Step by step
The question
Question: solve x² ≥ 3x.
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 3x 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 ≥ 0.
2 Take out the common factor
Every term has at least x.
So the common factor is x.
We divide every term by it.
What is left goes in the bracket.
So it is x(x − 3) ≥ 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 − 3 = 0, we add 3 on both sides.
The critical points are 0 and 3.
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 0 and 3 split the number line into 3 parts.
At 0, x is 0.
So the product is 0 there.
At 3, x − 3 is 0.
So the product is 0 there.
Left of 0, we test x = −1.
x is −1 and x − 3 is −4.
Minus times minus is plus.
Between 0 and 3, we test x = 1.
x is 1 and x − 3 is −2.
Plus times minus is minus.
Right of 3, we test x = 4.
x is 4 and x − 3 is 1.
Plus times plus is plus.
5 Read the answer
Greater than or equal to 0 means positive or 0.
So we want the plus parts, and the 0s.
So x ≤ 0 or x ≥ 3.
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 square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
7 Check
Let's check with a picture: the graph of y = x² − 3x.
It is a parabola.
It crosses the x-axis at the critical points, 0 and 3.
Above the x-axis, y is plus.
That is x ≤ 0 or x ≥ 3: our answer.
Let's look back.
First: get 0 on one side.
Then: take out the common factor.
Then: find the critical points.
Then: make a sign table.
Then: read the answer.
Then: show it on a number line.
Last: check.