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# Absolute values

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Let x, y, and z be nonzero real numbers. Find all possible values of x/ | x | + y/ | y | + z/ | z |. List your values in increasing order, separated by commas.

Dec 15, 2021

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If  x ,  y , and  z  are positive...

$$\frac{x}{|x|}+\frac{y}{|y|}+\frac{z}{|z|}+\frac{xyz}{|xyz|}\,=\,\frac{x}{x}+\frac{y}{y}+\frac{z}{z}+\frac{xyz}{xyz}\,=\,1+1+1+1\,=\,4$$

If only  x  is negative...

$$\frac{x}{|x|}+\frac{y}{|y|}+\frac{z}{|z|}+\frac{xyz}{|xyz|}\,=\,\frac{-|x|}{|x|}+\frac{y}{y}+\frac{z}{z}+\frac{-|x|yz}{|xyz|}\,=\,-1+1+1-1\,=\,0$$

If only  y  or only  z  is negative, the result will also be  0 .

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

$$\frac{x}{|x|}+\frac{y}{|y|}+\frac{z}{|z|}+\frac{xyz}{|xyz|}\,=\,\frac{-|x|}{|x|}+\frac{-|y|}{|y|}+\frac{z}{z}+\frac{(-|x|)(-|y|)z}{|xyz|}\,=\,-1-1+1+1\,=\,0$$

If any two are negative, the result will also be  0  .

If  x ,  y , and  z  are negative...

$$\frac{x}{|x|}+\frac{y}{|y|}+\frac{z}{|z|}+\frac{xyz}{|xyz|}\,=\,\frac{-|x|}{|x|}+\frac{-|y|}{|y|}+\frac{-|z|}{|z|}+\frac{(-|x|)(-|y|)(-|z|)}{|xyz|}\,=\,-1-1-1-1\,=\,-4$$

So, the answer is -4, 0, and 4!!

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Dec 15, 2021