Step by step
The question
Question: solve x − 1 < −3 or x + 2 > 6.
"Or" means one of the parts is enough.
1 Solve the first part
First part: x − 1 < −3.
We want x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides, so the sign stays true.
− 1 and + 1 make zero.
Cross them out.
−3 + 1 = −2.
So x is less than −2.
2 Solve the second part
Second part: x + 2 > 6.
We want x by itself.
The + 2 is in the way.
To remove + 2, we take away 2.
We do it on both sides, so the sign stays true.
+ 2 and − 2 make zero.
Cross them out.
6 − 2 = 4.
So x is greater than 4.
3 Show each part on a number line
4 Put the parts together
"Or": we keep every number that is shaded on either line.
So x < −2 or x > 4.
Careful!
Many people write 4 < x < −2.
But a chain means between, and no number is between like that.
So we keep the word "or".
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.
∞ means it goes on for ever.
∪ means or.
5 Check
Let's test one number from each piece.
First −3.
−4 is less than −3.
True!
Now 5, from the other piece.
7 is greater than 6.
True!
So the answer works.
Let's look back.
First: solve the first part.
Then: solve the second part.
Then: show each part on a number line.
Then: put the parts together.
Last: check.