Simplify (x - 3)/(2x) ÷ (x - 3)/(x(x + 4)) Step by Step

Step by step

The question

Question: simplify (x − 3)/2x ÷ (x − 3)/x(x + 4).

x − 32x ÷ x − 3x(x + 4)

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 − 3x(x + 4) → × x² + 4xx − 3

So we work out (x − 3)/2x × x(x + 4)/(x − 3).

= x − 32x × x(x + 4)x − 3

2 Factor every part

First the top of the second fraction: x(x + 4).

= x(x + 4)

It is already factored: x(x + 4) means x times (x + 4).

We just read the factors.

x(x + 4) = x × (x + 4)

So the top of the second fraction is x(x + 4).

= x(x + 4)

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 − 3)~~2x × x(x + 4)~~(x − 3)~~

(x − 3)/(x − 3) 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 (x + 4)/2.

= x + 42

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

So x can't be −4, 0 or 3.

x ≠ −4, 0, 3

5 Check

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

First in the question, flipped.

We get 3.

Question: 2 − 32(2) × 2(2 + 4)2 − 3 = −14 × 12−1 = 3

Now in our answer.

We get 3 too.

Answer: 2 + 42 = 3

Both are 3.

So (x + 4)/2 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