Step by step
The question
Question: solve |x + 2| = |x − 4|.
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 − 4.
Every sign in the bracket changes.
1 Two cases
So x + 2 = x − 4, or x + 2 = −(x − 4).
We keep the bracket: the minus is for all of it.
2 Solve the first case
First case: x + 2 = x − 4.
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.
x − x = 0.
2 = −4 is false for every x.
So this case gives no x.
4 Solve the second case
Second case: x + 2 = −(x − 4).
5 Open the brackets
A minus in front of a bracket changes the sign of every part inside.
x becomes −x.
−4 becomes +4.
6 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.
x + x = 2x.
Now we have 2x + 2 = 4.
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.
7 Remove the + 2
To remove + 2, we take away 2.
We do it on both sides, so they stay equal.
+ 2 and − 2 make zero.
Cross them out.
4 − 2 = 2.
Now we have 2x = 2.
8 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!
9 Write the answers
Only one case works.
So x = 1.
10 Check
Let's check: put x = 1 in.
The left bars give 3.
The right bars give 3 too.
The same!
Let's look back.
First: two cases.
Then: solve the first case.
Then: get x on one side.
Then: solve the second case.
Then: open the brackets.
Then: get x on one side.
Then: remove the + 2.
Then: remove the 2.
Then: write the answers.
Last: check.