Write 3/(x − 1) − 2/(x + 2) as a single fraction. Step by Step

Step by step

The question

Question: work out 3/(x − 1) − 2/(x + 2).

3x − 1 − 2x + 2

1 Find the common bottom

The bottoms are (x − 1) and (x + 2).

bottoms: (x − 1) and (x + 2)

To take away fractions, we need one bottom that both bottoms go into.

(x − 1) is in a bottom, so we take it.

(x + 2) is in a bottom, so we take it.

We multiply them together.

So the common bottom is (x − 1)(x + 2).

common bottom = (x − 1)(x + 2)

Each bottom goes into it.

(x − 1) times (x + 2) is (x − 1)(x + 2).

(x + 2) times (x − 1) is (x − 1)(x + 2).

(x − 1) × (x + 2) = (x − 1)(x + 2)

(x + 2) × (x − 1) = (x − 1)(x + 2)

Careful!

We can't take away the tops and take away the bottoms.

That would give 1 over −3: wrong.

Fractions need the same bottom first.

Careful: not 1−3 ✗

2 Make the bottoms the same

The first fraction is missing (x + 2): (x − 1) times (x + 2) is (x − 1)(x + 2).

We multiply its top and bottom by (x + 2).

The second fraction is missing (x − 1): (x + 2) times (x − 1) is (x − 1)(x + 2).

We multiply its top and bottom by (x − 1).

(x + 2)/(x + 2) and (x − 1)/(x − 1) are 1, so the value stays the same.

= 3(x + 2)(x − 1)(x + 2) − 2(x − 1)(x + 2)(x − 1)

3 Take away the tops

Both bottoms are now (x − 1)(x + 2).

Now one fraction: (3(x + 2) − 2(x − 1)) over (x − 1)(x + 2).

= 3(x + 2) − 2(x − 1)(x − 1)(x + 2)

4 Open the brackets

The minus sign in front of the 2 goes with it.

So we multiply by −2.

What is in front of a bracket multiplies every part inside.

a(b + c) = ab + ac

So we write each part times what is in front.

= 3(x) + 3(2) + (−2)(x) + (−2)(−1)(x − 1)(x + 2)

Minus times minus is plus.

− × − = +

3 times x is 3x.

3 times 2 is 6.

(−2) times x is −2x.

(−2) times (−1) is 2.

= 3x + 6 − 2x + 2(x − 1)(x + 2)

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

Plain numbers go together.

Each colour is one kind.

like terms = same letters,

same powers

We move like terms next to each other.

Each term takes its sign with it.

= 3x − 2x + 6 + 2(x − 1)(x + 2)

3x − 2x is x.

It is like 3 apples take away 2 apples: that is 1 apple.

1x is just written x.

6 + 2 is 8.

3x − 2x = x

6 + 2 = 8

So the expression is x + 8.

= x + 8(x − 1)(x + 2)

6 Cross out the same

A factor is a part that multiplies.

Is a whole factor on top the same as one on the bottom?

No.

So nothing crosses out.

We leave the bottom as factors.

So the answer is (x + 8)/(x − 1)(x + 2).

The bottoms can't be 0: we can't divide by 0.

x − 1 is 0 when x is 1.

x + 2 is 0 when x is −2.

So x can't be −2 or 1.

x ≠ −2, 1

7 Check

Let's check with a number: x = 2.

We pick a small number that makes no bottom 0.

First in the question.

3 take away 2/4.

We get 5/2.

Question: 32 − 1 − 22 + 2 = 31 − 24 = 52

Now in our answer.

We get 5/2 too.

Answer: 2 + 8(2 − 1)(2 + 2) = 104 = 52

Both are 5/2.

So (x + 8)/(x − 1)(x + 2) is right!

Let's look back.

First: find the common bottom.

Then: make the bottoms the same.

Then: take away the tops.

Then: open the brackets.

Then: group like terms.

Then: cross out the same.

Last: check.

Answer