Simplify (x + 2)/(x(x - 3)) ÷ (x + 2)/(x

Step by step

The question

Question: simplify (x + 2)/x(x − 3) ÷ (x + 2)/(x − 3).

x + 2x(x − 3) ÷ x + 2x − 3

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 + 2x − 3 → × x − 3x + 2

So we work out (x + 2)/x(x − 3) × (x − 3)/(x + 2).

= x + 2x(x − 3) × x − 3x + 2

2 Factor every part

First the bottom of the first fraction: x(x − 3).

= x(x − 3)

It is already factored: x(x − 3) means x times (x − 3).

We just read the factors.

x(x − 3) = x × (x − 3)

So the bottom of the first fraction is x(x − 3).

= x(x − 3)

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 + 2)~~x~~(x − 3)~~ × ~~(x − 3)(x + 2)~~

(x + 2)/(x + 2) is 1.

Cross them out.

(x − 3)/(x − 3) is 1.

Cross them out.

4 Multiply what is left

Top times top, bottom times bottom.

What is left is 1 over x.

= 1x

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

So x can't be −2, 0 or 3.

x ≠ −2, 0, 3

5 Check

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

First in the question, flipped.

We get 1/2.

Question: 2 + 22(2 − 3) × 2 − 32 + 2 = 4−2 × −14

= 12

Now in our answer.

We get 1/2 too.

Answer: 12

Both are 1/2.

So 1/x 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