Step by step
The question
Question: solve −2/(x + 3) > 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 + 3 0.
The bottom counts too: there the fraction has no value.
In x + 3 = 0, we take away 3 on both sides.
The critical points are −3.
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 −3 split the number line into 2 parts.
At −3 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
Left of −3, we test x = −4.
x + 3 is −1.
Minus over minus is plus.
Right of −3, we test x = 0.
x + 3 is 3.
Minus over plus is minus.
3 Read the answer
Greater than 0 means positive.
So we want the parts with a plus.
So x < −3.
Careful!
At x = −3 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, −4, put into the question.
2 is greater than 0.
True!
And a number outside the answer, 0.
−2/3 is not greater than 0.
False, as it should be: 0 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.