Prove $(1+2/x)^{ -1}= x/x+2 = x/2(1+x/2)^{ -1}$

Question

Prove $(1+2/x)^{ -1}= x/x+2 = x/2(1+x/2)^{ -1}$ I am stuck on this question I do not know how these three are related and tried working it out but got no where i will attack some of my work.And none of my collages know how to solve it Any help is much-appricated. thank you.

Answer 1

$\forall x \in \mathbb{R} \setminus \{0,-2\},$ $$\frac{1}{1+\frac{2}{x}} = \frac{1}{1+\frac{2}{x}} \times \frac{x}{x} = \frac{x}{x+2}$$ $$\frac{x}{2} \times \frac{1}{1+\frac{x}{2}} = \frac{x}{2\times(1+\frac{x}{2})} = \frac{x}{2+x}$$

Answer 2

Welcome to MSE!

Here are some hints:

$(1+2/x)^{-1} = \frac{1}{1+2/x} = \frac{1}{x/x+2/x} = \frac{1}{(x+2)/x} = \frac{x}{x+2}$ which is related to your 2nd expression (but in yours the parentheses are missing).

Here is the other equality:

$\frac{x}{2}\cdot (1+x/2)^{-1} = \frac{x}{2}\cdot \frac{1}{1+x/2} = \frac{x}{2}\cdot \frac{1}{2/2+x/2} = \frac{x}{2}\cdot \frac{1}{(x+2)/2} = \frac{x}{2}\cdot \frac{2}{x+2} = \frac{x}{x+2}$