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# help algebra

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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

#1
+398
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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
+398
+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.

hairyberry Mar 27, 2024