adverb
1
BAD : I was very alone at first but then I made some friends.
GOOD : I was very lonely at first but then I made some friends.
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alone = without other people around you: 'I've thought about getting married, but I prefer living alone.'
lonely = sad because you are alone and feel that nobody loves you or cares about you: 'I didn't know anyone in Boston and felt very lonely.' 'Sarah hated the long lonely days in the empty house.'
2
BAD : Children learn a lot by doing things alone.
GOOD : Children learn a lot by doing things on their own.
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on your own = without anyone's help or supervision; independently: 'He built the car all on his own.'