Data Sufficiency

dotnetuncle wrote:121. In the xy-plane, region R consists of all the points (x,y) such that 2x+3y<=6. Is the point (r,s) in the region R?

1) 3r + 2s = 6
2) r<=3 and s<=2

Whats the approach & solution?

Cheers
du

so, first, rephrase. you can simplify the question to, "is 2r + 3s < 6?"

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statement (2) is easier, so let's start with that one.

takeaway: if you're given RANGES or INEQUALITIES in a data sufficiency problem, you should TEST THE EXTREMES.

the biggest that r and s can be are, respectively, 3 and 2.
therefore, the largest possible value of (2r + 3s) is 12. this is a "no" to the question.

the smallest that r and s can be are huge negative numbers.
therefore, it's possible for (2r + 3s) to be a huge negative number. this is a "yes" to the question.

insufficient.

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statement (1):

there's a really quick VISUAL STRATEGY if you have enough visual intuition about these sorts of things.
namely:
the line 2x + 3y = 6 (the line that produces the inequality in the prompt) is not parallel to the line 3x + 2y = 6 (the line in statement 1).
therefore, the lines will intersect. therefore, each line will have points on both sides of the other one.
therefore, since the inequality is only on one side of the line 2x + 3y = 6, statement (1) will have points that are both inside and outside the region defined by that inequality.
so, insufficient.

if not, then, you can consider this equation and inequality to be simultaneous, and you can SUBSTITUTE.
get a substitution: 3r + 2s = 6 --> s = 3 - 1.5r. (note that you have to get the substitution from the equation - you can't generally get a substitution from an inequality)
therefore, any point on the line 3r + 2s = 6 can be defined by (r, 3 - 1.5r).
plug these into the prompt:
is 2r + 3s < 6?
is 2r + 3(3 - 1.5r) < 6?
is 9 - 4.5r < 6?
insufficient, since (9 - 4.5r) can have any value at all.

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together:

from (1), the question is now "is 9 - 4.5r < 6?"

from (2), r is anything < 3.
this means that (9 - 4.5r) can be anything > 9 - 4.5(3), or -4.5.
some of these values are less than 6; others are greater than 6.
insufficient.

ans (e)

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...or you could just do what they do in the back of the official guide, and find points that prove that the two statements together are insufficient.