Worked examples
The question
Question: simplify (x + 3)/x(x + 3).
A fraction with x in it is called a rational expression.
x + 3x(x + 3)
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
We do the same with brackets.
69 = 2 × 33 × 3 = 23
1 Factor the bottom
We look at the bottom: x(x + 3).
= x(x + 3)
It is already factored: x(x + 3) means x times (x + 3).
We just read the factors.
x(x + 3) = x × (x + 3)
So the bottom is x(x + 3).
= x(x + 3)
2 Cross out the same
Careful!
We cross out only a whole bracket, or a whole x, that multiplies everything.
Never one part of a bracket.
Careful: only whole brackets cross out
Now we write the fraction with the factors: (x + 3)/x(x + 3).
= (x + 3)x(x + 3)
(x + 3)/(x + 3) is 1.
Cross them out.
3 Write what is left
What is left is 1 over x.
= 1x
One more thing: the bottom of the question can't be 0.
So x can't be −3 or 0.
x ≠ −3, 0
4 Check
Let's check with a number: x = 2.
First in the question.
We get 1/2.
Question: 2 + 32(2 + 3) = 510 = 12
Now in our answer.
We get 1/2 too.
Answer: 12
Both are 1/2.
So 1/x is right!
Let's look back.
First: factor the bottom.
Then: cross out the same.
Then: write what is left.
Last: check.
▶ Watch it solved
The question
Question: simplify 2x(x + 1)/4x².
A fraction with x in it is called a rational expression.
2x(x + 1)4x2
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
We do the same with brackets.
69 = 2 × 33 × 3 = 23
1 Factor the top
We look at the top: 2x(x + 1).
= 2x(x + 1)
It is already factored: 2x(x + 1) means 2x times (x + 1).
We just read the factors.
2x(x + 1) = 2x × (x + 1)
So the top is 2x(x + 1).
= 2x(x + 1)
2 Cross out the same
Careful!
We cross out only a whole bracket, or a whole x, that multiplies everything.
Never one part of a bracket.
Careful: only whole brackets cross out
Now we write the fraction with the factors: 2x(x + 1)/4xx.
= 2x(x + 1)4xx
x/x is 1.
Cross them out.
2 and 4 can both be divided by 2.
2 becomes 1, 4 becomes 2.
3 Write what is left
What is left is (x + 1)/2x.
= x + 12x
One more thing: the bottom of the question can't be 0.
So x can't be 0.
x ≠ 0
4 Check
Let's check with a number: x = 2.
First in the question.
We get 3/4.
Question: 2(2)(2 + 1)4(2)2 = 1216 = 34
Now in our answer.
We get 3/4 too.
Answer: 2 + 14 = 34
Both are 3/4.
So (x + 1)/2x is right!
Let's look back.
First: factor the top.
Then: cross out the same.
Then: write what is left.
Last: check.
▶ Watch it solved
The question
Question: simplify (x² − 9)/x(x + 3) × x/(x − 3).
x2 − 9x(x + 3) × xx − 3
1 Factor every part
First the top of the first fraction: x² − 9.
= x2 − 9
x² − 9 is x² minus 3 squared.
A square minus a square is the plus bracket times the minus bracket.
a² − b² = (a + b)(a − b)
So the top of the first fraction is (x + 3)(x − 3).
= (x + 3)(x − 3)
Now the bottom of the first fraction: x(x + 3).
= x(x + 3)
It is already factored: x(x + 3) means x times (x + 3).
We just read the factors.
x(x + 3) = x × (x + 3)
So the bottom of the first fraction is x(x + 3).
= x(x + 3)
2 Cross out the same
Careful!
We cross out only a whole bracket, or a whole x, that multiplies everything.
Never one part of a bracket.
Careful: only whole brackets cross out
Now every part is factored.
A bracket on any top and the same bracket on any bottom make 1.
= (x + 3)(x − 3)x(x + 3) × x(x − 3)
(x + 3)/(x + 3) is 1.
Cross them out.
(x − 3)/(x − 3) is 1.
Cross them out.
x/x is 1.
Cross them out.
3 Multiply what is left
Top times top, bottom times bottom.
What is left is 1.
= 1
One more thing: the bottom of the question can't be 0.
So x can't be −3, 0 or 3.
x ≠ −3, 0, 3
4 Check
Let's check with a number: x = 2.
First in the question.
We get 1.
Question: 22 − 92(2 + 3) × 22 − 3 = −510 × 2−1 = 1
Now in our answer.
We get 1 too.
Answer: 1
Both are 1.
So 1 is right!
Let's look back.
First: factor every part.
Then: cross out the same.
Then: multiply what is left.
Last: check.
▶ Watch it solved
