Step by step
The question
Question: solve |x − 3| + 2 < 6.
1 Get the bars alone
We want the bars by themselves.
The + 2 is in the way, so we take away 2 on both sides.
+ 2 and − 2 make zero.
Cross them out.
6 minus 2 is 4.
2 Draw the distance
3 Write it without bars
So x − 3 is between −4 and 4: −4 < x − 3 < 4.
Now x is in the middle.
4 Remove the − 3
We want x by itself.
The − 3 is in the way.
To remove − 3, we add 3.
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.
− 3 and + 3 make zero.
Cross them out.
−4 plus 3 is −1.
4 plus 3 is 7.
Now we have −1 < x < 7.
So x is between −1 and 7: −1 < x < 7.
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: 4.
We put it in the question.
The inside is 1, and the bars make it 1.
3 is less than 6.
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 − 3.
Then: show it on a number line.
Last: check.