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OG Data sufficiency prob no.129

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by srcc25anu » Thu Apr 04, 2013 8:50 pm
let r = x and t = y for simplicity
q: whether 2x + 3y <=6 that is represented by area under the blue line.

ST1. its represented by ALL POINTS ONLY ON THE RED LINE.
Since some part of red line lies within the blue curve and some lies outside of it, not sufficient

St1: orange shaded area.
Again as some area lies within the blue line region and others dont, not sufficient.

Taking 1 and 2 together: Blue portion on the line satifies St1 and St2 and lies within the required region but BLACK portion of the line satifies both ST1 and St2 but DOES nOT LIE within the required region

Hence E
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by Anju@Gurome » Thu Apr 04, 2013 11:54 pm
While solving coordinate geometry problems, specifically DS problems, try to draw the scenario and solve by visualizing possible situations. In this case, the following diagram will help us to solve the problem:
Image

Statement 1: (r, s) may or may not lie in region R.

Not sufficient

Statement 2: If we take r = 3 and s = 2, then the point (3, 2) does not lie in region R.
r ≤ 3 and s ≤ 2 implies we can also take negative values for r and s. If r = -2, s = -3, then (-2, -3) lies in region R.

Not sufficient

1 & 2 Together: If r = 2, s = 0 then (2, 0) lies in region R. But if r = 2/3 and s = 2 then (2/3, 2) lies above the line 2x + 3y = 6, which means (2/3, 2) does not lie in region R. Combining also doesn't give a unique answer.

The correct answer is E.
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by GMATGuruNY » Fri Apr 05, 2013 1:47 am
In the XY plane, region R consists of all the points (x,y) such that 2x+3y≤6. Is the point (r,s) in region R?
1. 3r+2s=6
2. r≤3 and s≤2
Region R is composed of all the points on or below y=(-2/3)x + 2.

Statement 1: s = (-3/2)r + 3.
Image
The figure above shows that some points on s=(-3/2)r + 3 lie BELOW y=(-2/3)x + 2, while others lie ABOVE y=(-2/3)x + 2.
INSUFFICIENT.

Statement 2: r≤3 and s≤2.
Image
Inside the green box are points such that r≤3 and s≤2.
Some of the points inside the green box lie BELOW y=(-2/3)x + 2, while others lie ABOVE y=(-2/3)x + 2.
INSUFFICIENT.

Statements 1 and 2 combined:
Image
Inside the green box are points on s=(-3/2)r + 3 such that r≤3 and s≤2.
Some of these points lie BELOW y=(-2/3)x + 2, while others lie ABOVE y=(-2/3)x + 2.
INSUFFICIENT.

The correct answer is E.

An alternate way to combine the two statements is to treat this as MAX/MIN problem.

R MAXIMIZED:
In statement 2, the maximum possible value of r is 3.
If r=3 and s=0, both statements 1 and 2 are satisfied.
Check whether (3,0) is within the region defined by y ≤ (-2/3)x + 2:
0 = (-2/3)(3) + 2
0 ≤ 0.
YES.

S MAXIMIZED:
In statement 2, the maximum possible value of s is 2.
If s=2 and r=(2/3), both statements 1 and 2 are satisfied.
Check whether (2/3, 2) is within the region defined by y ≤ (-2/3)x + 2:
2 ≤ (-2/3)(2/3) + 2
2 ≤ 2/3
NO.

Since in the first case (r,s) is within the required region, but in the second case (r,s) is not within the required region, the two statements combined are INSUFFICIENT.
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