Worked examples
The question
Question: solve (x − 2)/(x + 3) ≥ 0.
x − 2x + 3 ≥ 0
There is an x at the bottom.
The bottom can never be 0.
the bottom can never be 0
1 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.
The bottom counts too: there the fraction has no value.
x − 2 = 0 x + 3 = 0
In x − 2 = 0, we add 2 on both sides.
In x + 3 = 0, we take away 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
2 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 2 split the number line into 3 parts.
0
At −3 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
✗
0
At 2, x − 2 is 0.
So the fraction is 0 there.
0
Left of −3, we test x = −4.
−4
x − 2 is −6 and x + 3 is −1.
−6
−1
Minus over minus is plus.
Between −3 and 2, we test x = 0.
0
x − 2 is −2 and x + 3 is 3.
−2
3
Minus over plus is minus.
Right of 2, we test x = 3.
3
x − 2 is 1 and x + 3 is 6.
1
6
Plus over plus is plus.
3 Read the answer
Greater than or equal to 0 means positive or 0.
So we want the plus parts, and the 0s.
But where the bottom is 0, the fraction has no value.
So that point is never part of the answer.
So x < −3 or x ≥ 2.
x < −3 or x ≥ 2
≥ 0 → the + parts, the 0
Careful!
At x = −3 the bottom is 0.
The fraction has no value there, so it is not in the answer, even with "or equal to".
Careful: not x = −3 is in ✗
4 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.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
interval: (−∞, −3) ∪ [2, ∞)
5 Check
Let's check: a number from the answer, −4, put into the question.
x = −4: (−4) − 2(−4) + 3 = −6−1 = 6
6 is greater than or equal to 0.
True!
6 ≥ 0 true
And a number outside the answer, 0.
x = 0: 0 − 20 + 3 = −23
−2/3 is not greater than or equal to 0.
False, as it should be: 0 is not in the answer.
−23 ≥ 0 false
Let's look back.
First: 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 = −3 is in ✗
▶ Watch it solved
The question
Question: solve (2x − 1)/(x + 1) ≥ 0.
2x − 1x + 1 ≥ 0
There is an x at the bottom.
The bottom can never be 0.
the bottom can never be 0
1 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.
The bottom counts too: there the fraction has no value.
2x − 1 = 0 x + 1 = 0
In 2x − 1 = 0, we add 1 on both sides.
In x + 1 = 0, we take away 1 on both sides.
2x − 1 + 1 = 0 + 1 x + 1 − 1 = 0 − 1
Now we have 2x = 1 and x = −1.
2x = 1 x = −1
To remove times 2, we divide both sides by 2.
The critical points are 1/2 and −1.
x = 12 x = −1
2 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 −1 and 1/2 split the number line into 3 parts.
0
At −1 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
✗
0
At 1/2, 2x − 1 is 0.
So the fraction is 0 there.
0
Left of −1, we test x = −2.
−2
2x − 1 is −5 and x + 1 is −1.
−5
−1
Minus over minus is plus.
Between −1 and 1/2, we test x = 0.
0
2x − 1 is −1 and x + 1 is 1.
−1
1
Minus over plus is minus.
Right of 1/2, we test x = 1.
1
2x − 1 is 1 and x + 1 is 2.
1
2
Plus over plus is plus.
3 Read the answer
Greater than or equal to 0 means positive or 0.
So we want the plus parts, and the 0s.
But where the bottom is 0, the fraction has no value.
So that point is never part of the answer.
So x < −1 or x ≥ 1/2.
x < −1 or x ≥ 12
≥ 0 → the + parts, the 0
Careful!
At x = −1 the bottom is 0.
The fraction has no value there, so it is not in the answer, even with "or equal to".
Careful: not x = −1 is in ✗
4 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.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
interval: (−∞, −1) ∪ [12, ∞)
5 Check
Let's check: a number from the answer, −2, put into the question.
x = −2: 2(−2) − 1(−2) + 1 = −5−1 = 5
5 is greater than or equal to 0.
True!
5 ≥ 0 true
And a number outside the answer, 0.
x = 0: 2(0) − 10 + 1 = −11 = −1
−1 is not greater than or equal to 0.
False, as it should be: 0 is not in the answer.
−1 ≥ 0 false
Let's look back.
First: 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 = −1 is in ✗
▶ Watch it solved
The question
Question: solve (x − 1)/(x + 2) ≤ 2.
x − 1x + 2 ≤ 2
There is an x at the bottom.
The bottom can never be 0.
the bottom can never be 0
1 Get 0 on one side
Careful!
Don't times both sides by x + 2.
It can be negative, and then the sign would turn.
So we get 0 on one side instead.
Careful: never times both sides by the bottom
We take away 2 on 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.
x − 1x + 2 − 2 ≤ 2 − 2
Now we have (x − 1)/(x + 2) − 2 ≤ 0.
x − 1x + 2 − 2 ≤ 0
2 Make one fraction
To take away, both need the same bottom.
2 is 2 times (x + 2), over (x + 2).
2 = 2(x + 2)x + 2
Same bottom, so we put the tops together.
x − 1 − 2(x + 2)x + 2 ≤ 0
We open the bracket: minus 2 times each part.
x − 1 − 2x − 4x + 2 ≤ 0
Like terms together: the top is −x − 5.
Now we have (−x − 5)/(x + 2) ≤ 0.
−x − 5x + 2 ≤ 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.
The bottom counts too: there the fraction has no value.
−x − 5 = 0 x + 2 = 0
In −x − 5 = 0, we add 5 on both sides.
In x + 2 = 0, we take away 2 on both sides.
−x − 5 + 5 = 0 + 5 x + 2 − 2 = 0 − 2
Now we have −x = 5 and x = −2.
x = 5 x = −2
To remove the minus, we times both sides by −1.
The critical points are −5 and −2.
x = −5 x = −2
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 −5 and −2 split the number line into 3 parts.
0
At −5, −x − 5 is 0.
So the fraction is 0 there.
0
0
At −2 the bottom is 0.
Dividing by 0 is not allowed: no value, so we write a cross.
✗
Left of −5, we test x = −6.
−6
−x − 5 is 1 and x + 2 is −4.
1
−4
Plus over minus is minus.
Between −5 and −2, we test x = −4.
−4
−x − 5 is −1 and x + 2 is −2.
−1
−2
Minus over minus is plus.
Right of −2, we test x = 0.
0
−x − 5 is −5 and x + 2 is 2.
−5
2
Minus over plus is minus.
5 Read the answer
Less than or equal to 0 means negative or 0.
So we want the minus parts, and the 0s.
But where the bottom is 0, the fraction has no value.
So that point is never part of the answer.
So x ≤ −5 or x > −2.
x ≤ −5 or x > −2
≤ 0 → the − parts, the 0
6 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.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
interval: (−∞, −5] ∪ (−2, ∞)
7 Check
Let's check: a number from the answer, −6, put into the question.
x = −6: (−6) − 1(−6) + 2 = −7−4 = 74
7/4 is less than or equal to 2.
True!
74 ≤ 2 true
And a number outside the answer, −4.
x = −4: (−4) − 1(−4) + 2 = −5−2 = 52
5/2 is not less than or equal to 2.
False, as it should be: −4 is not in the answer.
52 ≤ 2 false
Let's look back.
First: get 0 on one side.
Then: make one fraction.
Then: find the critical points.
Then: make a sign table.
Then: read the answer.
Then: show it on a number line.
Last: check.
Answer ≤ 0 → the − parts, the 0
▶ Watch it solved
