Step by step
The question
Question: work out x/(x − 3) − 3/(x − 3).
xx − 3 − 3x − 3
1 Take away the tops
The bottoms are already the same.
We take away the tops.
The bottom stays.
ac − bc = a − bc
Careful!
The bottoms are the same.
Only the tops are added.
The bottom stays.
Careful: not x − 3x − 3 + x − 3 ✗
Now one fraction: (x − 3)/(x − 3).
= x − 3(x − 3)
2 Cross out the same
We write the top as factors too: (x − 3).
= (x − 3)(x − 3)
The same bracket on top and bottom is 1.
Cross them out.
What is left is 1.
= 1
The bottoms can't be 0.
So x can't be 3.
x ≠ 3
3 Check
Let's check with a number: x = 1.
First in the question.
1 over −2 take away 3 over −2 is 1.
Question: 11 − 3 − 31 − 3 = 1−2 − 3−2 = 1
Now in our answer.
We get 1 too.
Answer: 1 − 3(1 − 3) = −2−2 = 1
Both are 1.
So 1 is right!
Let's look back.
First: take away the tops.
Then: cross out the same.
Last: check.
Answer
= 1