jmc

$\lfloor |-4.2|\rfloor =\lfloor 4.2\rfloor =4$ because the greatest integer less than $4.2$ is $4$. $|\lfloor -4.2\rfloor |=|-5|=5$ because the greatest integer less than $-4.2$ is $-5$. Therefore, the answer is $4+5=\boxed{9}.$