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

Step by step

The question

Question: simplify this fraction.

On top: 1 + 1 over x.

At the bottom: 1 − 1 over x.

A fraction with fractions inside is called a complex fraction.

1 + 1x1 − 1x

1 Find the common bottom

The small fraction has the bottom x.

The smallest thing they all go into is x.

small bottoms: x

common bottom: x

Careful!

We multiply EVERY part by x, also the part with no small bottom.

Leaving it out is the usual mistake.

Careful: every part times x, the whole numbers too

2 Times x on top and bottom

x/x is 1.

Times 1 does not change the value.

xx = 1

We multiply the top and the bottom by x.

Then the small fractions go away.

= x × (1 + 1x)x × (1 − 1x)

3 Multiply every part

Every part times x.

x times 1 is x.

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

x × 1 = x

x × 1x = 1

x × 1 = x

x × (−1x) = −1

So we get (x + 1)/(x − 1).

No more small fractions!

= x + 1x − 1

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 + 1)/(x − 1).

nothing goes into every part

One more thing: no bottom can be 0.

So x can't be 0 or 1.

x ≠ 0, 1

5 Check

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

In the question, the long way: the top is 3/2, the bottom is 1/2.

x = 2: top 1 + 12 = 32, bottom 1 − 12 = 12

Top divided by bottom: 3.

question: 32 ÷ 12 = 3

Our answer with x = 2 gives 3 too.

It works!

answer: 2 + 12 − 1 = 3

Let's look back.

First: find the common bottom.

Then: times x on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer