Range of function Involving Modulus Quantity. 
If $x,y,z\in \mathbb{R}\;,$ Then Range of $$\frac{|x+y|}{|x|+|y|}+\frac{|y+z|}{|y|+|z|}+\frac{|z+x|}{|z|+|x|}\,$$ 
  $\bf{My\; Try::}$ Here $x,y,z$ Not all Zero Simultaneously.

Now Using $\bf{\triangle\;  Inequality}\;,$ We get $|x+y|\leq |x|+|y|$
Similarly $|y+z|\leq |y|+|z|$ and $|z+x|\leq |z|+|x|$
So we get $$\frac{|x+y|}{|x|+|y|}+\frac{|y+z|}{|y|+|z|}+\frac{|z+x|}{|z|+|x|}\leq 3$$
Now I did not understand How can I calculate $\min$ of that expression, Help Required
Thanks 
 A: No two of the numbers may vanish simultaneously, otherwise one of the terms becomes $\frac{0}{0}$.
If all three numbers have the same sign, then the expression becomes $1+1+1 = 3$, so the upper bound you found is attained.
If one of the numbers - by symmetry let's say $x$ - is $0$, you have
$$\frac{\lvert 0+y\rvert}{0 + \lvert y\rvert} + \frac{\lvert y+z\rvert}{\lvert y\rvert + \lvert z\rvert} + \frac{\lvert z+0\rvert}{\lvert z\rvert + 0} = 2 + \frac{\lvert y+z\rvert}{\lvert y\rvert + \lvert z\rvert} \geqslant 2.$$
So let's look at all numbers nonzero, but not all having the same sign. Since the expression is invariant under multiplication with $-1$, and by symmetry, we can assume that $x < 0 < y \leqslant z$. Then $\lvert y+z\rvert = y + z = \lvert y\rvert + \lvert z\rvert$, whence
$$\frac{\lvert x+y\rvert}{\lvert x\rvert + \lvert y\rvert} + \frac{\lvert y+z\rvert}{\lvert y\rvert + \lvert z\rvert} + \frac{\lvert z+x\rvert}{\lvert z\rvert + \lvert x\rvert} \geqslant \frac{\lvert y+z\rvert}{\lvert y\rvert + \lvert z\rvert} = 1.$$
Choosing $y = z = 1$ and $x = -1$ attains that lower bound. So the range is contained in $[1,3]$. By connectedness of the domain and continuity, the range is exactly $[1,3]$.
A: We need not just that $x$, $y$ and $z$ are not all zero simultaneously, but that no two of them are zero; otherwise the expression is not defined.
At least two variables have the same sign, so at least one term is $1$. The others are non-negative, so the minimum is at least $1$. Since $1$ is attained (e.g. for $1,1,-1$), this is the minimum.
