ashish1354 wrote:1) is not sufficient since we do not know value of x
2) is not sufficient as it does not tell value of m & n
problem is combining 1) and 2)
Q) should we pick numbers for m n and x plug in and figure out (it takes too long to solve the problem) or is there a faster way ??
ok, so the first thing you have to know here - and that you
should know, for all future inequality problems - is that
there's really no such thing as "cross multiply".
that's right - there is no magic operation that allows you to multiply across an equals sign. in fact, what you know as "cross multiply" is really just
multiplication by both denominators.
try it for yourself:
take a/b = c/d and multiply both sides by bd (the product of the denominators). you'll get ad = bc, the very result of so-called "cross multiplication".
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therefore, we have the following result:
you cannot "cross multiply" an INEQUALITY unless you know the sign of the product of the denominators.
in the equation above, a/b = c/d, this means you can't cross multiply unless you know the sign of the product bd.
notice that this doesn't mean you have to have the signs of b and d themselves; all you have to know is whether those signs are the same (which would make bd positive) or opposite (which would make bd negative). if the latter is the case, then, when you cross multiply, you'll have to reverse the sign of the inequality, just as with any other multiplication/division by a negative number.
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as long as statement (2) is true - either by itself or in concert with statement (1) - m, n, and x are all positive, and therefore both denominators are positive.
this means that, as long as statement (2) is true, you can "cross multiply" the inequality to give "is mn + nx > mn + mx?", from which mn can be subtracted to give "is nx > mx?"
once you have that, you can divide both sides by x (which is known to be positive) to give the final rephrase: "is n > m?"
statement (2) isn't sufficient to answer this question; statement (1), which does indeed declare that n > m, isn't good enough by itself because you can't cross multiply the inequality if you only have that statement.
if you have both statements together, then, as described, the question can be rephrased as "is n > m?", and statement (1) tells you that n > m. that's sufficient.
