Step by step
The question
Question: solve x² − x − 12 < 0.
1 Find two numbers
To see where it is plus or minus, we write x² − x − 12 as brackets times brackets.
Then the sign rules for times tell us.
The rule: we need two numbers.
They multiply to the last number, −12.
They add to the middle number, −1.
Why?
Open two brackets, x plus m, times x plus n.
We get x squared, then m plus n lots of x, then m times n.
So the two numbers multiply to −12 and add to −1.
Look at the sign in front of 12: it is minus.
So one number is plus and one is minus.
Now the sign in front of x: it is minus.
So the bigger one is minus.
When we add a plus and a minus, the bigger one wins.
A plus and a minus number added: that is the bigger take away the smaller.
We list the pairs that make 12, smallest first.
For each pair, we check two things: they multiply to 12, and the bigger take away the smaller is 1.
One is minus, so adding them is a take away.
1 × 12 = 12.
12 − 1 = 11.
Not 1.
2 × 6 = 12.
6 − 2 = 4.
Not 1.
3 × 4 = 12.
4 − 3 = 1.
Both work!
So the two numbers are 3 and 4.
With the signs: 3 and −4.
The two numbers go in the brackets.
So we get (x + 3)(x − 4) < 0.
2 Find the critical points
A critical point is an x where the side with x is 0.
A times only changes from plus to minus where one bracket passes 0.
We make each bracket 0.
In x + 3 = 0, we take away 3 on both sides.
In x − 4 = 0, we add 4 on both sides.
The critical points are −3 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 −3 and 4 split the number line into 3 parts.
At −3, x + 3 is 0.
So the product is 0 there.
At 4, x − 4 is 0.
So the product is 0 there.
Left of −3, we test x = −4.
Any number in that part works.
We pick an easy one.
x + 3 is −1 and x − 4 is −8.
Minus times minus is plus.
Between −3 and 4, we test x = 0.
x + 3 is 3 and x − 4 is −4.
Plus times minus is minus.
Right of 4, we test x = 5.
x + 3 is 8 and x − 4 is 1.
Plus times plus is plus.
4 Read the answer
Less than 0 means negative.
So we want the parts with a minus.
So −3 < x < 4.
The critical points give 0, so they are not in.
Careful!
Not the plus parts!
Less than 0 means minus.
5 Show it on a number line
An open circle at −3.
−3 itself is not part of the answer.
An open circle at 4.
4 itself is not part of the answer.
We shade between −3 and 4.
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.
6 Check
Let's check with a picture: the graph of y = x² − x − 12.
It is a parabola.
It crosses the x-axis at the critical points, −3 and 4.
Below the x-axis, y is minus.
That is −3 < x < 4: our answer.
Let's check: a number from the answer, 0, put into the question.
−12 is less than 0.
True!
And a number outside the answer, −4.
8 is not less than 0.
False, as it should be: −4 is not in the answer.
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.