Step by step
The question
Question: solve x/(x − 3) = 3/(x − 3) + 2.
First: a bottom can never be 0.
x = 3 makes x − 3 0.
So x ≠ 3.
1 Remove the fractions
The bottom is (x − 3).
Times (x − 3) will remove it.
We multiply every part by (x − 3).
We do it on both sides, so they stay equal.
(x − 3) divided by (x − 3) is 1.
(x − 3) divided by (x − 3) is 1.
(x − 3) times 2 is 2(x − 3).
Write what is left.
No more fractions!
2 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
Work out each multiplication.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
3 − 6 = −3.
Now we have x = −3 + 2x.
4 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the x from the left side.
To remove x, we take away x.
We do it on both sides, so they stay equal.
x and − x make zero.
Cross them out.
2x − x = x.
Now we have 0 = −3 + x.
Let's swap the two sides, so x is on the left.
5 Remove the − 3
We want x by itself.
The − 3 is in the way.
To remove − 3, we add 3.
We do it on both sides, so they stay equal.
− 3 and + 3 make zero.
Cross them out.
0 + 3 = 3.
So x = 3!
But x = 3 is the x that is not allowed: it makes a bottom 0.
So there is no solution.
Let's look back.
First: remove the fractions.
Then: open the brackets.
Then: group like terms.
Then: get x on one side.
Last: remove the − 3.