Step by step
The question
Question: work out 5/x(x − 2) − 2 over x.
1 Find the common bottom
The bottoms are x(x − 2) and x.
The common bottom must have every factor of both bottoms, each one once.
It is x(x − 2).
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 already has it.
The second fraction is missing (x − 2).
We multiply its top and bottom by (x − 2).
(x − 2)/(x − 2) is 1, so the value stays the same.
3 Take away the tops
Now one fraction: (5 − 2(x − 2)) over x(x − 2).
4 Open the brackets
The (−2) in front of the bracket multiplies every part inside.
So we write (−2) times each part.
Watch the signs.
(−2) times x is −2x.
(−2) times (−2) is 4.
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
5 + 4 is 9.
So the expression is −2x + 9.
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 (−2x + 9)/x(x − 2).
The bottoms can't be 0.
So x can't be 0 or 2.
7 Check
Let's check with a number: x = 1.
First in the question.
5 over −1 take away 2 is −7.
Now in our answer.
We get −7 too.
Both are −7.
So (−2x + 9)/x(x − 2) 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.