Simplify 6/((x + 1)(x - 2)) + 2/(x + 1) Step by Step

Step by step

The question

Question: add 6/(x + 1)(x − 2) + 2/(x + 1).

6(x + 1)(x − 2) + 2x + 1

1 Find the common bottom

The bottoms are (x + 1)(x − 2) and (x + 1).

bottoms: (x + 1)(x − 2) and (x + 1)

The common bottom must have every factor of both bottoms, each one once.

It is (x + 1)(x − 2).

common bottom = (x + 1)(x − 2)

Careful!

We can't add the tops and add the bottoms.

Fractions need the same bottom first.

Careful: not 8x² − 1 ✗

2 Make the bottoms the same

The first fraction already has it.

The second fraction is missing (x − 2).

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

(x − 2)/(x − 2) is 1, so the value stays the same.

= 6(x + 1)(x − 2) + 2(x − 2)(x + 1)(x − 2)

3 Add the tops

Now one fraction: (6 + 2(x − 2)) over (x + 1)(x − 2).

= 6 + 2(x − 2)(x + 1)(x − 2)

4 Open the brackets

The 2 in front of the bracket multiplies every part inside.

a(b + c) = ab + ac

So we write 2 times each part.

= 6 + 2(x) + 2(−2)(x + 1)(x − 2)

− × + = −

Watch the signs.

2 times x is 2x.

2 times (−2) is −4.

= 6 + 2x − 4(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.

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 + 6 − 4(x + 1)(x − 2)

6 − 4 is 2.

6 − 4 = 2

So the expression is 2x + 2.

= 2x + 2(x + 1)(x − 2)

6 Cross out the same

We write the top as factors too: 2(x + 1).

= 2~~(x + 1)(x + 1)~~(x − 2)

The same bracket on top and bottom is 1.

Cross them out.

What is left is 2/(x − 2).

= 2x − 2

The bottoms can't be 0.

So x can't be −1 or 2.

x ≠ −1, 2

7 Check

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

First in the question.

6 over −2 plus 2/2 is −2.

Question: 6(1 + 1)(1 − 2) + 21 + 1

= 6−2 + 22 = −2

Now in our answer.

We get −2 too.

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

Both are −2.

So 2/(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