Simplify x/(x - 1) + 1/(x(x - 1)) Step by Step

Step by step

The question

Question: add x/(x − 1) + 1/x(x − 1).

xx − 1 + 1x(x − 1)

1 Find the common bottom

The bottoms are (x − 1) and x(x − 1).

bottoms: (x − 1) and x(x − 1)

The common bottom must have every factor of both bottoms, each one once.

It is x(x − 1).

common bottom = x(x − 1)

Careful!

We can't add the tops and add the bottoms.

Fractions need the same bottom first.

Careful: not x + 1−1 + x² ✗

2 Make the bottoms the same

The first fraction is missing x.

We multiply its top and bottom by x.

The second fraction already has it.

x/x is 1, so the value stays the same.

= x × x(x − 1)x + 1x(x − 1)

3 Add the tops

Now one fraction: (x × x + 1)/x(x − 1).

= x × x + 1x(x − 1)

On top: x × x + 1 is x² + 1.

x × x + 1 = x² + 1

So the top is x² + 1.

= x² + 1x(x − 1)

4 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 (x² + 1)/x(x − 1).

The bottoms can't be 0.

So x can't be 0 or 1.

x ≠ 0, 1

5 Check

Let's check with a number: x = 2.

First in the question.

2 plus 1/2 is 5/2.

Question: 22 − 1 + 12(2 − 1) = 21 + 12 = 52

Now in our answer.

We get 5/2 too.

Answer: 2² + 12(2 − 1) = 52

Both are 5/2.

So (x² + 1)/x(x − 1) is right!

Let's look back.

First: find the common bottom.

Then: make the bottoms the same.

Then: add the tops.

Then: cross out the same.

Last: check.

Answer