Step by step
The question
Question: solve |x − 5| = |x + 1|.
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 + 1.
Every sign in the bracket changes.
1 Two cases
So x − 5 = x + 1, or x − 5 = −(x + 1).
We keep the bracket: the minus is for all of it.
2 Solve the first case
First case: x − 5 = x + 1.
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.
−5 = 1 is false for every x.
So this case gives no x.
4 Solve the second case
Second case: x − 5 = −(x + 1).
5 Open the brackets
A minus in front of a bracket changes the sign of every part inside.
x becomes −x.
1 becomes −1.
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 − 5 = −1.
We want x by itself.
Two things are stuck to x: times 2 and − 5.
The − 5 is on the outside, so it comes off first.
Then the times 2.
7 Remove the − 5
To remove − 5, we add 5.
We do it on both sides, so they stay equal.
− 5 and + 5 make zero.
Cross them out.
−1 + 5 = 4.
Now we have 2x = 4.
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.
4 divided by 2 is 2.
So x = 2!
9 Write the answers
Only one case works.
So x = 2.
10 Check
Let's check: put x = 2 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 − 5.
Then: remove the 2.
Then: write the answers.
Last: check.