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Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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10 May 2010, 14:07
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Are x and y both positive?
(1) \(2x-2y=1\). Well this one is clearly insufficient. You can do it with number plugging OR consider the following: both x and y are positive means that point (x,y) is in the I quadrant. \(2x-2y=1\) --> \(y=x-\frac{1}{2}\). We know it's an equation of a line and basically the question asks whether this line (all (x,y) points of this line) is only in I quadrant. This line for sure passes I quadrant (for example, x = 1.5 and y = 1) but it cannot entirely be only in I quadrant, so there must be some (x, y) points whose coordinates are not both positive. Not sufficient.
(2) \(\frac{x}{y}>1\) --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.
(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally. One of the approaches:
From (1): \(2x-2y=1\), so \(x=y+\frac{1}{2}\)
From (2): \(\frac{x}{y}>1\);
Substitute x into (2): \(\frac{y + \frac{1}{2}}{y}>1\);
\(1 + \frac{1}{2y}>1\);
\(\frac{1}{2y}>0\);
\(y > 0\). \(y\) is positive. Thus, \(x=y+\frac{1}{2}=positive+positive=positive\). So, \(x\) is positive too. Sufficient.
Answer: C.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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11 May 2010, 04:40
I'm still not clear why X and Y has to be positive when X/Y > 1. Can you please explain the way you combined taking both X and Y to be positive and also X and Y as negative. Since in either case X/Y will be > 1.
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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11 May 2010, 04:59
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harikattamudi wrote:
I'm still not clear why X and Y has to be positive when X/Y > 1. Can you please explain the way you combined taking both X and Y to be positive and also X and Y as negative. Since in either case X/Y will be > 1.
Thanks -H
From (2) \(\frac{x}{y}>1\), we can only deduce that x and y have the same sigh (either both positive or both negative).
When we consider two statement together:
From (1): \(2x-2y=1\) --> \(x=y+\frac{1}{2}\)
From (2): \(\frac{x}{y}>1\) --> \(\frac{x}{y}-1>0\) --> \(\frac{x-y}{y}>0\) --> substitute \(x\) from (1) --> \(\frac{y+\frac{1}{2}-y}{y}>0\)--> \(\frac{1}{2y}>0\) (we can drop 2 as it won't affect anything here and write as I wrote \(\frac{1}{y}>0\), but basically it's the same) --> \(\frac{1}{2y}>0\) means \(y\) is positive, and from (2) we know that if y is positive x must also be positive.
OR: as \(y\) is positive and as from (1) \(x=y+\frac{1}{2}\), \(x=positive+\frac{1}{2}=positive\), hence \(x\) is positive too.
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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20 Jul 2010, 12:41
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I found this one easiest to solve by drawing a graph. Clearly 1) and 2) alone are not sufficient as discussed, so what remains to be seen is if 2) adds enough information to 1) to determine if both x and y are positive.
Drawing a quick graph of the line y=x-1/2 we find that the x-intercept of the line is (0.5,0) and the y-intercept is (0,-0.5). From this graph we can clearly see that we don't need to worry about anything in the 4th quadrant (+x/-y is not >1) or the 3rd quadrant (|x|<|y|, therefore x/y is not >1). All that is left is the 1st quadrant, in which x and y are both positive.
Sufficient.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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30 Sep 2010, 01:03
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zerotoinfinite2006 wrote:
Manbehindthecurtain wrote:
Are x and Y both positive?
1) 2X-2Y = 1 2) (x/y) > 1
I guessed and got it right with a 50/50 guess at the end.
What I have done here is this
1) 2x - 2y = 1 hence x - y = \frac{1}{2} {Dividing both side by 2} In sufficient
2) x > y Alone in sufficient
When (1) + (2) We can say that if X is greater than y than x-y will yield a positive result.
Please correct me if I am wrong
First of all: the question is "are x and Y both positive?" not whether "x-y will yield a positive result".
Next, the red part is not correct.
\(\frac{x}{y}>1\) does not mean that \(x>y\). If both \(x\) and \(y\) are positive, then \(x>y\), BUT if both are negative, then \(x<y\). What you are actually doing when writing \(x>y\) from \(\frac{x}{y}>1\) is multiplying both parts of inequality by \(y\): never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.
So from (2) \(\frac{x}{y}>1\), we can only deduce that \(x\) and \(y\) have the same sigh (either both positive or both negative).
See the complete solution of this problem in my previous post.
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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04 Jan 2012, 00:28
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C is the answer. Question: Is x > 0 AND y > 0?
Statement 1: 2x - 2y = 1 => 2(x - y) = 1 => x - y = 1/2 This just tells us that the difference is positive. But this can be true for cases when both x and y are positive, and when both x and y are negative. For instance, x = 1.5, y = 1 => x - y = 0.5; also, x = -1, y = -1.5 => x - y = 0.5. Thus, INSUFFICIENT.
Statement 2: x/y > 1 This just tells us that x and y have the same sign. That is, both are positive or both are negative. INSUFFICIENT.
Combining these statements, we can use the same numbers used in Statement 1 to find out that both the cases together do not work for negative numbers. For instance, x = -1, y = -1.5 => x - y = 0.5. However, x/y < 1. This violates statement 2.
Thus, the combination of the given statements tells us that x and y both have to be positive. => x > 0 AND y > 0. SUFFICIENT.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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12 May 2012, 01:36
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Statement (1): x-y = 1/2. We can have x=1,y=1/2. Can also have x=0,y=-1/2. Insufficient. Statement (2): x/y>1. We can have x=3,y=2. Can also have x=-3,y=-2. Insufficient.
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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17 Jan 2013, 04:55
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Manbehindthecurtain wrote:
Are x and y both positive?
(1) 2x-2y = 1 (2) x/y > 1
1. x-y = 1/2 This means that the distance between x and y is 1/2 unit and that x is greater than y. But x and y could be positive such as x=5 and y=4.5, OR x and y could be both negative such as x=-4 and y=-4.5
INSUFFICIENT.
2. x/y > 1 This shows that x and y must be positive meaning they are either both (+) or both (-). ex) x/y = 5/2 OR x/y = -5/-2 = 5/2 still > 1
INSUFFICIENT.
Combine. Let x = 5 and y=9/2: 5/(9/2) = 10/9 > 1 - This means when x and y are both positive it could be a solution to x/y > 1 Let x = -4 and y=-9/2: -4/(-9/2) = 8/9 < 1 - This means when x and y are negative it could not be a solution to x/y > 1
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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14 Sep 2013, 22:23
Hello Bunuel, Request you to please provide your comments on the doubt posted here-
Usually, whenever I see combining an inequality and equation, I substitute the value of one of the variable in the inequality and then analyze the effect. So, going by that approach;
x-y=1/2 ---(1) x/y>1 --(2) Substituting the value of x in equation(2)
(y+1/2)/y>1
Lets assume that y is positive-
(y+1/2) > y
1/2>0 --This means that our assumption is true since 1/2 is greater than Zero. Hence, y > 0
Now, Lets assume that y is negative-
Now, here I'm stuck, I know that multiplying by a negative number changes the sign of the inequality. I'm sure that the sign will be changed but what would be the resulting equation. I mean, do we need to replace y with "-y" in the whole equation. Please clarify. Which of the following would be correct then
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1
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14 Sep 2013, 22:39
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imhimanshu wrote:
Hello Bunuel, Request you to please provide your comments on the doubt posted here-
Usually, whenever I see combining an inequality and equation, I substitute the value of one of the variable in the inequality and then analyze the effect. So, going by that approach;
x-y=1/2 ---(1) x/y>1 --(2) Substituting the value of x in equation(2)
(y+1/2)/y>1
Lets assume that y is positive-
(y+1/2) > y
1/2>0 --This means that our assumption is true since 1/2 is greater than Zero. Hence, y > 0
Now, Lets assume that y is negative-
Now, here I'm stuck, I know that multiplying by a negative number changes the sign of the inequality. I'm sure that the sign will be changed but what would be the resulting equation. I mean, do we need to replace y with "-y" in the whole equation. Please clarify. Which of the following would be correct then
a) y+1/2 <y b) y+1/2 < -y c) -y+1/2 < -y
Please help. Thanks
Refer to the highlighted portion : Actually you don't have to take 2 cases at this point: The expression you have is : \(\frac{y+0.5}{y}>1 \to 1+\frac{0.5}{y}>1 \to \frac{1}{y}>0\)--> Hence, y>0.
As for your doubt, if y is negative, we cross-multiply it and get : \(y+0.5<y \to 0>0.5\), which is absurd.
If y is negative, then -y would be positive, and for multiplying a positive quantity, you don't need to flip signs. So , yes expression a is correct.
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