Simplify (1/x + 1/3)/(1/x - 1/3) Step by Step

Step by step

The question

Question: simplify this fraction.

On top: 1/x + 1/3.

At the bottom: 1/x − 1/3.

A fraction with fractions inside is called a complex fraction.

1x + 131x − 13

1 Find the common bottom

The small fractions have bottoms x and 3.

The smallest thing they all go into is 3x.

small bottoms: x, 3

common bottom: 3x

2 Times 3x on top and bottom

3x/3x is 1.

Times 1 does not change the value.

3x3x = 1

We multiply the top and the bottom by 3x.

Then the small fractions go away.

= 3x × (1x + 13)3x × (1x − 13)

3 Multiply every part

Every part times 3x.

3x times 1/x: the bottom x goes into 3x, so we get 3.

3x times 1/3: the bottom 3 goes into 3x, so we get x.

3x × 1x = 3

3x × 13 = x

3x × 1x = 3

3x × (−13) = −x

So we get (3 + x)/(3 − x).

No more small fractions!

= 3 + x3 − x

4 Simplify

Is there a number or an x that goes into every part, top and bottom?

No.

So it is already simple.

So the answer is (3 + x)/(3 − x).

nothing goes into every part

One more thing: no bottom can be 0.

So x can't be 0 or 3.

x ≠ 0, 3

5 Check

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

In the question, the long way: the top is 5/6, the bottom is 1/6.

x = 2: top 12 + 13 = 56, bottom 12 − 13 = 16

Top divided by bottom: 5.

question: 56 ÷ 16 = 5

Our answer with x = 2 gives 5 too.

It works!

answer: 3 + 23 − 2 = 5

Let's look back.

First: find the common bottom.

Then: times 3x on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer