Step by step
The question
Question: solve |4 − x| < 6.
1 Draw the distance
2 Write it without bars
So 4 − x is between −6 and 6: −6 < 4 − x < 6.
Now x is in the middle.
3 Remove the 4
We want x by itself.
The 4 is in the way.
To remove 4, we take away 4.
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.
4 and − 4 make zero.
Cross them out.
−6 minus 4 is −10.
6 minus 4 is 2.
Now we have −10 < −x < 2.
4 Remove the −1
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
We do it to all three parts.
We divide by a negative number.
That turns both signs around.
−1 divided by −1 is 1.
So just x is left.
−10 divided by −1 is 10.
2 divided by −1 is −2.
Now we have 10 > x > −2.
We turn it around, so the small number is on the left.
It says the same thing.
So x is between −2 and 10: −2 < x < 10.
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: 5.
We put it in the question.
The inside is −1, and the bars make it 1.
1 is less than 6.
True!
So the answer works.
Let's look back.
First: draw the distance.
Then: write it without bars.
Then: remove the 4.
Then: remove the −1.
Then: show it on a number line.
Last: check.