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

Step by step

The question

Question: simplify (x + 1)/x(x − 1) ÷ (x + 1) over x.

x + 1x(x − 1) ÷ x + 1x

1 Flip and multiply

Dividing by a fraction is the same as times its flip.

Keep the first fraction.

Change divide to times.

Turn the second fraction upside down.

÷ x + 1x → × xx + 1

So we work out (x + 1)/x(x − 1) × x/(x + 1).

= x + 1x(x − 1) × xx + 1

2 Factor every part

First the bottom of the first fraction: x(x − 1).

= x(x − 1)

It is already factored: x(x − 1) means x times (x − 1).

We just read the factors.

x(x − 1) = x × (x − 1)

So the bottom of the first fraction is x(x − 1).

= x(x − 1)

3 Cross out the same

Careful!

We cross out only a whole bracket, or a whole x, that multiplies everything.

Never one part of a bracket.

Careful: only whole brackets cross out

Now every part is factored.

A bracket on any top and the same bracket on any bottom make 1.

= ~~(x + 1)~~x(x − 1) × x~~(x + 1)~~

(x + 1)/(x + 1) is 1.

Cross them out.

x/x is 1.

Cross them out.

4 Multiply what is left

Top times top, bottom times bottom.

What is left is 1/(x − 1).

= 1x − 1

One more thing: the bottoms of the question can't be 0.

So x can't be −1, 0 or 1.

x ≠ −1, 0, 1

5 Check

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

First in the question, flipped.

We get 1.

Question: 2 + 12(2 − 1) × 22 + 1 = 32 × 23 = 1

Now in our answer.

We get 1 too.

Answer: 12 − 1 = 1

Both are 1.

So 1/(x − 1) is right!

Let's look back.

First: flip and multiply.

Then: factor every part.

Then: cross out the same.

Then: multiply what is left.

Last: check.

Answer