Minimising a complex modulus Suppose $u, v$ and $w$ are complex numbers such that $|u|=5$, $|v| = 7$ and $|w| = 19$. What is the smallest value that $|u+v+w|$ can attain? I used the inequality $|u+v+w| \ge |u| - |v| - |w|$ to attain the value $-21$ however how do I know this is the minimum possible value for the total modulus and how would I justify this?

The inequalities $$ |u+v+w| \ge |u| - |v| - |w| = 5 - 7 - 19 = -21 \\\ |u+v+w| \ge |v| - |u| - |w| = 7 - 5 - 19 = -17 $$ are both true. So $-21$ and $-17$ are both _lower bounds_ for $|u+v+w|$, but that does not give anything useful because $|u+v+w| \ge 0$ holds in general. So these lower bounds cannot be the _minimum_ (the value is not attained).

You need to utilize the fact that $|w|$ is larger than $|u| + |v|$ and put $w$ in front: $$ |u+v+w| = |w+(u+v)| \ge |w| - |u + v| \ge |w| - (|u| + |v|) = |w| - |u| - |v| = 19 - 5 - 7 = 7 $$ Now we have a _non-negative_ estimate for $|u+v+w|$. That value is attained for $u = -5, v = -7, w = 19$, so it is indeed the minimum value.