-지 그래요? — how about, can we

This expression is used to suggest that someone do something.

It is a much weaker and softer expression than -도록 하다.

Details

1. When stating a reason in the preceding clause, -(으)니까 is used, not -아/어서

더워서 창문을 열지 그래요? ✕

2. The past tense form 지 그랬어요? is used to express one's disapproval or regret about the fact that another person did not do something despite it having been the preferable course of action.

아픈데 참으면서 계속 일을 한 거예요? 많이 아프면 좀 쉬지 그랬어요?

Did you keep working even though you were in pain? If you were in that much pain, why didn't you get
some rest?

This sentence expresses the speaker's feeling of pity about why another person did not take a break despite being in a lot of pain.

Kimchi Reader

Used as -지 그래, -지 그래요, -지 그러세요 depending on the level of speech to suggest that the listener do something. -지 그래 and -지 그래요 are less direct / more gentle compared to other kinds of suggestions in 반말 and 존댓말 respectively, but -지 그러세요 is probably on the more direct side and should not be used to those above you.