Step by step
The question
Question: solve −x < −3 or −x > 1.
Two inequalities, joined by "or".
"Or" means: x can make one of them true, or the other.
We solve both, side by side.
1 Remove the −1
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
In each one, on both sides.
We divide by a negative number.
That turns the signs around.
−1 divided by −1 is 1.
So just x is left.
−3 divided by −1 is 3.
1 divided by −1 is −1.
So x > 3 or x < −1.
2 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.
∞ means it goes on for ever.
∪ means or.
3 Check
Let's test one number from each part.
First −2, from the left part.
2 is greater than 1.
True!
Now 4, from the right part.
−4 is less than −3.
True!
Both parts work!
Let's look back.
First: remove the −1.
Then: show it on a number line.
Last: check.