Source: Veritas Prep
In the coordinate geometry plane, region P is defined by all the points (x, y) for which 3y + 12 > 2x. Does point (a, b) lie within region P?
1) 4b = a - 8
2) b < 0 and a > 3
The OA is E.
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In the coordinate geometry plane, region P is defined by all
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BTGmoderatorLU
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Jay@ManhattanReview
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To test whether the point (a, b) lie within region P, we can plug in few test values of a and b in 3y + 12 > 2x. If we consistently find the inequality 3y + 12 > 2x is true or false but not sometimes true and sometime false, we get the answer.BTGmoderatorLU wrote:Source: Veritas Prep
In the coordinate geometry plane, region P is defined by all the points (x, y) for which 3y + 12 > 2x. Does point (a, b) lie within region P?
1) 4b = a - 8
2) b < 0 and a > 3
The OA is E.
Let's take each statement one by one.
1) 4b = a - 8
Case 1: Say a = 0, thus b = -2.
Plugging-in the value of a and b in the inequality 3y + 12 > 2x for x and y, respectively, we get 3*-2 + 12 > 2*0 => -6 + 12 > 2 => 6 > 2. The answer is Yes.
Case 2: Say b = 0, thus a = 8.
Plugging-in the value of a and b in the inequality 3y + 12 > 2x for x and y, respectively, we get 3*0 + 12 > 2*8 => 12 < 16. The answer is No.
No unique answer. Insufficient.
2) b < 0 and a > 3
Case 1: Say a = 4, and b = -1.
Plugging-in the value of a and b in the inequality 3y + 12 > 2x for x and y, respectively, we get 3*-1 + 12 > 2*4 => -3 + 12 > 8 => 9 > 8. The answer is Yes.
Case 2: Say a = 6, and b = -1/2.
Plugging-in the value of a and b in the inequality 3y + 12 > 2x for x and y, respectively, we get 3*(-1/2) + 12 > 2*6 => -3/2 + 12 > 12 => 10.5 < 12. The answer is No.
No unique answer. Insufficient.
(1) and (2) together
Both the cases discussed in Statement 2 are applicable considering Statement 1 also, leading to no unique answer. Insufficient.
The correct answer: E
Hope this helps!
-Jay
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fskilnik@GMATH
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$$3y + 12 > 2x\,\,\,\,\, \Leftrightarrow \,\,\,\,\,2x - 3y < 12$$BTGmoderatorLU wrote:Source: Veritas Prep
In the coordinate geometry plane, region P is defined by all the points (x, y) for which 3y + 12 > 2x. Does point (a, b) lie within region P?
1) 4b = a - 8
2) b < 0 and a > 3
$$\left( {a,b} \right)\,\,\,\mathop \in \limits^? \:\,{\text{region}}\,P\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\boxed{\,\,\,2a - 3b\,\,\,\mathop < \limits^? \,\,\,\,12\:\,}$$
Let´s BIFURCATE (1+2), that is, present two EXPLICIT VIABLE scenarios satisfying both statements together, each scenario answering our FOCUS differently!
$$\left( {1 + 2} \right)\,\,\,\left( * \right)\,\,\,\left\{ \matrix{
\,a - 4b = 8 \hfill \cr
\,b < 0\,\,,\,\,a > 3 \hfill \cr} \right.\,\,\,\,$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {4, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( * \right)\,\,\,{\rm{ok}}\,\,\,\, \Rightarrow \,\,\,\,\,2\left( 4 \right) - 3\left( { - 1} \right)\,\,\,\,\mathop < \limits^? \,\,\,\,12\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \, \hfill \cr
\,\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {6, - {1 \over 2}} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( * \right)\,\,\,{\rm{ok}}\,\,\,\, \Rightarrow \,\,\,\,\,2\left( 6 \right) - 3\left( { - {1 \over 2}} \right)\,\,\,\,\mathop < \limits^? \,\,\,\,12\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \, \hfill \cr} \right.$$
The correct answer is therefore (E).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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