Worked examples
The question
Question: solve x² − x − 6 > 0.
There is an x², so the answer can be two parts.
x2 − x − 6 > 0
1 Find two numbers
We need two numbers.
They multiply to −6, and they add to −1.
? × ? = −6
? + ? = −1
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.
× is − → one +, one −; bigger one −
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: 2 and −3.
1 × 6: 6 − 1 = 5
2 × 3: 3 − 2 = 1 ✓
The two numbers go in the brackets.
So we get (x + 2)(x − 3) > 0.
(x + 2)(x − 3) > 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.
critical point: where it is 0 (or ÷ 0)
We make each bracket 0.
x + 2 = 0 x − 3 = 0
In x + 2 = 0, we take away 2 on both sides.
In x − 3 = 0, we add 3 on both sides.
x + 2 − 2 = 0 − 2 x − 3 + 3 = 0 + 3
The critical points are −2 and 3.
x = −2 x = 3
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 −2 and 3 split the number line into 3 parts.
0
At −2, x + 2 is 0.
So the product is 0 there.
0
0
At 3, x − 3 is 0.
So the product is 0 there.
0
Left of −2, we test x = −3.
−3
x + 2 is −1 and x − 3 is −6.
−1
−6
Minus times minus is plus.
Between −2 and 3, we test x = 0.
0
x + 2 is 2 and x − 3 is −3.
2
−3
Plus times minus is minus.
Right of 3, we test x = 4.
4
x + 2 is 6 and x − 3 is 1.
6
1
Plus times plus is plus.
4 Read the answer
Greater than 0 means positive.
So we want the parts with a plus.
So x < −2 or x > 3.
x < −2 or x > 3
> 0 → the + parts
Careful!
Not the minus part!
Greater than 0 means plus.
Careful: not −2 < x < 3 ✗
5 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.
interval: (−∞, −2) ∪ (3, ∞)
6 Check
Let's check with a picture: the graph of y = x² − x − 6.
It is a parabola.
It crosses the x-axis at the critical points, −2 and 3.
Above the x-axis, y is plus.
That is x < −2 or x > 3: our answer.
Let's check: a number from the answer, 5, put into the question.
x = 5: 52 − 5 − 6 = 25 − 5 − 6 = 14
14 is greater than 0.
True!
14 > 0 true
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.
Answer Careful: not −2 < x < 3 ✗
▶ Watch it solved
The question
Question: solve x² + 2x ≤ 8.
There is an x², so the answer can be two parts.
x2 + 2x ≤ 8
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.
x2 + 2x − 8 ≤ 8 − 8
Now we have x² + 2x − 8 ≤ 0.
x2 + 2x − 8 ≤ 0
2 Find two numbers
We need two numbers.
They multiply to −8, and they add to 2.
? × ? = −8
? + ? = 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.
× is − → one +, one −; bigger one +
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.
1 × 8: 8 − 1 = 7
2 × 4: 4 − 2 = 2 ✓
The two numbers go in the brackets.
So we get (x − 2)(x + 4) ≤ 0.
(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.
critical point: where it is 0 (or ÷ 0)
We make each bracket 0.
x − 2 = 0 x + 4 = 0
In x − 2 = 0, we add 2 on both sides.
In x + 4 = 0, we take away 4 on both sides.
x − 2 + 2 = 0 + 2 x + 4 − 4 = 0 − 4
The critical points are 2 and −4.
x = 2 x = −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.
0
At −4, x + 4 is 0.
So the product is 0 there.
0
0
At 2, x − 2 is 0.
So the product is 0 there.
0
Left of −4, we test x = −5.
−5
x − 2 is −7 and x + 4 is −1.
−7
−1
Minus times minus is plus.
Between −4 and 2, we test x = 0.
0
x − 2 is −2 and x + 4 is 4.
−2
4
Minus times plus is minus.
Right of 2, we test x = 3.
3
x − 2 is 1 and x + 4 is 7.
1
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.
−4 ≤ x ≤ 2
≤ 0 → the − parts and the 0s
Careful!
Not the plus parts!
Less than 0 means minus.
Careful: not x ≤ −4 or x ≥ 2 ✗
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.
interval: [−4, 2]
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.
x = 0: 02 + 2(0) = 0 + 0 = 0
0 is less than or equal to 8.
True!
0 ≤ 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.
Answer Careful: not x ≤ −4 or x ≥ 2 ✗
▶ Watch it solved
The question
Question: solve 2x² + x − 3 ≤ 0.
There is an x², so the answer can be two parts.
2x2 + x − 3 ≤ 0
1 Find two numbers
The number in front of x² is 2.
So we multiply 2 by −3: that is −6.
We need two numbers that multiply to −6 and add to 1.
2 × (−3) = −6
? × ? = −6, ? + ? = 1
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.
× is − → one +, one −; bigger one +
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: −2 and 3.
1 × 6: 6 − 1 = 5
2 × 3: 3 − 2 = 1 ✓
2 Split the middle
We write x as −2x + 3x.
It is the same amount.
2x2 − 2x + 3x − 3 ≤ 0
3 Group in pairs
We put the first two together, and the last two together.
(2x2 − 2x) + (3x − 3) ≤ 0
From the first pair we take out 2x.
From the second pair we take out 3.
Both leave the same bracket.
(2x2 − 2x) = 2x(x − 1)
(3x − 3) = 3(x − 1)
Now both parts have x − 1 in them.
2x(x − 1) + 3(x − 1) ≤ 0
4 Same bracket out
x − 1 is in both parts, so we take it out once.
What is left, 2x + 3, goes in the other bracket.
ab + cb = b(a + c)
So we get (x − 1)(2x + 3) ≤ 0.
(x − 1)(2x + 3) ≤ 0
5 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.
critical point: where it is 0 (or ÷ 0)
We make each bracket 0.
x − 1 = 0 2x + 3 = 0
In x − 1 = 0, we add 1 on both sides.
In 2x + 3 = 0, we take away 3 on both sides.
x − 1 + 1 = 0 + 1 2x + 3 − 3 = 0 − 3
Now we have x = 1 and 2x = −3.
x = 1 2x = −3
To remove times 2, we divide both sides by 2.
The critical points are 1 and −3/2.
x = 1 x = −32
6 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 −3/2 and 1 split the number line into 3 parts.
0
At −3/2, 2x + 3 is 0.
So the product is 0 there.
0
0
At 1, x − 1 is 0.
So the product is 0 there.
0
Left of −3/2, we test x = −2.
−2
x − 1 is −3 and 2x + 3 is −1.
−3
−1
Minus times minus is plus.
Between −3/2 and 1, we test x = 0.
0
x − 1 is −1 and 2x + 3 is 3.
−1
3
Minus times plus is minus.
Right of 1, we test x = 2.
2
x − 1 is 1 and 2x + 3 is 7.
1
7
Plus times plus is plus.
7 Read the answer
Less than or equal to 0 means negative or 0.
So we want the minus parts, and the 0s.
So −3/2 ≤ x ≤ 1.
−32 ≤ x ≤ 1
≤ 0 → the − parts and the 0s
Careful!
Not the plus parts!
Less than 0 means minus.
Careful: not x ≤ −32 or x ≥ 1 ✗
8 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.
interval: [−32, 1]
9 Check
Let's check with a picture: the graph of y = 2x² + x − 3.
It is a parabola.
It crosses the x-axis at the critical points, −3/2 and 1.
Below the x-axis, y is minus.
That is −3/2 ≤ x ≤ 1: our answer.
Let's check: a number from the answer, 0, put into the question.
x = 0: 2(0)2 + 0 − 3 = 0 + 0 − 3 = −3
−3 is less than or equal to 0.
True!
−3 ≤ 0 true
Let's look back.
First: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: find the critical points.
Then: make a sign table.
Then: read the answer.
Then: show it on a number line.
Last: check.
Answer Careful: not x ≤ −32 or x ≥ 1 ✗
▶ Watch it solved
