Step by step
The question
Question: simplify (x² − 16)/(2x + 8) ÷ (x − 4)/6.
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.
So we work out (x² − 16)/(2x + 8) × 6/(x − 4).
2 Factor every part
First the top of the first fraction: x² − 16.
x² − 16 is x² minus 4 squared.
A square minus a square is the plus bracket times the minus bracket.
So the top of the first fraction is (x + 4)(x − 4).
Now the bottom of the first fraction: 2x + 8.
2 and 8 can both be divided by 2.
We take 2 out.
What is left goes in the bracket.
So the bottom of the first fraction is 2(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.
Now every part is factored.
A bracket on any top and the same bracket on any bottom make 1.
(x + 4)/(x + 4) is 1.
Cross them out.
(x − 4)/(x − 4) is 1.
Cross them out.
6 and 2 can both be divided by 2.
6 becomes 3, 2 becomes 1.
4 Multiply what is left
Top times top, bottom times bottom.
What is left is 3.
One more thing: the bottoms of the question can't be 0.
So x can't be −4 or 4.
5 Check
Let's check with a number: x = 2.
First in the question, flipped.
We get 3.
Now in our answer.
We get 3 too.
Both are 3.
So 3 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.