Step by step
The question
Question: solve |3x − 2| = |x + 6|.
Both bars give how far from 0.
Two numbers the same distance from 0 are equal, or opposite.
Like 3 and minus 3.
Careful!
The minus is for the whole of x + 6.
Every sign in the bracket changes.
1 Two cases
So 3x − 2 = x + 6, or 3x − 2 = −(x + 6).
We keep the bracket: the minus is for all of it.
2 Solve the first case
First case: 3x − 2 = x + 6.
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 right 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 2x − 2 = 6.
We want x by itself.
Two things are stuck to x: times 2 and − 2.
The − 2 is on the outside, so it comes off first.
Then the times 2.
4 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.
6 + 2 = 8.
Now we have 2x = 8.
5 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.
8 divided by 2 is 4.
So x = 4!
6 Solve the second case
Second case: 3x − 2 = −(x + 6).
7 Open the brackets
A minus in front of a bracket changes the sign of every part inside.
x becomes −x.
6 becomes −6.
8 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 right side.
To remove −x, we add x.
We do it on both sides, so they stay equal.
− x and + x make zero.
Cross them out.
3x + x = 4x.
Now we have 4x − 2 = −6.
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.
9 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.
−6 + 2 = −4.
Now we have 4x = −4.
10 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.
−4 divided by 4 is −1.
So x = −1!
11 Write the answers
So x = 4 or x = −1.
12 Check
Let's check: put x = 4 in.
The left bars give 10.
The right bars give 10 too.
The same!
Now put x = −1 in.
The left bars give 5.
The right bars give 5 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: 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.