Step by step
The question
Question: solve x/(x − 2) > 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 each bracket 0.
The bottom counts too: there the fraction has no value.
In x − 2 = 0, we add 2 on both sides.
The critical points are 0 and 2.
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 0 and 2 split the number line into 3 parts.
At 0, x is 0.
So the fraction is 0 there.
At 2 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
Left of 0, we test x = −1.
x is −1 and x − 2 is −3.
Minus over minus is plus.
Between 0 and 2, we test x = 1.
x is 1 and x − 2 is −1.
Plus over minus is minus.
Right of 2, we test x = 3.
x is 3 and x − 2 is 1.
Plus over plus is plus.
3 Read the answer
Greater than 0 means positive.
So we want the parts with a plus.
So x < 0 or x > 2.
Careful!
At x = 2 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.
∪ means or.
5 Check
Let's check: a number from the answer, −1, put into the question.
1/3 is greater than 0.
True!
And a number outside the answer, 1.
−1 is not greater than 0.
False, as it should be: 1 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.