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