Step by step
The question
Question: solve |x − 2| = |3x|.
Both bars give how far from 0.
Two numbers the same distance from 0 are equal, or opposite.
Like 3 and minus 3.
Careful!
Only one case loses an answer.
We always do both.
1 Two cases
So x − 2 = 3x, or x − 2 = −(3x).
We keep the bracket: the minus is for all of it.
2 Solve the first case
First case: x − 2 = 3x.
3 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.
3x − x = 2x.
Now we have −2 = 2x.
Let's swap the two sides, so x is on the left.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
−2 divided by 2 is −1.
So x = −1!
5 Solve the second case
Second case: x − 2 = −(3x).
6 Open the brackets
A minus in front of a bracket changes the sign of every part inside.
3x becomes −3x.
7 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the −3x from the right side.
To remove −3x, we add 3x.
We do it on both sides, so they stay equal.
− 3x and + 3x make zero.
Cross them out.
x + 3x = 4x.
Now we have 4x − 2 = 0.
We want x by itself.
Two things are stuck to x: times 4 and − 2.
The − 2 is on the outside, so it comes off first.
Then the times 4.
8 Remove the − 2
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 make zero.
Cross them out.
0 + 2 = 2.
Now we have 4x = 2.
9 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
We can make 2/4 simpler.
It is the same as 1/2.
So x = 1/2!
10 Write the answers
So x = −1 or x = 1/2.
11 Check
Let's check: put x = −1 in.
The left bars give 3.
The right bars give 3 too.
The same!
Now put x = 1/2 in.
The left bars give 3/2.
The right bars give 3/2 too.
The same!
Both answers work!
Let's look back.
First: two cases.
Then: solve the first case.
Then: get x on one side.
Then: remove the 2.
Then: solve the second case.
Then: open the brackets.
Then: get x on one side.
Then: remove the − 2.
Then: remove the 4.
Then: write the answers.
Last: check.