Step by step
The question
Question: solve x + 4 < 2 or x − 5 ≥ 0.
"Or" means one of the parts is enough.
1 Solve the first part
First part: x + 4 < 2.
We want x by itself.
The + 4 is in the way.
To remove + 4, we take away 4.
We do it on both sides, so the sign stays true.
+ 4 and − 4 make zero.
Cross them out.
2 − 4 = −2.
So x is less than −2.
2 Solve the second part
Second part: x − 5 ≥ 0.
We want x by itself.
The − 5 is in the way.
To remove − 5, we add 5.
We do it on both sides, so the sign stays true.
− 5 and + 5 make zero.
Cross them out.
0 + 5 = 5.
So x is greater than or equal to 5.
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 ≥ 5.
Careful!
Many people write 5 ≤ 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.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
5 Check
Let's test one number from each piece.
First −3.
1 is less than 2.
True!
Now 6, from the other piece.
1 is greater than or equal to 0.
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.