Step by step
The question
Question: solve 1 < 2x + 4 ≤ 9.
x is in the middle, between two numbers.
1 Remove the + 4
We want x by itself.
The + 4 is in the way.
To remove + 4, we take away 4.
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.
+ 4 and − 4 make zero.
Cross them out.
1 minus 4 is −3.
9 minus 4 is 5.
Now we have −3 < 2x ≤ 5.
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.
−3 divided by 2 is −3/2.
5 divided by 2 is 5/2.
Now we have −3/2 < x ≤ 5/2.
So x is between −3/2 and 5/2: −3/2 < x ≤ 5/2.
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.
6 is greater than 1, and less than or equal to 9.
True!
So the answer works.
Let's look back.
First: remove the + 4.
Then: remove the 2.
Then: show it on a number line.
Last: check.