Step by step
The question
Question: solve −3 < x/2 + 1 < 4.
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.
4 minus 1 is 3.
Now we have −4 < x/2 < 3.
2 Remove the fraction
The bottom number is 2.
Times 2 will remove it.
We multiply all three parts by 2.
2 is a positive number, so the signs stay the same.
2/2 is 1.
The fraction is gone.
2 times −4 is −8.
2 times 3 is 6.
Now we have −8 < x < 6.
So x is between −8 and 6: −8 < x < 6.
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.
4 Check
Let's test one number from the answer: −2.
We put it in place of x in the middle part.
0 is greater than −3, and less than 4.
True!
So the answer works.
Let's look back.
First: remove the + 1.
Then: remove the fraction.
Then: show it on a number line.
Last: check.