josemnz83 wrote:
In the xy-plane, region A consists of all the points (x,y) such that 1 - x < 2y. Is the point (a,b) in region A?
(1) a > 2b
(2) b = 1
Solution:Statement One Alone:a > 2b
If (a, b) = (5, 2), then a > 2b is satisfied. Substituting x = 5 and y = 2 in the inequality 1 - x < 2y, we get 1 - 5 < 2(2), which is equivalent to -4 < 4. Since the point (5, 2) satisfies the inequality 1 - x < 2y, this point lies in region A.
If (a, b) = (-1, -1), then again the inequality a > 2b is satisfied. Substituting x = y = -1 in 1 - x < 2y, we get 1 - (-1) < 2(-1), which is equivalent to 2 < -2. Since (-1, -1) does not satisfy the inequality 1 - x < 2y, this point does not lie in region A.
We see that we get different answers depending on the values of a and b, so statement one alone is not sufficient.
Eliminate answer choices A and D.
Statement Two Alone:
b = 1
First, suppose that a = 2. Substituting (2, 1) into 1 - x < 2y, we obtain 1 - 2 < 2(1), which is equivalent to -1 < 2. Since the point (2, 1) satisfies 1 - x < 2y, (2, 1) is in region A. In this case, the answer to the question is yes.
On the other hand, if a = -2, then (-2, 1) does not satisfy 1 - x < 2y (because 1 - (-2) = 3 is not greater than 2(1) = 2). In this scenario, the answer to the question is no.
Since we get different answers for the question depending on the value of a, statement two alone is not sufficient.
Eliminate answer choice B.
Statements One and Two Together:Using statement two, we obtain (a, b) = (a, 1). Using statement one, we obtain a > 2(1), i.e. a > 2.
Let’s substitute (a, 1) into 1 - x < 2y:
1 - a < 2(1)
1 - a < 2
-a < 1
a > -1
Since we know a is greater than 2, then a > -1 holds. Thus, the point (a, 1) satisfies the inequality 1 - x < 2y and it belongs to the region defined by this inequality. Statements one and two together are sufficient.
Answer: C
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