Step by step
The question
Question: work out 3/x − 1/(x + 4).
1 Find the common bottom
The bottoms are x and (x + 4).
The common bottom must have every factor of both bottoms, each one once.
It is x(x + 4).
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 + 4).
We multiply its top and bottom by (x + 4).
The second fraction is missing x.
We multiply its top and bottom by x.
(x + 4)/(x + 4) is 1, so the value stays the same.
3 Take away the tops
Now one fraction: (3(x + 4) − x)/x(x + 4).
4 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 times x is 3x.
3 times 4 is 12.
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.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
3x − x is 2x.
It is like 3 apples take away 1 apple: that is 2 apples.
So the expression is 2x + 12.
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 + 12)/x(x + 4).
The bottoms can't be 0.
So x can't be −4 or 0.
7 Check
Let's check with a number: x = 1.
First in the question.
3 take away 1/5 is 14/5.
Now in our answer.
We get 14/5 too.
Both are 14/5.
So (2x + 12)/x(x + 4) 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.