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Recently I studied triangle inequality and the proof using textbook precalculus by David Cohen.

My question is whats the benefit of this inequality ? One benefit I found is to solve inequality of the form |x+a| + |x+b| < c which make the solution much easier than taking cases. I assume this inequality can be used in proof? the beauty of this inequality is to separate absolute of sum to sum of absolutes which - supposedly - will make proving (whatever the proof is) much easier.

Are there any other benefits ?

Are there any important inequality other triangle and AM-GM inequality that quite famous ?

Thanks.