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|}
Note that for any non-zero number a, the quantity a|a| can only take two values: -1 when a is negative, and 1 when is positive.
Therefore, 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 x|x|+y|y|+z|z|+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 x|x|+y|y|+z|z|+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 x|x|+y|y|+z|z|+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 x|x|+y|y|+z|z|+x+y+z|x+y+z|=−2.
Case 5: x, y, z are all negative, making x + y + z also negative. Then x|x|+y|y|+z|z|+x+y+z|x+y+z|=−4.
Therefore, the possible values of x|x|+y|y|+z|z|+x+y+z|x+y+z| are −4,−2,0,2,4.