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

Step by step

The question

Question: add 2/(x + 1) + 3/(x − 2).

2x + 1 + 3x − 2

1 Find the common bottom

The bottoms are (x + 1) and (x − 2).

bottoms: (x + 1) and (x − 2)

To add 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 add the tops and add the bottoms.

That would give 5/(2x − 1): wrong.

Fractions need the same bottom first.

Careful: not 52x − 1 ✗

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.

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

3 Add the tops

Both bottoms are now (x + 1)(x − 2).

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

= 2(x − 2) + 3(x + 1)(x + 1)(x − 2)

4 Open the brackets

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.

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

Plus times minus is minus.

+ × − = −

2 times x is 2x.

2 times (−2) is −4.

3 times x is 3x.

3 times 1 is 3.

= 2x − 4 + 3x + 3(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.

= 2x + 3x − 4 + 3(x + 1)(x − 2)

2x + 3x is 5x.

It is like 2 apples plus 3 apples: that is 5 apples.

−4 + 3 is −1.

2x + 3x = 5x

−4 + 3 = −1

So the expression is 5x − 1.

= 5x − 1(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 (5x − 1)/(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 −1 or 2.

x ≠ −1, 2

7 Check

Let's check with a number: x = 1.

We pick a small number that makes no bottom 0.

First in the question.

2/2 plus 3 over −1.

We get −2.

Question: 21 + 1 + 31 − 2 = 22 + 3−1 = −2

Now in our answer.

We get −2 too.

Answer: 5(1) − 1(1 + 1)(1 − 2) = 4−2 = −2

Both are −2.

So (5x − 1)/(x + 1)(x − 2) is right!

Let's look back.

First: find the common bottom.

Then: make the bottoms the same.

Then: add the tops.

Then: open the brackets.

Then: group like terms.

Then: cross out the same.

Last: check.

Answer