Simplify 4/(x - 2) + 1/x Step by Step

Step by step

The question

Question: add 4/(x − 2) + 1 over x.

4x − 2 + 1x

1 Find the common bottom

The bottoms are (x − 2) and x.

bottoms: (x − 2) and x

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

It is x(x − 2).

common bottom = x(x − 2)

Careful!

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

Fractions need the same bottom first.

Careful: not 52x − 2 ✗

2 Make the bottoms the same

The first fraction is missing x.

We multiply its top and bottom by x.

The second fraction is missing (x − 2).

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

x/x is 1, so the value stays the same.

= 4x(x − 2)x + (x − 2)x(x − 2)

3 Add the tops

Now one fraction: (4x + (x − 2)) over x(x − 2).

= 4x + (x − 2)x(x − 2)

4 Open the brackets

A bracket with a plus or nothing in front can just be removed.

( ) away

Now we have 4x + x − 2.

= 4x + x − 2x(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.

Each colour is one kind.

like terms = same letters,

same powers

4x + x is 5x.

It is like 4 apples plus 1 apple: that is 5 apples.

4x + x = 5x

So the expression is 5x − 2.

= 5x − 2x(x − 2)

6 Cross out the same

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 − 2)/x(x − 2).

The bottoms can't be 0.

So x can't be 0 or 2.

x ≠ 0, 2

7 Check

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

First in the question.

4 over −1 plus 1 is −3.

Question: 41 − 2 + 11 = 4−1 + 11 = −3

Now in our answer.

We get −3 too.

Answer: 5(1) − 21(1 − 2) = 3−1 = −3

Both are −3.

So (5x − 2)/x(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