I was thinking about using something like this If $( c > 0 )$, $ ( |a + b| < c ) $ implies $( -c < a + b < c )$.
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1Just consider the possible cases for the signs of $a$ and $b$. Both directions of the iff can be proved this way. – Brevan Ellefsen Aug 28 '23 at 04:32
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4Does this answer your question? Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero - found using an Approach0 search. Note there are also other duplicates, e.g., ... – John Omielan Aug 28 '23 at 04:35
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2(cont.) if $a,b\in R$ and $b\neq 0$, show that $\lvert a + b\rvert = \lvert a\rvert+\lvert b\rvert$ if an only if $ab\geq 0$, and several quite similar questions, e.g., Why does solution to $|a| + |b| = |a+b|$, where $a$ and $b$ are linear equations, satisfy $ab\ge 0$, Can we extend the idea that if $|a+b|=|a|+|b|$, then $ab \geq 0$?, etc. – John Omielan Aug 28 '23 at 04:37