Step by step
The question
Question: solve x² + 2x ≤ 8.
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 8 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 + 2x − 8 ≤ 0.
2 Find two numbers
We need two numbers.
They multiply to −8, and they add to 2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 8, smallest first.
We take the smaller from the bigger.
We need 2.
2 and 4 give 2.
Yes!
With the signs: −2 and 4.
The two numbers go in the brackets.
So we get (x − 2)(x + 4) ≤ 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 add 2 on both sides.
In x + 4 = 0, we take away 4 on both sides.
The critical points are 2 and −4.
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 −4 and 2 split the number line into 3 parts.
At −4, x + 4 is 0.
So the product is 0 there.
At 2, x − 2 is 0.
So the product is 0 there.
Left of −4, we test x = −5.
x − 2 is −7 and x + 4 is −1.
Minus times minus is plus.
Between −4 and 2, we test x = 0.
x − 2 is −2 and x + 4 is 4.
Minus times plus is minus.
Right of 2, we test x = 3.
x − 2 is 1 and x + 4 is 7.
Plus times plus is plus.
5 Read the answer
Less than or equal to 0 means negative or 0.
So we want the minus parts, and the 0s.
So −4 ≤ x ≤ 2.
Careful!
Not the plus parts!
Less than 0 means minus.
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.
7 Check
Let's check with a picture: the graph of y = x² + 2x − 8.
It is a parabola.
It crosses the x-axis at the critical points, −4 and 2.
Below the x-axis, y is minus.
That is −4 ≤ x ≤ 2: our answer.
Let's check: a number from the answer, 0, put into the question.
0 is less than or equal to 8.
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.