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

Step by step

The question

Question: simplify this fraction.

On top: 1/x + 1/2.

At the bottom: 1/x − 1/2.

A fraction with fractions inside is called a complex fraction.

1x + 121x − 12

1 Find the common bottom

The small fractions have bottoms x and 2.

The smallest thing they all go into is 2x.

small bottoms: x, 2

common bottom: 2x

2 Times 2x on top and bottom

2x/2x is 1.

Times 1 does not change the value.

2x2x = 1

We multiply the top and the bottom by 2x.

Then the small fractions go away.

= 2x × (1x + 12)2x × (1x − 12)

3 Multiply every part

Every part times 2x.

2x times 1/x: the bottom x goes into 2x, so we get 2.

2x times 1/2: the bottom 2 goes into 2x, so we get x.

2x × 1x = 2

2x × 12 = x

2x × 1x = 2

2x × (−12) = −x

So we get (2 + x)/(2 − x).

No more small fractions!

= 2 + x2 − 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 (2 + x)/(2 − x).

nothing goes into every part

One more thing: no bottom can be 0.

So x can't be 0 or 2.

x ≠ 0, 2

5 Check

Let's check with a number: x = 3.

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

x = 3: top 13 + 12 = 56, bottom 13 − 12 = −16

Top divided by bottom: −5.

question: 56 ÷ (−16) = −5

Our answer with x = 3 gives −5 too.

It works!

answer: 2 + 32 − 3 = −5

Let's look back.

First: find the common bottom.

Then: times 2x on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer