Step by step
The question
Question: solve |x − 5| ≤ 1.
1 Draw the distance
2 Write it without bars
So x − 5 is between −1 and 1: −1 ≤ x − 5 ≤ 1.
Now x is in the middle.
3 Remove the − 5
We want x by itself.
The − 5 is in the way.
To remove − 5, we add 5.
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.
− 5 and + 5 make zero.
Cross them out.
−1 plus 5 is 4.
1 plus 5 is 6.
Now we have 4 ≤ x ≤ 6.
So x is between 4 and 6: 4 ≤ x ≤ 6.
4 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A square bracket means the end is in, like the filled circle.
5 Check
Let's test one number from the answer: 6.
We put it in the question.
The inside is 1, and the bars make it 1.
1 is less than or equal to 1.
True!
So the answer works.
Let's look back.
First: draw the distance.
Then: write it without bars.
Then: remove the − 5.
Then: show it on a number line.
Last: check.