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. 

In 2D space, consider a rectangular container filled with $1$ unit of incompressible fluid. If its width is $z$, then its height is $1/z$, and its perimeter is $2 + 2/z$. Suppose the walls of the container are elastic, so that it tries to minimize its perimeter, but it is still constrained to take the shape of a rectangle.
Imagine that the rectangle starts with dimensions $0.2 \times 5$. What will it do? It will snap inwards along its longer dimension, bulging outwards along its shorter dimension, until it settles on a $1\times 1$ square. That square attains the minimum perimeter, which is $4$, representing the inequality $2 + 2/z\geq4$.
Can you visualize the rectangle going KSSSschlllurp! and finding that minimum? Hopefully you can, because I'm too lazy to produce the right computer animation. :)
Bonus: Once you have that animation, you can also overlay the plots described in the other answers! In particular, you can show the hyperbola along which a corner of the rectangle travels, and you can show the gradient of the potential $x+y$. This connects the visualization to a more algebraic approach.
