Below the age required by law to do something, such as drink alcohol or vote.
From 'under-' meaning below or less than, plus 'age.' It's a compound word created in English when legal age requirements became important in modern society, probably gaining prominence in the early 20th century as drinking and voting laws were codified.
The legal definition of 'underage' is actually different in every country and for different activities—you can be old enough to drive but too young to drink, or old enough to work but too young to vote. This shows that 'age' isn't one real thing; society just draws lines wherever it decides it wants them.
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