Step by step
The question
Question: solve x² − 16 ≥ 0.
There is an x², so the answer can be two parts.
1 Two squares
This is one square take away another square.
x² is x squared.
16 is 4 squared.
A square minus a square is the plus bracket times the minus bracket.
So we get (x + 4)(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 + 4 = 0, we take away 4 on both sides.
In x − 4 = 0, we add 4 on both sides.
The critical points are −4 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 4 split the number line into 3 parts.
At −4, x + 4 is 0.
So the product is 0 there.
At 4, x − 4 is 0.
So the product is 0 there.
Left of −4, we test x = −5.
x + 4 is −1 and x − 4 is −9.
Minus times minus is plus.
Between −4 and 4, we test x = 0.
x + 4 is 4 and x − 4 is −4.
Plus times minus is minus.
Right of 4, we test x = 5.
x + 4 is 9 and x − 4 is 1.
Plus times plus is plus.
4 Read the answer
Greater than or equal to 0 means positive or 0.
So we want the plus parts, and the 0s.
So x ≤ −4 or x ≥ 4.
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 square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
6 Check
Let's check with a picture: the graph of y = x² − 16.
It is a parabola.
It crosses the x-axis at the critical points, −4 and 4.
Above the x-axis, y is plus.
That is x ≤ −4 or x ≥ 4: our answer.
Let's check: a number from the answer, 6, put into the question.
20 is greater than or equal to 0.
True!
Let's look back.
First: two squares.
Then: find the critical points.
Then: make a sign table.
Then: read the answer.
Then: show it on a number line.
Last: check.