Visualize $z+\frac{1}{z} \ge 2$ As we know, always
$$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$
However, is there any geometric way to visualize this equation for some one who is not that expert in math?
I know this question might have various answers. 

Well, what you wrote is clearly not true, it is only true for $x\in (0,\infty)$.
You could just try plotting the function $x\mapsto x+\frac1x$ and see that it is always above $2$.
http://www.wolframalpha.com/input/?i=plot+z+%2B+1%2Fz
Or, you can plot $x\mapsto x$ and $x\mapsto \frac1x$ and try to understand what is happening on $(0,\infty)$. It's clear you only need to look at the interval $(\frac12, 2)$, since the equality obviously holds outside it.
https://www.wolframalpha.com/input/?i=plot+x+and+1%2Fx+from+x%3D1%2F2+to+2
Now, on this graph, I think it's possible to explain what is happening:


*

*It is clear that for $x=1$, the sum of the two function is precisely $1$.

*If you go to the right of the point $x=1$, then the bottom function $\frac1x$ is decreasing more slowly than the top function $x$ is increasing, so, their sum will be above $2$, because it will be $$(1 + \text{something}) + (1 - \text{something smaller}) = 2 + (\text{something} - \text{something smaller})$$ so it will be $2$ plus something positive.

*If you go to the left of the point $x=1$, then the bottom function $x$ is decreasing more slowly than the top function $\frac1x$ is increasing, so, their sum will be above $2$, because it will be $$(1 + \text{something}) + (1 - \text{something smaller}) = 2 + (\text{something} - \text{something smaller})$$
so it will be $2$ plus something positive. 

