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When the answer is

x must be at least less than 0.5 (i.e., x<0.5).

where 0.5 is the specific answer I am looking for, what is the proper question?

x must be at least less than what value?

does not sound like a proper sentence. "What is the maximum possible value for x?" does not work because x cannot be 0.5.

I also feel "at least" is used improperly here but "x must be less than what value?" cannot work (whether grammatically or otherwise) because for this question 0.4 is a correct answer, but I want it to be wrong.

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  • You are right. at least is at best redundant, and probably confusing. Commented Jan 10, 2018 at 3:55

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x must be less than what value? is a proper, valid, question for an answer of x must be less than 0.5.

You have said that it cannot work because 0.4 would be a correct answer, but 0.4 would not be a correct answer as all values between 0.4 and 0.5 are in your range of allowed values, so x can be less than 0.4 but x could also be greater than 0.4.

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There is nothing wrong with:

x must be less than what value?

But if you're looking for alternative ways of asking the same:

  • What is the exclusive upper bound of x?
  • For what minimum value of x is x no longer valid?
  • x cannot equal or exceed what value?

But honestly I believe your version is the most concise. Sometimes when you're dealing with less than or equal to or greater than or equal to, it is easier to express yourself by describing the opposite case (in this case, rather than specify at what value of x is no longer valid and beyond, specify under which value x is valid as in your example).

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The term you want is “limit”.

[This was invented by Newton (?) when he invented [differential] calculus. He wanted the gradient of a curve; he had a system where, at the actual point in question on the curve, it was a divide-by-zero error, but he needed a magic way to get an equation that did not have a 0 for the denominator.]

The sentence would be, “What is the limit, as x approaches from below, for the function x<0.5?”. The answer to that is “simple reading-and-comprehension” — 0.5. The more complicated version is, “What is the limit, as x approaches y from below, for the function x < y, where y = 0.5?” Of course, with more complicated mathematics, the limit value would not be so obvious.

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