Bunuel wrote:

If ∼(x) represents the absolute value of the difference between the value of x and the value of x rounded to the nearest integer, which of the following has the greatest value?

A. ~(√2)

B. ∼(π)

C. ∼(−19/7)

D. ∼(√80)

E. ∼(−6.1)

Take some time to translate this one before you start trying to work with the answer choices. A good way to translate these 'strange function' problems is to try putting some simple numbers of your own into them, and see what happens.

For instance, let's try 1.5. 'The absolute value of the difference between the value of x and the value of x rounded to the nearest integer'... Well, I don't know what the 'nearest integer' is here, because 1.5 is halfway between 1 and 2. Let's try 1.6 instead.

The difference between 1.6 and 2 is 0.4.

If I tried 1.9, the difference between 1.9 and 2 is 0.1.

And if I tried 1.99... the difference is 0.01.

It looks like what this function does, is it measures 'how close to an integer' a number is. If a number is very close to an integer (like if it's 3.000001) then the value of this function will be very small. If the number is almost halfway between integers (like if it's 2.49999) then the value of the function will be large. So, we need to find the answer choice that's furthest away from an integer, and eliminate answer choices that are close to an integer.

A. ~(√2) = 1.4ish. That's pretty far from an integer; keep it.

B. ∼(π) = 3.1ish. That's closer than (A) was. Eliminate.

C. ∼(−19/7) = the nearest integer will be -21/7, so this is 2/7 away from an integer. Is 2/7 a smaller distance than 0.4? Yes, it's only 0.29. Eliminate.

D. ∼(√80) = very close to the square root of 81, so it'll be very close to an integer. Eliminate.

E. ∼(−6.1) = only 0.1 away from an integer. Eliminate.

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