text: I would try to pull out the inherent properties in the problem as they relate to the possible choices.  For example, in this one, we have:\n\nx < 0\n0 < y < 1\nz > 1\n\nFor (a), we are told that x^2 - z^2 is always negative.  In this case, we want to either note that x^2 - z^2 factors into (x+z)(x-z) or that x^2 = (-x)^2.  Both would suggest that we can make x^2 - z^2 ZERO by choosing any x = -z (e.g. z = 2, x = -2) so (a) is immediately shown FALSE.\n\nFor (b), we note that if z > 1, 0 < 1/z < 1 so that we can let y = 1/z.\n\nFor (c), we simply note that the product of 2 nonzero Real numbers can never be zero.\n\nFor (d), we either note that y^2 - z^2 = (y+z)(y-z) so that we have the product (positive + positive)(smaller # - bigger #) = (positive)(negative) = negative or that squaring a positive number less than 1 will only make the number smaller while squaring a positive number greater than 1 will only make the number bigger so that we get smaller # - bigger # = -#.\n\nTherefore some sort of algebraic analysis is usually the fastest way to attack these problems.  Obviously though, seeing all these tricks takes an understanding and familiarity with algebra.  However, many of the algebraic techniques you need for standardized tests is actually captured in this one problem.\n\nIt'll also help to know when to reverse inequality signs.  For this, understanding increasing/decreasing functions is critical.  For example, if you're subtracting 10 from both sides of an inequality, you look at the graph of y = x - 10 and see that it is strictly increasing (i.e. the larger the values you plug in, the larger the ones you get).  Therefore, in this case, you would not reverse the inequality sign.  If you were multiplying by -2 you would see a strictly decreasing graph and would therefore reverse the inequality sign.\n\nIf you need to guess, plugging in values such as -1, 0, and 1 where appropriate are best for counterexamples.  Also in cases with questions such as above as "which one is false", look for answers that appear to be broader in scope.  For example, (c) is pretty narrow.  All it says is that xy cannot obtain one specific value so it is likely to be true.  We can also look at the wording for clues.  (b) uses weak wording: CAN (without the never) is not very restrictive, and in fact, it can be argued on a technicality that (b) is never false.  For example, "pigs can fly" is true in an imaginative person's dreams.  (a) and (d) both use strong working, but (a) is broader.  x can assume ANY value less than 0 whereas y must be between 0 and 1, exclusively.  Lastly, look for choices that are equivalent.  If x > 0 and x^2 > 0 are both possibilities, we can eliminate both since they are equivalent conditions. and they can't both be TRUE/FALSE when we are only allowed to pick one or the other.