3)(x − 3) is right!
Let's look back.
First: find the common bottom.
Then: make the bottoms the same.
Then: take away the tops.
Then: open the brackets.
Then: group like terms.
Then: cross out the same.
Last: check.