In the xy-plane, region A consists of all the points (x,y)

Hi,
How u got 1 not sufficient?
why 1-x>2y changed to x>2y?
Can u plz explain me.
Thankz in advance,
rrsnathan.

You are confusing the two inequalities. The inequality 1 - x <2y is the inequality that defines region A.

The inequality x >2y is the inequality that defines statement 1.

I'm certainly not an expert on coordinate geometry but will try to explain the problem to you. I will refer to the graph that Zorralou kindly provided. The graphical representation of the inequalities helped me understand the problem and I hope that it helps you as well

The first thing that you want to do is rewrite the given inequality (1-x <2y) in slope intercept form. The inequality will then look like this y >-1/2x + 1/2. Recognize that the y intercept is positive 1/2 and that the inequality has a negative slope that is half of 1, or 1/2. Use this information to help you sketch the inequality on the coordinate plane. See Zorrolou's graph; he has represented this inequality with the black line.

Any point above the "black line" is part of region A so the question is essentially asking whether point (a,b) is above the black line.

Statement 1 says that a>2b
You first want to rewrite the variables in terms of x and y. a >2b becomes x >2y. Again I found it helpful to rewrite the inequality in slope intercept form. Divide both sides by 2 and you get x/2 > y or y< x/2.

Graph this inequality on the coordinate plane. See Zorrolou's diagram again. The inequality y <x/2 is represented by the red line. Notice that the inequality states that y is LESS than 1/2x which means that the region of this inequality is BELOW the red line.

Back to our original question. Is point (a,b) part of region A? We now know that region A is above the black line and below the red line. This is helpful but is not sufficient. As it is now point (a,b) could be above the black line or below the black line. For instance point (5,1) is above the black line (part of region A) and below the red line while point (-1,2) is also below the red line but is NOT part of region A (above the black line).

I hope that my explanation helps you. I'm still trying to learn all this myself! I took the GMAT 3 times and this is the first time that I'm actually trying to get some of these math concepts down. I have not taken any math since taking calculus in high school over a decade ago! Best of luck!

hey josemnz83 tkankz. Really ur explanation was really helpful

+kudos for ur great explanation

Similar questions to practices:
in-the-xy-plane-region-r-consists-of-all-the-points-x-y-102233.html (DS)
point-x-y-is-a-point-within-the-triangle-what-is-the-139419.html (PS)

Hope it helps.

Full marks to zarrolou, however here is an alternate approach which I felt was simpler.

Given inequality 1 - x < 2y does point (a,b) satisfy this equality
so is 1-a<2b ?

(1) a>2b

a= 10 and 2b= 5 putting the values in 1-a<2b we get, is -9<5 ? answer is yes
a=-5 and 2b=-10 ( Note: a>2b) putting the values in 1-a<2b we get is 6 <-10 answer is no
hence insufficient

(2) b=1
is 1-a<2 , clearly insufficient if a =-3 answer is no if a= 3 answer is yes

1+2
then a>2,
if a>2 then 1-a<2 is always true ,(LHS is always negative so always less than 2)Sufficient.

Hope it helps.

Well let me chip in with an alternative approach that I thought was even easier

So 2y>1-x
Is 2b>1-a?

(1) a>2b

If we add we have the question is 2a>1? Is a>1/2? We don't know hence insufficient

(2) b=1

Then we have is 2>1-a? Is a>-1?

This statement is insufficient on its own

Both together we have is a>-1, well we know that a>2 if we replace b=1 on the first statement hence sufficient

C

Asked: 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?
For (a, b) to be in regions A, following condition should be satisfied: -
1 - a < 2b
a > 1 - 2b

(1) a > 2b
a > 2b
For (a, b) to be in region A:
2b > 1- 2b
4b > 1
b>.25
But this is not mentioned
NOT SUFFICIENT

(2) b = 1
For (a, b) to be in region A:
a > 1- 2 = - 1
Which is not mentioned
NOT SUFFICIENT

(1) + (2)
(1) a > 2b
(2) b = 1
a > 2
1- 2b = - 1
a > 2 > -1
SUFFICIENT

IMO C

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