Step by step
The question
Question: solve x² − 3x > 10.
There is an x², so the answer can be two parts.
1 Get 0 on one side
We want 0 on one side.
So we take 10 away from both sides.
We do it on both sides.
Why both sides?
The sign compares the two sides.
If we change only one side, the answer can change.
Now we have x² − 3x − 10 > 0.
2 Find two numbers
We need two numbers.
They multiply to −10, and they add to −3.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 10, smallest first.
We take the smaller from the bigger.
We need 3.
2 and 5 give 3.
Yes!
With the signs: 2 and −5.
The two numbers go in the brackets.
So we get (x + 2)(x − 5) > 0.
3 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 + 2 = 0, we take away 2 on both sides.
In x − 5 = 0, we add 5 on both sides.
The critical points are −2 and 5.
4 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, x + 2 is 0.
So the product is 0 there.
At 5, x − 5 is 0.
So the product is 0 there.
Left of −2, we test x = −3.
x + 2 is −1 and x − 5 is −8.
Minus times minus is plus.
Between −2 and 5, we test x = 0.
x + 2 is 2 and x − 5 is −5.
Plus times minus is minus.
Right of 5, we test x = 6.
x + 2 is 8 and x − 5 is 1.
Plus times plus is plus.
5 Read the answer
Greater than 0 means positive.
So we want the parts with a plus.
So x < −2 or x > 5.
Careful!
Not the minus part!
Greater than 0 means plus.
6 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.
7 Check
Let's check with a picture: the graph of y = x² − 3x − 10.
It is a parabola.
It crosses the x-axis at the critical points, −2 and 5.
Above the x-axis, y is plus.
That is x < −2 or x > 5: our answer.
Let's check: a number from the answer, 7, put into the question.
28 is greater than 10.
True!
Let's look back.
First: get 0 on one side.
Then: 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.