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Math Expert V
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Re: Inequality and absolute value questions from my collection  [#permalink]

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3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Also insufficient as x,y, and a could be 0
Why did you assume that ALL COULD be zero.

Because they could be 0, why not? And IF x=y=a=0, then the answer would be NO but if they take some other values then the answer would be YES.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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I get super confused in solving the below kind of inequalities.
Can some expert help?

Are the below working's correct?

1/| n | >n
• Case 1: n > 0… n 2 < 1… - 1 < n < 1
• Case 2: n < 0… - n 2 < 1… n 2 > -1…

x/| x | < x
• Case 1: x > 0… x^2 > x… as x > 0, divide by x without changing signs… x > 1
• Case 2: x < 0… - x^2 > x… as x < 0, divide and only flip sign… - x < 1… x > -1
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Re: Inequality and absolute value questions from my collection  [#permalink]

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1
Bunuel wrote:
11. Is |x+y|>|x-y|?
(1) |x| > |y|
(2) |x-y| < |x|

To answer this question you should visualize it. We have comparison of two absolute values. Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they have the same sign will contribute to each other and x-y will not.

5+3=8 and 5-3=2
OR -5-3=-8 and -5-(-3)=-2.

So if we could somehow conclude that x and y have the same sign or not we would be able to answer the question.

(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.

(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin. Sufficient. (Note that vise-versa is not right, meaning that x and y can have the same sign but |x| can be less than |x-y|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)

Hi,

I didnt quite understand this part "(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin."

what in the second equation tells us that they both have to be the same sign?

Thanks Math Expert V
Joined: 02 Sep 2009
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Re: Inequality and absolute value questions from my collection  [#permalink]

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anishasjkaul wrote:
Bunuel wrote:
11. Is |x+y|>|x-y|?
(1) |x| > |y|
(2) |x-y| < |x|

To answer this question you should visualize it. We have comparison of two absolute values. Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they have the same sign will contribute to each other and x-y will not.

5+3=8 and 5-3=2
OR -5-3=-8 and -5-(-3)=-2.

So if we could somehow conclude that x and y have the same sign or not we would be able to answer the question.

(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.

(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin. Sufficient. (Note that vise-versa is not right, meaning that x and y can have the same sign but |x| can be less than |x-y|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)

Hi,

I didnt quite understand this part "(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin."

what in the second equation tells us that they both have to be the same sign?

Thanks If x and y have the opposite signs: ---x---0---y---- the distance between the origin and x will always be less than x and y.

Hope it helps.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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NS13983 wrote:
I get super confused in solving the below kind of inequalities.
Can some expert help?

Are the below working's correct?

1/| n | >n
• Case 1: n > 0… n 2 < 1… - 1 < n < 1
• Case 2: n < 0… - n 2 < 1… n 2 > -1…

x/| x | < x
• Case 1: x > 0… x^2 > x… as x > 0, divide by x without changing signs… x > 1
• Case 2: x < 0… - x^2 > x… as x < 0, divide and only flip sign… - x < 1… x > -1

1. $$\frac{1}{| n |}>n$$ --> multiply by $$|n|$$ (we can safely do that since |n|>0): $$n*|n| < 1$$.

If $$n>0$$, then we'll have $$n^2<1$$ --> $$-1<n<1$$. Since we consider the range when $$n>0$$, then for this range we'll have $$0<n<1$$.
If $$n<0$$, then we'll have $$-n^2<1$$ --> $$n^2>-1$$. Which is true for any n from the range we consider. So, $$n*|n| < 1$$ holds true for any negative value of n.

Thus $$\frac{1}{| n |}>n$$ holds true if $$n<0$$ and $$0<n<1$$.

2. $$\frac{x}{|x|} < x$$.

If $$x>0$$, then we'll have $$\frac{x}{x} < x$$ --> $$1<x$$.
If $$x<0$$, then we'll have $$\frac{x}{-x} < x$$ --> $$-1<x$$ and as we considering the range for which $$x<0$$ then completer range would be: $$-1<x<0$$.

Thus $$\frac{x}{|x|} < x$$ holds true if $$-1<x<0$$ or $$x>1$$.

Hope it helps.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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Bunuel wrote:
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Is this also a right way to solve this:
x^2+2xy+y^2=9a --> If we have to find x^2 + y^2 > 4a we should know whether 2xy>5a or 2xy<5a...No info on this.Hence Insuff
x^2-2xy+y^2=a---> If we have to find x^2 + y^2 > 4a we should know whether 2xy>3a or 2xy<3a...No info..Hence Insuff

Combining both..we still dont know as both the cases are possible and hence E..

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Re: Inequality and absolute value questions from my collection  [#permalink]

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JusTLucK04 wrote:
Bunuel wrote:
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Is this also a right way to solve this:
x^2+2xy+y^2=9a --> If we have to find x^2 + y^2 > 4a we should know whether 2xy>5a or 2xy<5a...No info on this.Hence Insuff
x^2-2xy+y^2=a---> If we have to find x^2 + y^2 > 4a we should know whether 2xy>3a or 2xy<3a...No info..Hence Insuff

Combining both..we still dont know as both the cases are possible and hence E..

Yes, the question from (1) becomes: is 2xy < 5a and from (2) it becomes: is 2xy > 3a. So, when we combine the question becomes is 3a < 2xy < 5a. It's unclear how you deduced that we cannot answer that question without plugging values.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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Bunuel wrote:
4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) 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:
$$2x-2y=1$$ --> $$x=y+\frac{1}{2}$$
$$\frac{x}{y}>1$$ --> $$\frac{x-y}{y}>0$$ --> substitute x --> $$\frac{1}{y}>0$$ --> $$y$$ is positive, and as $$x=y+\frac{1}{2}$$, $$x$$ is positive too. Sufficient.

Hi Bunnel,

I don't quite get the part on "substitute x". Does it mean $$\frac{1}{y}(x-y)>0$$ and so (x-y) is zero?

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pretzel wrote:
Bunuel wrote:
4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) 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:
$$2x-2y=1$$ --> $$x=y+\frac{1}{2}$$
$$\frac{x}{y}>1$$ --> $$\frac{x-y}{y}>0$$ --> substitute x --> $$\frac{1}{y}>0$$ --> $$y$$ is positive, and as $$x=y+\frac{1}{2}$$, $$x$$ is positive too. Sufficient.

Hi Bunnel,

I don't quite get the part on "substitute x". Does it mean $$\frac{1}{y}(x-y)>0$$ and so (x-y) is zero?

No, we are substituting $$x=y+\frac{1}{2}$$ into $$\frac{x-y}{y}>0$$.

From (1): $$2x-2y=1$$ --> $$x=y+\frac{1}{2}$$

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 (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.

Graphic approach for this question: inequality-and-absolute-value-questions-from-my-collection-86939-260.html#p1269802

Hope this helps.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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Bunuel wrote:
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

Question basically asks is -4<n<4 true.

(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.

(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.

hi
m a little confused here. How is A sufficient

isn't n^2>16 ----> n^2-16>0 ---> (N^2-4^2) >0 ----> n+4>0 or n-4>0 ----> n> -4 or n>4.....how did u get n<-4....

kindly correct me if m wrong
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sugand wrote:
Bunuel wrote:
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

Question basically asks is -4<n<4 true.

(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.

(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.

hi
m a little confused here. How is A sufficient

isn't n^2>16 ----> n^2-16>0 ---> (N^2-4^2) >0 ----> n+4>0 or n-4>0 ----> n> -4 or n>4.....how did u get n<-4....

kindly correct me if m wrong

n^2 > 16 means |n| > 4, which means that n > 4 or n < -4.

Notice that n > -4 or n > 4 does not make any sense. What are the possible value of n in this case?
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Re: Inequality and absolute value questions from my collection  [#permalink]

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Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

Great question.
I looked at S1 and as usual thought of special numbers, but then proceeding with -0=|-0| did not make sense to me because I thought "0 is neither +ve nor -ve so why negate it and then find absolute value". Is there some logic i'm missing or is it just a 'possible calculation'?
Thank you
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Re: Inequality and absolute value questions from my collection  [#permalink]

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deeuk wrote:
Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

Great question.
But -0=|-0| did not make sense to me because I thought "0 is neither +ve nor -ve so why negate it and then find absolute value". I did not know that was a possible direction. Is there some logic i'm missing or is it just a 'possible calculation'?
Thank you

Yes, 0 is neither negative nor positive but there is nothing wrong in writing -0, or |0|, because -0=0 and |0| = 0.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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Bunuel wrote:
13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Last one.

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.

(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

About S1: can i say |x-1|<=1 after rooting the given and putting it in modulus? I recall reading about squaring being used to remove modulus, so I thought to reverse that. Please correct me.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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deeuk wrote:
Bunuel wrote:
13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Last one.

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.

(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

About S1: can i say |x-1|<=1 after rooting the given and putting it in modulus? I recall reading about squaring being used to remove modulus, so I thought to reverse that. Please correct me.

Yes, that;s correct. We CAN take the square root from both sides of (x-1)^2 <= 1 because both sides are non-negative.

For more on inequalities check here: inequalities-tips-and-hints-175001.html

Hope it helps.
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Re: Inequality and absolute value questions from my collection  [#permalink]

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Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

(1) -n=|-n|, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.

(2) n^2=16 --> n=4 or n=-4. Not sufficient.

(1)+(2) n is negative OR n equals to zero from (1), n is 4 or -4 from (2). --> n=-4, hence it's negative, sufficient.

Bunuel

Cant get thru stmt 1. Opening the mod we get either -n=-n or -n=n. First case gives null so am left with n=0 and hence sufficient. Where am I wrong in opening up the mod.?
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sinhap07 wrote:
Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

(1) -n=|-n|, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.

(2) n^2=16 --> n=4 or n=-4. Not sufficient.

(1)+(2) n is negative OR n equals to zero from (1), n is 4 or -4 from (2). --> n=-4, hence it's negative, sufficient.

Bunuel

Cant get thru stmt 1. Opening the mod we get either -n=-n or -n=n. First case gives null so am left with n=0 and hence sufficient. Where am I wrong in opening up the mod.?

First of all |-n| = |n|, so we are given that -n = |n|, which implies that n <= 0.
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Bunuel wrote:
sinhap07 wrote:
Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

(1) -n=|-n|, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.

(2) n^2=16 --> n=4 or n=-4. Not sufficient.

(1)+(2) n is negative OR n equals to zero from (1), n is 4 or -4 from (2). --> n=-4, hence it's negative, sufficient.

Bunuel

Cant get thru stmt 1. Opening the mod we get either -n=-n or -n=n. First case gives null so am left with n=0 and hence sufficient. Where am I wrong in opening up the mod.?

First of all |-n| = |n|, so we are given that -n = |n|, which implies that n <= 0.

How did we arrive at |-n| = |n|?
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sinhap07 wrote:
How did we arrive at |-n| = |n|?

It's true for ANY x that |-x| = |x|: |-1| = |1|.
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lagomez wrote:
Bunuel wrote:
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0

First let's simplify given expression $$6*x*y = x^2*y + 9*y$$:

$$y*(x^2-6x+9)=0$$ --> $$y*(x-3)^2=0$$. Note here that we CAN NOT reduce this expression by $$y$$, as some of you did. Remember we are asked to determine the value of $$xy$$, and when reducing by $$y$$ you are assuming that $$y$$ doesn't equal to $$0$$. We don't know that.

Next: we can conclude that either $$x=3$$ or/and $$y=0$$. Which means that $$xy$$ equals to 0, when y=0 and x any value (including 3), OR $$xy=3*y$$ when y is not equal to zero, and x=3.

(1) $$y-x=3$$. If y is not 0, x must be 3 and y-x to be 3, y must be 6. In this case $$xy=18$$. But if y=0 then x=-3 and $$xy=0$$. Two possible scenarios. Not sufficient.

(2) $$x^3<0$$. x is negative, hence x is not equals to 3, hence y must be 0. So, xy=0. Sufficient.

This one was quite tricky and was solved incorrectly by all of you.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

Never multiply (or reduce) inequality by variable (or 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.

would you think this is a 700+ question?

Hi Bunuel,

Can you please comment if my approach is correct?

Given: 6*x*y = x^2*y + 9*y, which is reduced to y*(x-3)^2 = 0.......(1)

Statement  y-x=3 or y = x+3
sub'ing y in (1)
(x+3)(x-3)(x-3) = 0
(x^2-9)(x-3)=0
so, x^2 = 9 or x = +/-3
and x=3
therefore, y = 6 or 0 for x = 3 and -3 respectively..........Not sufficient

Statement ....says x < 0, so (x-3)^2 will never be = 0, y has to be it.....therefore xy=0 so sufficient!

So my question to you is whether I handled  properly? and whether substituting y into the original problem equation and arriving at x = +/- 3 is correct?
So far, I've learnt that Sqrt(x) will only have +x as solution as GMAT doesn;t consider -ve solution, but x^2 =25 will have +/-5 as solution so we should consider both solutions for x^2 = 25 in our analysis.....Did I get this right?
Thanks! Re: Inequality and absolute value questions from my collection   [#permalink] 01 Jun 2015, 01:35

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