Step by step
The question
Question: solve 2|x − 1| + 3 < 11.
1 Get the bars alone
We want the bars by themselves.
The + 3 is in the way, so we take away 3 on both sides.
+ 3 and − 3 make zero.
Cross them out.
11 minus 3 is 8.
To remove times 2, we divide both sides by 2.
2 is a positive number, so the sign stays the same.
2 divided by 2 is 1.
So just the bars is left.
8 divided by 2 is 4.
2 Draw the distance
3 Write it without bars
So x − 1 is between −4 and 4: −4 < x − 1 < 4.
Now x is in the middle.
4 Remove the − 1
We want x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it to all three parts.
Why all three?
The signs compare the three parts.
If we change only one part, the answer can change.
− 1 and + 1 make zero.
Cross them out.
−4 plus 1 is −3.
4 plus 1 is 5.
Now we have −3 < x < 5.
So x is between −3 and 5: −3 < x < 5.
5 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A round bracket means the end is not in, like the open circle.
6 Check
Let's test one number from the answer: 2.
We put it in the question.
The inside is 1, and the bars make it 1.
5 is less than 11.
True!
So the answer works.
Let's look back.
First: get the bars alone.
Then: draw the distance.
Then: write it without bars.
Then: remove the − 1.
Then: show it on a number line.
Last: check.