Hmm. I think there’s two senses of care both the respondents and I are conflating.
Care (1) as in concern for welfare of X
Care (2) as in interested/engaged/emotionally invested in X
I often care (2) but rarely care (1). Both at once is vanishingly rare.
Quote Tweet
What’s something you care about and I probably don’t? Can you make me care in 1 tweet? 
