Step by step
The question
Question: solve −3 < 2x + 1 ≤ 7.
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 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 make zero.
Cross them out.
−3 minus 1 is −4.
7 minus 1 is 6.
Now we have −4 < 2x ≤ 6.
2 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.
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 −3, and less than or equal to 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.