In a manner that shows failure to do what is required; in a way that breaks rules or fails to meet obligations.
From 'delinquent' (adjective) plus the suffix '-ly,' which converts adjectives into adverbs showing manner or condition. This is the standard English adverb formation pattern.
Saying someone acts 'delinquently' sounds formal and official—it's the word you'd hear in a court case or legal document rather than just saying 'irresponsibly,' showing how language gets more serious when laws are involved.
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