[INST] Let x and y be nonzero real numbers. Let m and M be the minimium value and maximum value of

 (|x + y|)/(|x| + |y|),
respectively. Find M - m./n [/INST]  ###Response: We're given two nonzero real numbers x and y, and asked to find the minimum m and maximum M of the expression:

(|x + y|)/(|x| + |y|)

To find the minimum, we want to make the expression as small as possible. Since both |x + y| and |x| + |y| are nonnegative, (|x + y|)/(|x| + |y|) must also be nonnegative. We can set x = 1 and y = -1 to get:

(|x + y|)/(|x| + |y|) = (|0|)/(|1| + |-1|) = 0/2 = 0

This shows that the minimum value of the expression is 0.

To find the maximum, we want to make the expression as large as possible. By the Triangle Inequality, we know that:

|x + y| ≤ |x| + |y|

So we have:

(|x + y|)/(|x| + |y|) ≤ (|x| + |y|)/(|x| + |y|) = 1

This shows that the maximum value of the expression is 1. Equality occurs when x = y.

Therefore, the range of the expression is [0, 1], and the difference between the maximum and minimum values is:

M - m = 1 - 0 = 1