Step by step
The question
Question: solve (x − 5)/x ≥ 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 − 5 = 0, we add 5 on both sides.
The critical points are 5 and 0.
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 5 split the number line into 3 parts.
At 0 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
At 5, x − 5 is 0.
So the fraction is 0 there.
Left of 0, we test x = −1.
x − 5 is −6 and x is −1.
Minus over minus is plus.
Between 0 and 5, we test x = 1.
x − 5 is −4 and x is 1.
Minus over plus is minus.
Right of 5, we test x = 6.
x − 5 is 1 and x is 6.
Plus over plus is plus.
3 Read the answer
Greater than or equal to 0 means positive or 0.
So we want the plus parts, and the 0s.
But where the bottom is 0, the fraction has no value.
So that point is never part of the answer.
So x < 0 or x ≥ 5.
Careful!
At x = 0 the bottom is 0.
The fraction has no value there, so it is not in the answer, even with "or equal to".
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.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
5 Check
Let's check: a number from the answer, −1, put into the question.
6 is greater than or equal to 0.
True!
And a number outside the answer, 1.
−4 is not greater than or equal to 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.