(1/x)+(x/x+2)=1

To answer this question you will need to break down this equation.

First of all lets try and simplify all that can be simplified.

$${\frac{{\mathtt{1}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{x}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} = {\mathtt{1}}$$

First of all we can see a fraction that can be simplified. That is ofcourse, X Over X. Any number, if the Numerator and Denominator is identical, then it will always equal to 1. So being that, we will substitute X over X with 1.

$${\frac{{\mathtt{1}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} = {\mathtt{1}}$$ 

Let us move over some numbers across the = sign.

$${\frac{{\mathtt{1}}}{{\mathtt{x}}}} = {\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,-\,}}{\mathtt{1}}$$

Lets solve that:

$${\frac{{\mathtt{1}}}{{\mathtt{x}}}} = -{\mathtt{2}}$$

Now we must eradicate the denominator. To do that we will multiply both sides by x.

$${\frac{{\mathtt{1}}}{{\mathtt{x}}}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}\right)$$

Lets solve that!

$${\frac{{\mathtt{1}}}{{\mathtt{x}}}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}\right) \Rightarrow {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}} \Rightarrow {\mathtt{x}} = -{\mathtt{0.5}}$$

We have jumped a step, however, you should be able to see what has been done.

x=-0.5

Lets substitute that in!

$${\frac{{\mathtt{1}}}{-{\mathtt{0.5}}}}{\mathtt{\,\small\textbf+\,}}{\frac{\left(-{\mathtt{0.5}}\right)}{-{\mathtt{0.5}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} = {\mathtt{1}}$$

Lets solve it now and see if it work!

$${\frac{{\mathtt{1}}}{-{\mathtt{0.5}}}} = -{\mathtt{2}}$$

$${\frac{\left(-{\mathtt{0.5}}\right)}{-{\mathtt{0.5}}}} = {\mathtt{1}}$$

$${\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} = {\mathtt{1}}$$

Is it correct?

Yes!

Therefore $${\mathtt{x}} = -{\mathtt{0.5}}$$

Good job, Takahiro !!!

Points and a Thumbs-Up !!

Hi Takahiro,

Thanks for that great answer,

Welcome to web2.0calc forum.

I have sent you a private message.

Melody.