Solve x <= -3 or x > 1

Step by step

The question

Question: solve x ≤ −3 or x > 1.

"Or" means one of the parts is enough.

Both parts are solved already: x is by itself in each one.

1 Show each part on a number line

2 Put the parts together

"Or": we keep every number that is shaded on either line.

So x ≤ −3 or x > 1.

Careful!

Many people write 1 < x ≤ −3.

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.

3 Check

Let's test one number from each piece.

First −4.

−4 is less than or equal to −3.

True!

Now 2, from the other piece.

2 is greater than 1.

True!

So the answer works.

Let's look back.

First: show each part on a number line.

Then: put the parts together.

Last: check.