Step by step
The question
Question: solve −3 ≤ 2x + 1 < 7.
x is in the middle, between two numbers.
Is x alone now?
Not yet: + 1 is still with x.
How do we remove it?
1 Remove the + 1
We want x by itself.
The + 1 is in the way.
First we remove the + 1, so 2x is alone.
Then we remove the times 2.
To remove + 1, we take away 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 cancel each other.
They make zero.
−3 minus 1 is −4.
7 minus 1 is 6.
Now we have −4 ≤ 2x < 6.
So after step 1: we took away 1, and now we have −4 ≤ 2x < 6.
2 Remove the 2
2x means 2 times 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.
They cancel each other.
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.
3 Show it on a number line
A filled circle at −2.
−2 is part of the answer, because of the "or equal to".
An open circle at 3.
3 itself is not part of the answer.
We shade between them.
These are all the numbers between −2 and 3.
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.
It is between −2 and 3.
We put it in place of x in the middle part.
3 is greater than or equal to −3, and less than 7.
True!
So the answer works.
Let's look back.
First: remove the + 1.
Then: remove the 2.
Then: show it on a number line.
Last: check.