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# M27-18

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:27
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95% (hard)

Question Stats:

32% (01:54) correct 68% (01:44) wrong based on 237 sessions

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If $$x$$ and $$y$$ are negative numbers, is $$x \lt y$$?

(1) $$3x + 4 \lt 2y + 3$$

(2) $$2x - 3 \lt 3y - 4$$

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16 Sep 2014, 00:27
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Official Solution:

(1) $$3x + 4 \lt 2y + 3$$. Re-arrange: $$3x \lt 2y-1$$. $$x$$ can be some very small number, for instance -100 and $$y$$ some large enough number for instance -3 and the answer would be YES, $$x \lt y$$ BUT if $$x=-2$$ and $$y=-2.1$$, then the answer would be NO, $$x \gt y$$. Not sufficient.

(2) $$2x - 3 \lt 3y - 4$$. Re-arrange: $$x \lt 1.5y-\frac{1}{2}$$. Re-write as $$x \lt y+(0.5y-\frac{1}{2}) = y + \text{negative}$$. So, $$x \lt y$$ (as $$y+\text{negative}$$ is "more negative" than $$y$$). Sufficient.

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14 Nov 2015, 22:53
Hello,

Is there any specific approach to solve this type of questions. I find to solve these questions very difficult.

Thanks
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05 Mar 2016, 21:55
Bunuel
why can we apply same logic, which you applied in option B, in option A. We have x<2/3 y -1/3. please tell me if I am missing anything. y is negative and we are subtracting and -ve number.
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05 Mar 2016, 22:54
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Bunuel
why can we apply same logic, which you applied in option B, in option A. We have x<2/3 y -1/3. please tell me if I am missing anything. y is negative and we are subtracting and -ve number.

Hi,
I'll try to answer this for you ..

We know x and y are negative numbers ..

and the answer to the Q lies in

1)knowing that if a -ive number is multiplied with a bigger positive number/integer, The product will become even smaller..
meaning 3x<2x etc
2) role of < and > signs
so if we want to check on the relative values of x and y, they have to brought in some way to same Coeff..

Can you spot the difference in two statements given keeping the above info in mind..

the statements are..

(1) 3x+4<2y+3
So, 3x+1<2y..
3x<2y-1
now x and y are negative integers..
so 3x< 2x..
but we cannot make a relation between 2x and 2y-1..
Insuff

(2) 2x−3<3y−4
2x<3y-1..
now 3x<2x..
so we can write 3x<2x<3y-1..
or 3x<3y-1
x<y-1/3..
since both x and y are negative numbers and x is lesser than sum of y, a -ive number, and another -ive number.. x<y
Suff

B

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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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13 Jun 2016, 07:35
(1) 3x+4<2y+3

3x+1<2y

1.5x+0.5<y

If 1.5 times x (i.e. a -ve term) and 0.5 added to it is still less than y, then x should be less that y..isn't it?

what am I doing wrong?

(2) 2x−3<3y−4

2x-3+4<3y

2x+1<3y

.66x+.33<y

I used the same logic as stated above....X being -ve, .66 times x is also -ve and 0.33 added to it is still less than y. Hence x should be less than y.

What is wrong with my approach in (1) ?
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07 Mar 2017, 16:37
2
1
Graphically
One easy way to do this is to see then the lines intersect by graphing them. In the attached, red is (1), blue is (2), and yellow is the prompt. You can see that (1) intersects with y>x when x, y negative, and thus you can pick options that allow the red line to be true, but can literally go on either side of y > x. For the blue line, that is also possible, but only when y and x are positive. Thus, we know that (2) is sufficient for all x,y negative.

Algebraically
Another easy way to do this is to see when the lines intersect, but by using algebra.

(1) 2y > 3x+1
Pretend this is 2y=3x+1.
Set this equal to what we are comparing it to, y=x, or, 2y=2x.
2y=3x+1=2x
They intersect at x = -1. Either side of this, you will a Y or N answer when seeing if (1) is sufficient.

(2) 3y > 2x+1
Pretend this is 3y=2x+1.
Set this equal to what we are comparing it to, y=x, or, 3y=3x.
3y=2x+1=3x
They intersect at x=1. If less than 1, you will always get the same answer when seeing if (2) is sufficient. Since x < 0 < 1, (2) is sufficient.
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14 May 2017, 11:28
Why are we using picking numbers for statement 1 and solving the second statement using inequalities and algebra?
Both inequalities are of the same nature. Can someone please explain the sufficiency of the first statement using algebra?
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03 Oct 2017, 11:52
Bunuel wrote:
Official Solution:

(1) $$3x + 4 \lt 2y + 3$$. Re-arrange: $$3x \lt 2y-1$$. $$x$$ can be some very small number, for instance -100 and $$y$$ some large enough number for instance -3 and the answer would be YES, $$x \lt y$$ BUT if $$x=-2$$ and $$y=-2.1$$, then the answer would be NO, $$x \gt y$$. Not sufficient.

(2) $$2x - 3 \lt 3y - 4$$. Re-arrange: $$x \lt 1.5y-\frac{1}{2}$$. Re-write as $$x \lt y+(0.5y-\frac{1}{2}) = y + \text{negative}$$. So, $$x \lt y$$ (as $$y+\text{negative}$$ is "more negative" than $$y$$). Sufficient.

Question: Bunuel
case-1) 3x < 2y -1
y=0 => 3x < -1 => x < -0.33
x=0 => y > 0.5 (but y < 0 given). Therefore, we can have both x < y (-0.33 < -0.1) and x > y (-0.33 > -0.4). Thus 1 NS

case-2) x < (3/2)y - 1/2
y = 0 => x < -1/2
x = 0 => y > 1/3 (but y < 0 given). Therefore, we can have both x < y (-0.5 < -0.1) and x > y (-0.5 > -0.6). Thus 2 NS

Case-1+2) x < -0.33 & x < -0.5 => x < -0.33 & y < 0. Thus 1+2 NS => E

What am i doing wrong? Can you please clarify Bunuel...
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03 Oct 2017, 11:57
manishtank1988 wrote:
Bunuel wrote:
Official Solution:

(1) $$3x + 4 \lt 2y + 3$$. Re-arrange: $$3x \lt 2y-1$$. $$x$$ can be some very small number, for instance -100 and $$y$$ some large enough number for instance -3 and the answer would be YES, $$x \lt y$$ BUT if $$x=-2$$ and $$y=-2.1$$, then the answer would be NO, $$x \gt y$$. Not sufficient.

(2) $$2x - 3 \lt 3y - 4$$. Re-arrange: $$x \lt 1.5y-\frac{1}{2}$$. Re-write as $$x \lt y+(0.5y-\frac{1}{2}) = y + \text{negative}$$. So, $$x \lt y$$ (as $$y+\text{negative}$$ is "more negative" than $$y$$). Sufficient.

Question: Bunuel
case-1) 3x < 2y -1
y=0 => 3x < -1 => x < -0.666
x=0 => y > 0.5 (but y < 0 given). Therefore, we can have both x < y (-0.33 < -0.1) and x > y (-0.33 > 0.4). Thus 1 NS

case-2) x < (3/2)y - 1/2
y = 0 => x < -1/2
x = 0 => y > 1/3 (but y < 0 given). Therefore, we can have both x < y (-0.33 < -0.1) and x > y (-0.33 > 0.4). Thus 2 NS

What am i doing wrong? Can you please clarify Bunuel...

We are told that BOTH x and y are negative numbers. Can you give an example when that condition and $$2x - 3 \lt 3y - 4$$ are satisfied and x is not less than y?
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03 Oct 2018, 23:07
[url]@chetan2u[/url] I noted that you mentioned that the 2nd witll equate to x<y-1/3 but if I take x=-2 and y=-2 then x<y does not hold true
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03 Oct 2018, 23:59
Superg8 wrote:
[url]@chetan2u[/url] I noted that you mentioned that the 2nd witll equate to x<y-1/3 but if I take x=-2 and y=-2 then x<y does not hold true

Hi..
If you take x=y,how can x be less than y..
X and y are NEGATIVE..
AND statement II tells us x<y-1/3..( this is true we don't have to prove this)
You can't take x=y=-2..
If y =-2, x<-2-1/3=-5/3 so x<-5/3 it can be -3,-10
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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05 Oct 2018, 10:30
Bunuel wrote:
Official Solution:

(1) $$3x + 4 \lt 2y + 3$$. Re-arrange: $$3x \lt 2y-1$$. $$x$$ can be some very small number, for instance -100 and $$y$$ some large enough number for instance -3 and the answer would be YES, $$x \lt y$$ BUT if $$x=-2$$ and $$y=-2.1$$, then the answer would be NO, $$x \gt y$$. Not sufficient.

(2) $$2x - 3 \lt 3y - 4$$. Re-arrange: $$x \lt 1.5y-\frac{1}{2}$$. Re-write as $$x \lt y+(0.5y-\frac{1}{2}) = y + \text{negative}$$. So, $$x \lt y$$ (as $$y+\text{negative}$$ is "more negative" than $$y$$). Sufficient.

How did you conclude that (0.5y−0.5) is more negative. if y is positive then that may not be the case.
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05 Oct 2018, 10:33
sakuac wrote:
Bunuel wrote:
Official Solution:

(1) $$3x + 4 \lt 2y + 3$$. Re-arrange: $$3x \lt 2y-1$$. $$x$$ can be some very small number, for instance -100 and $$y$$ some large enough number for instance -3 and the answer would be YES, $$x \lt y$$ BUT if $$x=-2$$ and $$y=-2.1$$, then the answer would be NO, $$x \gt y$$. Not sufficient.

(2) $$2x - 3 \lt 3y - 4$$. Re-arrange: $$x \lt 1.5y-\frac{1}{2}$$. Re-write as $$x \lt y+(0.5y-\frac{1}{2}) = y + \text{negative}$$. So, $$x \lt y$$ (as $$y+\text{negative}$$ is "more negative" than $$y$$). Sufficient.

How did you conclude that (0.5y−0.5) is more negative. if y is positive then that may not be the case.

It is given that y is negative so 0.5y is negative and add another negative term to it -0.5, you get smaller negative number
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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25 Nov 2018, 09:18
Keeping it SIMPLE -

One trick to remember is that if u can prove $$x = y$$ then u can answer YES or NO to a $$x<y$$ or $$x>y$$ question.

Statement 1 -
$$3x + 4 < 2y + 3$$
Rearrange : $$3x - 2y < -1$$
Simply try$$x = y = -2$$
We see that above condition is satisfied.
But answer to Is $$x < y?$$ is NO.

Try $$x = -3$$ and $$y = -2$$
We see that above condition is satisfied.
Hence answer to Is$$x < y?$$ is YES.

Statement 2 -
$$2x - 3 < 3y - 4$$
Rearrange : $$2x - 3y < -1$$
Before testing values. Just observe the equation.
The ($$-3y$$) part has will always be positive since y is negative.
Also, we need to get a negative value so ($$2x)$$ part has to be more than the $$(-3y)$$ part.
Is this possible if $$x$$ and $$y$$ are equal? No.
Hence we have different values of $$x$$ and $$y$$.
This should be sufficient.

M27-18 &nbs [#permalink] 25 Nov 2018, 09:18
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# M27-18

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