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