How do I pre-determine mutually exclusive comparisons?

The human eye can see that no value x satisfies the condition

x<1 & x>2

but how can I make R see that. I want to use this in a function which gets passed comparisons (say as strings) and not necessarily data. Let's say I want to write a function that checks whether a combination of comparisons can ever be fulfilled anyway, like this

areTherePossibleValues <- function(someString){
    someCode
}

areTherePossibleValues("x<1 & x>2")
[1] FALSE

I mean one could do that by interpreting the substrings that are comparison signs and so on, but I feel like there's got to be a better way. The R comparison functions ('<','>','=' and so on) themselves actually might be the answer to this, right?