For non-zero real numbers $a, b, c$, we calculate the value of

$$
\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a b c}{|a b c|}
$$

How many different results can we obtain?
Solution. The definition of absolute value: $|x|=x$, if $x \geq 0$ and $|x|=-x$, if $x<0$. We can distinguish the following cases:

If $a, b$ and $c$ are all positive, then (1) has a value of 4, and if they are all negative, then the value is -4.

If two of $a, b$ and $c$ are negative and the third is positive, then two of the first three fractions have a value of -1, the third has a value of +1, and the fourth fraction has a value of +1, so their sum is 0. The same result is obtained if two of $a, b$ and $c$ are positive.

The expression takes on three different values, and these are: $4, -4$ and 0.