SOLUTION: Let x, y, and z be nonzero real numbers. Find all possible values of (x)/(|x|) + (y)/(|y|) + (z)/(|z|) + (xyz)/(|xyz|). List your values in increasing order, separated by com

Question 1033415: Let x, y, and z be nonzero real numbers. Find all possible values of
(x)/(|x|) + (y)/(|y|) + (z)/(|z|) + (xyz)/(|xyz|).
List your values in increasing order, separated by commas.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52794) (Show Source): You can put this solution on YOUR website!
.
Let x, y, and z be nonzero real numbers. Find all possible values of
(x)/(|x|) + (y)/(|y|) + (z)/(|z|) + (xyz)/(|xyz|).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let F(x,y,z) = + + + . It is clear that it is sufficient to estimate/calculate F(x,y,z) at x = +/-1, y = +/-1 and z = +/-1. In total, there are 8 inputs. If there are all "+" in the input, the result is 4. If there is precisely 1 "-" in the input, then the result is 0. If there are precisely 2 "-" in the input, then the result is 0. If there are all "-" in the input, then the result is -4. Answer. The list of all possible values is 4, 0, -4.

Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!

Case 1. All three variables positive: x>0, y>0, z>0 Case 2. Two of the variables positive, one negative By symmetry, it doesn't matter which one is negative, the answer will be the same. Say, only y is negative: x>0, y<0, z>0 Case 3. One of the variables positive, two negative Also by symmetry, it doesn't matter which one is positive, the answer will be the same. Say, only x is positive x>0, y<0, z<0 Case 4. All three negative: x<0, y<0, z<0 In increasing order, separated by commas: -4,0,4 Edwin

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