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Let xy, and z be nonzero real numbers. Find all possible values of\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{xyz}{|xyz|}.List your values in increasing order, separated by commas.

Guest Dec 14, 2014

Best Answer 

 #1
avatar+17744 
+10

For any number,  n,  |n|  is positive.

So, if  n  is positive, then  n/|n|  is  +1  and if  n  is negative, then  n/|n|  is  -1.

For xyz/|xyz|:

--  if all three, x, y, and z, are negative, then  xyz  is negative, and  xyz/|xyz|  =  -1.

-- if two of them are negative, then  xyz  is positive, and  xyz/|xyz|  =  +1.

-- if one of them is negative, then  xyz  is negative, and  xyz/|xyz|  =  -1.

-- if none of them is negative, then  xyz  is positive, and  xyz/|xyz|  =  +1.

If x is negative, y is negative, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + -1 + -1 + -1  =  -4

If x is positive, y is negative, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + -1 + -1 + +1  =  0

If x is negative, y is positive, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + +1 + -1 + +1  =  0

If x is negative, y is negative, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + -1 + +1 + +1  =  -2

If x is positive, y is positive, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + +1 + -1 + -1  =  0

If x is positive, y is negative, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + -1 + +1 + -1  =  0

If x is negative, y is positive, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + +1 + +1 + -1  =  0

If x is positive, y is positive, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + +1 + +1 + +1  =  4

geno3141  Dec 14, 2014
 #1
avatar+17744 
+10
Best Answer

For any number,  n,  |n|  is positive.

So, if  n  is positive, then  n/|n|  is  +1  and if  n  is negative, then  n/|n|  is  -1.

For xyz/|xyz|:

--  if all three, x, y, and z, are negative, then  xyz  is negative, and  xyz/|xyz|  =  -1.

-- if two of them are negative, then  xyz  is positive, and  xyz/|xyz|  =  +1.

-- if one of them is negative, then  xyz  is negative, and  xyz/|xyz|  =  -1.

-- if none of them is negative, then  xyz  is positive, and  xyz/|xyz|  =  +1.

If x is negative, y is negative, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + -1 + -1 + -1  =  -4

If x is positive, y is negative, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + -1 + -1 + +1  =  0

If x is negative, y is positive, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + +1 + -1 + +1  =  0

If x is negative, y is negative, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + -1 + +1 + +1  =  -2

If x is positive, y is positive, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + +1 + -1 + -1  =  0

If x is positive, y is negative, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + -1 + +1 + -1  =  0

If x is negative, y is positive, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + +1 + +1 + -1  =  0

If x is positive, y is positive, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + +1 + +1 + +1  =  4

geno3141  Dec 14, 2014
 #2
avatar+92751 
0

Thanks Geno, that looked like a challenge  

Melody  Dec 15, 2014

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