Step by step
The question
Question: solve x² + 5x + 4 > 0.
There is an x², so the answer can be two parts.
1 Find two numbers
We need two numbers.
They multiply to 4, and they add to 5.
They multiply to a plus number, so the two numbers have the same sign.
They add to a plus number, so both are plus.
We list the pairs that make 4, smallest first.
We add each pair.
We need 5.
1 and 4 give 5.
Yes!
The two numbers go in the brackets.
So we get (x + 1)(x + 4) > 0.
2 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.
In x + 1 = 0, we take away 1 on both sides.
In x + 4 = 0, we take away 4 on both sides.
The critical points are −1 and −4.
3 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 −4 and −1 split the number line into 3 parts.
At −4, x + 4 is 0.
So the product is 0 there.
At −1, x + 1 is 0.
So the product is 0 there.
Left of −4, we test x = −5.
x + 1 is −4 and x + 4 is −1.
Minus times minus is plus.
Between −4 and −1, we test x = −3.
x + 1 is −2 and x + 4 is 1.
Minus times plus is minus.
Right of −1, we test x = 0.
x + 1 is 1 and x + 4 is 4.
Plus times plus is plus.
4 Read the answer
Greater than 0 means positive.
So we want the parts with a plus.
So x < −4 or x > −1.
Careful!
Not the minus part!
Greater than 0 means plus.
5 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.
6 Check
Let's check with a picture: the graph of y = x² + 5x + 4.
It is a parabola.
It crosses the x-axis at the critical points, −4 and −1.
Above the x-axis, y is plus.
That is x < −4 or x > −1: our answer.
Let's check: a number from the answer, 1, put into the question.
10 is greater than 0.
True!
Let's look back.
First: find two numbers.
Then: find the critical points.
Then: make a sign table.
Then: read the answer.
Then: show it on a number line.
Last: check.