2017 AMC 8 Problems/Problem 21
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There are 
cases to consider:
Case 
: of 
, 
, and 
are positive and the other is negative. WLOG assume that and are positive and is negative. In this case, we have that ![\[\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=1+1-1-1=0.\]](http://latex.artofproblemsolving.com/1/1/2/1129f25a010f4ba94c660c13c8035a26721937b2.png)
Case : of , , and are negative and the other is positive. WLOG assume that and are negative and is positive. In this case, we have that ![\[\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=-1-1+1+1=0.\]](http://latex.artofproblemsolving.com/6/b/3/6b3f2333ac9e5b385d2d7c7342575cc5daa8b2ce.png)
In both cases, we get that the given expression equals 
.
~nukelauncher
