Step by step
The question
Question: solve (x + 6)/(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 + 6 = 0, we take away 6 on both sides.
In x + 2 = 0, we take away 2 on both sides.
The critical points are −6 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 −6 and −2 split the number line into 3 parts.
At −6, x + 6 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 −6, we test x = −7.
x + 6 is −1 and x + 2 is −5.
Minus over minus is plus.
Between −6 and −2, we test x = −5.
x + 6 is 1 and x + 2 is −3.
Plus over minus is minus.
Right of −2, we test x = 0.
x + 6 is 6 and x + 2 is 2.
Plus over plus is plus.
3 Read the answer
Less than 0 means negative.
So we want the parts with a minus.
So −6 < 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.
5 Check
Let's check: a number from the answer, −5, put into the question.
−1/3 is less than 0.
True!
And a number outside the answer, −7.
1/5 is not less than 0.
False, as it should be: −7 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.