Step by step
The question
Question: solve (3x + 6)/(x − 5) < 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 3x + 6 = 0, we take away 6 on both sides.
In x − 5 = 0, we add 5 on both sides.
Now we have 3x = −6 and x = 5.
To remove times 3, we divide both sides by 3.
The critical points are −2 and 5.
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 −2 and 5 split the number line into 3 parts.
At −2, 3x + 6 is 0.
So the fraction is 0 there.
At 5 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
Left of −2, we test x = −3.
3x + 6 is −3 and x − 5 is −8.
Minus over minus is plus.
Between −2 and 5, we test x = 0.
3x + 6 is 6 and x − 5 is −5.
Plus over minus is minus.
Right of 5, we test x = 6.
3x + 6 is 24 and x − 5 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 −2 < x < 5.
Careful!
At x = 5 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.
5 Check
Let's check: a number from the answer, 0, put into the question.
−6/5 is less than 0.
True!
And a number outside the answer, −3.
3/8 is not less than 0.
False, as it should be: −3 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.