Step by step
The question
Question: add 2/(x + 1) + 3/(x − 1).
1 Find the common bottom
The bottoms are (x + 1) and (x − 1).
The common bottom must have every factor of both bottoms, each one once.
It is (x + 1)(x − 1).
Careful!
We can't add the tops and add the bottoms.
Fractions need the same bottom first.
2 Make the bottoms the same
The first fraction is missing (x − 1).
We multiply its top and bottom by (x − 1).
The second fraction is missing (x + 1).
We multiply its top and bottom by (x + 1).
(x − 1)/(x − 1) is 1, so the value stays the same.
3 Add the tops
Now one fraction: (2(x − 1) + 3(x + 1)) over (x + 1)(x − 1).
4 Open the brackets
What is in front of a bracket multiplies every part inside.
So we write each part times what is in front.
Watch the signs.
2 times x is 2x.
2 times (−1) is −2.
3 times x is 3x.
3 times 1 is 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.
2x + 3x is 5x.
It is like 2 apples plus 3 apples: that is 5 apples.
−2 + 3 is 1.
So the expression is 5x + 1.
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 (5x + 1)/(x + 1)(x − 1).
The bottoms can't be 0.
So x can't be −1 or 1.
7 Check
Let's check with a number: x = 2.
First in the question.
2/3 plus 3 is 11/3.
Now in our answer.
We get 11/3 too.
Both are 11/3.
So (5x + 1)/(x + 1)(x − 1) is right!
Let's look back.
First: find the common bottom.
Then: make the bottoms the same.
Then: add the tops.
Then: open the brackets.
Then: group like terms.
Then: cross out the same.
Last: check.