Step by step
The question
Question: solve −5 < 4x − 1 ≤ 11.
x is in the middle, between two numbers.
1 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.
11 plus 1 is 12.
Now we have −4 < 4x ≤ 12.
2 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it to all three parts.
4 is a positive number, so the signs stay the same.
4 divided by 4 is 1.
So just x is left.
−4 divided by 4 is −1.
12 divided by 4 is 3.
Now we have −1 < x ≤ 3.
So x is between −1 and 3: −1 < x ≤ 3.
3 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.
A square bracket means the end is in, like the filled circle.
4 Check
Let's test one number from the answer: 1.
We put it in place of x in the middle part.
3 is greater than −5, and less than or equal to 11.
True!
So the answer works.
Let's look back.
First: remove the − 1.
Then: remove the 4.
Then: show it on a number line.
Last: check.