The phrase "careful about whether" is correct and commonly used in written English.
It is typically used to express caution or concern about a certain situation or decision. Example: "I want to make sure we are careful about whether we include all the necessary information in our report."
"Universities will have to be very careful about whether the way it is used.
That means we have to be extremely careful about whether things are truly anonymous, and not rely on our intuition alone".
Horizontal transfers do not necessarily destroy an indel argument, we just need to be careful about whether they actually affect the conclusion or not.
Be careful about whether he can see that you've checked your text/message.
A victory by JPMorgan's clients may mean that banks will have to be more careful about deciding whether to share — or silo — information that affects their clients' investments.
Not every punter understands that the bet works this way and even those who do are not always careful about checking whether or not their horses actually took part.
Be careful about handling them, whether they seem particularly dangerous or not.
Use it as a source of inspiration, but make your own careful choices about whether to believe what it says about the science of creativity.
Dispersants are detergents and at best are mildly toxic, so applying them requires a careful calculation about whether the dispersant-oil mixture will cause more or fewer problems than untreated crude oil would.
If the targeted students fail to reach their expected A-levels, and would otherwise fail to be admitted, the college has careful discussions about whether to go ahead and admit them anyway.
Roffe also warns against an assumption that boarding provision is right for all children, especially those from vulnerable backgrounds: "There is provision to fund free places for children in the state boarding sector but we make a very careful judgment about whether a child is suitable to board, will benefit and whether there are any safeguarding issues.
