Going off of this question:  How can we prove that $max(|a|,|b|)\geq \frac{1}{2}(|a+b|)$?
The link here:  shows that $\max(a,b)=\frac{a+b+|a-b|}{2}$.      Is it true that $\max(|a|,|b|)=\frac{1}{2}(|a|+|b|)$?
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