Step by step
The question
Question: solve 3/(x − 1) < 0.
There is an x at the bottom.
The bottom can never be 0.
1 Find the critical points
A critical point is an x where the side with x is 0, or has no value.
Only there can its sign change.
We make x − 1 0.
The bottom counts too: there the fraction has no value.
In x − 1 = 0, we add 1 on both sides.
The critical points are 1.
2 Make a sign table
Between the critical points the sign stays the same.
So one test number tells us the sign of each part.
We draw a table.
The critical points 1 split the number line into 2 parts.
At 1 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
Left of 1, we test x = 0.
x − 1 is −1.
Plus over minus is minus.
Right of 1, we test x = 2.
x − 1 is 1.
Plus over plus is plus.
3 Read the answer
Less than 0 means negative.
So we want the parts with a minus.
So x < 1.
Careful!
At x = 1 the bottom is 0.
Dividing by 0 is not allowed.
So that point is never in the answer.
4 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.
5 Check
Let's check: a number from the answer, 0, put into the question.
−3 is less than 0.
True!
And a number outside the answer, 2.
3 is not less than 0.
False, as it should be: 2 is not in the answer.
Let's look back.
First: find the critical points.
Then: make a sign table.
Then: read the answer.
Then: show it on a number line.
Last: check.