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

Step by step

The question

Question: simplify this fraction.

On top: 1/2 − 1 over x.

At the bottom: x/2.

A fraction with fractions inside is called a complex fraction.

12 − 1xx2

1 Find the common bottom

The small fractions have bottoms 2 and x.

The smallest thing they all go into is 2x.

small bottoms: 2, x

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 × (12 − 1x)2x × (x2)

3 Multiply every part

Every part times 2x.

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

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

2x × 12 = x

2x × (−1x) = −2

2x × x2 = x²

So we get (x − 2)/x².

No more small fractions!

= x − 2x²

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 (x − 2)/x².

nothing goes into every part

One more thing: no bottom can be 0.

So x can't be 0.

x ≠ 0

5 Check

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

In the question, the long way: the top is 0, the bottom is 1.

x = 2: top 12 − 12 = 0, bottom 1 = 1

Top divided by bottom: 0.

question: 0 ÷ 1 = 0

Our answer with x = 2 gives 0 too.

It works!

answer: 2 − 22² = 0

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