Step by step
The question
Question: solve |2x − 1| ≤ 5.
1 Draw the distance
2 Write it without bars
So 2x − 1 is between −5 and 5: −5 ≤ 2x − 1 ≤ 5.
Now x is in the middle.
3 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.
−5 plus 1 is −4.
5 plus 1 is 6.
Now we have −4 ≤ 2x ≤ 6.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it to all three parts.
2 is a positive number, so the signs stay the same.
2 divided by 2 is 1.
So just x is left.
−4 divided by 2 is −2.
6 divided by 2 is 3.
Now we have −2 ≤ x ≤ 3.
So x is between −2 and 3: −2 ≤ x ≤ 3.
5 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.
6 Check
Let's test one number from the answer: 1.
We put it in the question.
The inside is 1, and the bars make it 1.
1 is less than or equal to 5.
True!
So the answer works.
Let's look back.
First: draw the distance.
Then: write it without bars.
Then: remove the − 1.
Then: remove the 2.
Then: show it on a number line.
Last: check.