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Ho to solve this inequality problem DS question

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by srcc25anu » Wed May 08, 2013 5:43 am
We need to find whether x and Y are positive. that is whether they lie in first quadrant as highlighted by yellow shaded redion in graph attached.

ST1: 2x - 1 = 2y
this is the red line. Some part lies in Q1 and some does not.
So insufficient

St2: x/y > 1
this is denoted by blue shaded area. Some part lies in Q1 and some does not.
Hence insufficient.

Together: only the Red line in Q1 satifies both conditions and all of it lies in Q1. Both x and y are positive.
Hence Sufficient

Ans C
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by Brent@GMATPrepNow » Wed May 08, 2013 6:29 am
karan.7045 wrote:Are x and y both positive?
1) 2x - 2y =1
2) x/y >1
Target question: Are x and y both positive?

Statement 1: 2x - 2y = 1
There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case x and y are both positive
Case b: x = -0.5 and y = -1, in which case x and y are not both positive

Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case x and y are both positive
Case b: x = -4 and y = -2, in which case x and y are not both positive

Statements 1 and 2
Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2

Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then y must be positive.

Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that y is positive, it must be the case that x and y are both positive
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

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Brent
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by Atekihcan » Wed May 08, 2013 6:42 am
From statement 1 we can only conclude x = y + 1/2, i.e. x > y
So, statement 1 is not sufficient.

From statement 2, we can conclude that,
either x and y both positive, and x > y
or x and y both negative and x < y
So, statement 2 is also not sufficient.

Now, both statements together, from statement 1 we know x > y.
From statement 2, x can be greater than y only if both x and y are positive.
So, both statements together is sufficient.

Answer : C
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