+0  
 
0
6
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avatar+1071 

Let $x$, $y$, and $z$ be nonzero real numbers. Find all possible values of
\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|}

 Mar 27, 2024

Best Answer 

 #1
avatar+394 
+2

for \(\frac{a}{|a|}\) we have 1 if a>0, -1, if a <0. This is easy to see because | | either return the positive version of the number or the negative version of the number.

If all three of x, y, z are positive, then xyz>0, then all four terms are one, producing 4.

If 2 of x, y, z are positive, then xyz<0 then there are two ones and two negative ones resulting in 0.

If one of x, y, z is positive, then xyz>0, and there are two ones and two negative ones, resulting in 0.

If x, y, z are all negative, then xyz<0, and we get all four terms as negative one resulting in -4.

So, all possible values are -4, 0, 4.

 Mar 27, 2024
 #1
avatar+394 
+2
Best Answer

for \(\frac{a}{|a|}\) we have 1 if a>0, -1, if a <0. This is easy to see because | | either return the positive version of the number or the negative version of the number.

If all three of x, y, z are positive, then xyz>0, then all four terms are one, producing 4.

If 2 of x, y, z are positive, then xyz<0 then there are two ones and two negative ones resulting in 0.

If one of x, y, z is positive, then xyz>0, and there are two ones and two negative ones, resulting in 0.

If x, y, z are all negative, then xyz<0, and we get all four terms as negative one resulting in -4.

So, all possible values are -4, 0, 4.

hairyberry Mar 27, 2024

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