there are some signals you should learn to recognize IMMEDIATELY in these problems.
number plugging is a nice backup method, but some of the rephrases i'm going to mention are canonical; you MUST learn to recognize them.
ideally, you should reserve number plugging for problems that are at least a bit novel or unusual. for a problem such as the first one below, for instance, number plugging means a failure to recognize things that you're supposed to be able to recognize.
this doesn't mean, of course, that you
shouldn't use number plugging if you don't know how to interpret the problem; you should solve problems in whatever way you can conjure. however, to repeat, if you have to use number plugging on a problem like #1, you really need to study inequalities, positive/negative number properties, and rephrasing some more.
sabal wrote:Question 1. DS
If mv < pv <0, is v>0?
1. m<p
2. m<0
The answer is
D
when you see a statement such as "mv < pv < 0", you should IMMEDIATELY start thinking of CASES.
this statement provides a goldmine of information about the signs of m, p, and v. you should rephrase it,
on the spot, into a statement about the signs of the individual numbers. if you do, you'll often find that the statements themselves become very easy to interpret.
here's how to handle this one:
* recognize that pv < 0. this means that
p and v have opposite signs.
* recognize that mv < 0. this means that
m and v have opposite signs.
* based on these observations, there are only
2 possible cases:
(m) (p) (v)
(+) (+) (-) case 1
(-) (-) (+) case 2
* use the statement mv < pv.
in case 1, dividing this statement by v gives m > p (you have to flip the sign because v is negative).
in case 2, dividing this statement by v gives m < p (you don't flip the sign, because v is positive).
therefore, here's the complete listing of cases:
CASE 1: m(+) p(+) v(-), m > p
NUMBER LINE REPRESENTATION: ------v------0------p------m------
CASE 2: m(-) p(-) v(+), m < p
NUMBER LINE REPRESENTATION: ------m------p------0------v------
yes, this rephrasing takes a lot of time. however, you should realize the following fact.
IMPORTANT FACT: if a rephrasing takes a lot of time, then, once you've done that rephrasing, the statements should work out very quickly. you will not have to spend tons of time on a rephrase AND tons of time on the statements.
in other words,
either (a) the rephrase will be long and involved or (b) the statements will require lots of effort to evaluate, but not both.
once you've done this rephrasing, true to the promise above, the statements are very easy to evaluate.
statement (1) means that you have case 2, so the answer is "yes".
statement (2) means that you have case 2, so the answer is "yes".
done.
answer = (d).
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review:
TAKEAWAY: inequalities involving 0 should be treated as number properties statements about positives and negatives rather than inequalities. you should only treat them as actual inequalities if the number-properties route is an absolute dead end.
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i understand that a lot of this is already in cramya's post above, but i'm re-posting it in a manner that's more legible (at least to me; i find all that "-ve" and "+ve" stuff really, really hard to read).
