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# 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|}

Apr 30, 2024

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Note that for any non-zero number a, the quantity $$\dfrac a{|a|}$$ can only take two values: -1 when a is negative, and 1 when is positive.

Therefore, $$\dfrac a{|a|}$$ serves as an indicator of whether a is positive. So, there are only 5 cases:

Case 1: x, y, z are all positive, making x + y + z also positive. Then $$\displaystyle {\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|}} = 4$$.

Case 2: One of x, y, z, x + y + z is negative (for example, x = -1, y = 3, z = 4). Then $$\displaystyle\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|} = 2$$.

Case 3: Two of x, y, z, x + y + z are negative (for example, x = -1, y = -2, z = 4). Then $$\displaystyle\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|} = 0$$.

Case 4: Three of x, y, z x + y + z are negative (for example, x = -7, y = -5, z = 1). Then $$\displaystyle\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|} = -2$$.

Case 5: x, y, z are all negative, making x + y + z also negative. Then $$\displaystyle\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|} = -4$$.

Therefore, the possible values of $$\displaystyle\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|}$$ are $$-4,-2,0,2,4$$.

Apr 30, 2024