Step by step
The question
Question: solve |x − 2| < 3.
1 Draw the distance
2 Write it without bars
So x − 2 is between −3 and 3: −3 < x − 2 < 3.
Now x is in the middle.
3 Remove the − 2
We want x by itself.
The − 2 is in the way.
To remove − 2, we add 2.
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.
− 2 and + 2 make zero.
Cross them out.
−3 plus 2 is −1.
3 plus 2 is 5.
Now we have −1 < x < 5.
So x is between −1 and 5: −1 < x < 5.
4 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.
5 Check
Let's test one number from the answer: 3.
We put it in the question.
The inside is 1, and the bars make it 1.
1 is less than 3.
True!
So the answer works.
Let's look back.
First: draw the distance.
Then: write it without bars.
Then: remove the − 2.
Then: show it on a number line.
Last: check.