Is it correct to put 'the' before uncaring?

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Yes it is perfectly correct to put the definite article before "uncaring" because "uncaring" is an adjective describing "people".

If I wrote "To all the tall people", "To all the white people", "To all the Chinese people" and so on I would expect you to have no problem with that because the structure ARTICLE+ADJECTIVE+NOUN is a very common one in English, it's like saying "I like the blue vases" or "Look at the fast cars"

I suspect that you are confused because "uncaring" looks like a verb, but it isn't. It is related to the verb 'to care' which has the present participle 'caring' and which can be used as a verb in sentences like "He is caring for his sick wife" meaning that he is looking after her but 'caring' can also be used as an adjective in sentences like "She is a very caring person" which means that she is a person with a lot of empathy and love for others.

"Uncaring" is the opposite of "caring" when used as an adjective and so it is an adjective itself. It's not a verb because there would have to be an infinitive form "to uncare" which has not yet been used as far as I know and, if it has it is very rare and I'm not quite sure what it would mean.

So "To all the uncaring people" is perfectly OK, it just says that the writer is addressing the people who are the opposite of "caring". That is those who have a lack of empathy and love for some group of other people. "The suffering people" perhaps as 'suffering' can also be used as an adjective.