### Instruction: Let $x$ and $y$ be nonzero real numbers. Let $m$ and $M$ be the minimium value and maximum value of
\[\frac{|x + y|}{|x| + |y|},\]respectively.  Find $M - m.$.### Response: I notice that the expression is always between 0 and 1, because the numerator is always less than or equal to the denominator.
So 0 and 1 are possible lower and upper bounds for $m$ and $M$.
I wonder if I can make the expression equal to 0 or 1 by choosing some special values of $x$ and $y$.
If I can, then those would be the exact values of $m$ and $M$.
I try to make the numerator zero.
This happens when $x + y = 0$, or $x = -y$.
In this case, the expression becomes
\[\frac{0}{|x| + |y|} = 0.\]
So $m = 0$.
I try to make the numerator equal to the denominator.
This happens when $|x + y| = |x| + |y|$.
This is equivalent to saying that $x$ and $y$ have the same sign.
In this case, the expression becomes
\[\frac{|x + y|}{|x| + |y|} = \frac{|x| + |y|}{|x| + |y|} = 1.\]
So $M = 1$.
I conclude that the minimum and maximum values of the expression are 0 and 1, respectively.