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Senior Manager  Joined: 01 Aug 2008
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If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

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If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$

(2) $$|x - 3|\leq{-y}$$
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If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

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If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$. As given that $$y$$ is non negative value then $$|x - 3|$$ is more than (or equal to) some non negative value, (we could say the same ourselves as absolute value in our case ($$|x - 3|$$) is never negative). So we can not determine single numerical value of $$x$$. Not sufficient.

Or another way: to check $$|x - 3|\geq{y}\geq{0}$$ is sufficient or not just plug numbers:
A. $$x=5$$, $$y=1>0$$, and B. $$x=8$$, $$y=2>0$$: you'll see that both fits in $$|x - 3|>=y$$, $$y\geq{0}$$.

Or another way:
$$|x - 3|\geq{y}$$ means that:

$$x - 3\geq{y}\geq{0}$$ when $$x-3>0$$ --> $$x>3$$

OR (not and)
$$-x+3\geq{y}\geq{0}$$ when $$x-3<0$$ --> $$x<3$$

Generally speaking $$|x - 3|\geq{y}\geq{0}$$ means that $$|x - 3|$$, an absolute value, is not negative. So, there's no way you'll get a unique value for $$x$$. INSUFFICIENT.

(2) $$|x-3|\leq{-y}$$. Now, as $$|x-3|$$ is never negative ($$0\leq{|x-3|}$$) then $$0\leq{-y}$$ --> $$y\leq{0}$$ BUT stem says that $$y\geq{0}$$ thus $$y=0$$. $$|x-3|\leq{0}$$ --> $$|x-3|=0=y$$ (as absolute value, in our case |x-3|, can not be less than zero) --> $$x-3=0$$ --> $$x=3$$. SUFFICIENT

In other words:
$$-y$$ is zero or less, and the absolute value ($$|x-3|$$) must be at zero or below this value. But absolute value (in this case $$|x-3|$$) can not be less than zero, so it must be $$0$$.

Hope it helps..
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: value of x?  [#permalink]

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2
neoreaves wrote:
If y > = 0, what is the value of x?
1. |x - 3| >= y
2. |x - 3| <= - y

IMO B

Statement 1). |x - 3| >= y >=0
|x - 3| >= 0 , for different values of x, this is true.

Statement 2). |x - 3| <= - y since |x - 3| is always >=0 , and y>=0

|x - 3| <= - y will hold true only when y is 0

=> |x - 3| = 0 , only solution is x=3 hence sufficient.

Thus B
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If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

Show Tags First number line: y>=0
Second: From properties of absolute value, |x-3|>=0 is true no matter what x is.
Third: (2) tells us that |x-3|<=-y. We know nothing about y other than y>=0. Therefore |x-3|<=-y only tells us that |x-3|<=0.
Fourth: Since |x-3|<=0 AND |x-3|>=0, we now know the exact value of |x-3|. In the diagram it is the point on the number line that is both <=0 and >=0. The only value for which that is true is 0. So |x-3|=0. From there you know that x-3=0 and so x=3.
Senior Manager  Joined: 13 May 2013
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Re: If y>=0, What is the value of x?  [#permalink]

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If y >= 0, What is the value of x?

(1) |x-3| >= y
We are told that |x-3| ≥ y but all we know is that y is a positive # greater than or equal to zero. Therefore, all we know is that |x-3| is greater than or equal to zero and x could be an infinite number of possibilities.

(2) |x-3| <= -y

(I am having real difficulty understanding the rationale for (2) could someone explain it to me? (preferably like they would to a 5th grader )

Thanks!
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Re: If y>=0, What is the value of x?  [#permalink]

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WholeLottaLove wrote:
If y >= 0, What is the value of x?

(1) |x-3| >= y
We are told that |x-3| ≥ y but all we know is that y is a positive # greater than or equal to zero. Therefore, all we know is that |x-3| is greater than or equal to zero and x could be an infinite number of possibilities.

(2) |x-3| <= -y

(I am having real difficulty understanding the rationale for (2) could someone explain it to me? (preferably like they would to a 5th grader )

Thanks!

Hi WholeLottaLove

The important property concerning absolute value inequalities is:
|a| <= b <--> -b <= a <=b
[a is in the middle of -b and b, inclusive]

Apply to statement (2)
|x - 3| <= -y
<--> -(-y) <= (x-3) <= -y,
<--> y <= (x-3) <= -y

We know y is 0 or positive, -y is 0 or negative.
Because there is not any number that is both negative and positive.Thus, there is ONLY one number that is both >= y AND <= -y. That is zero.
Therefore, (x-3) must be zero. ==> x = 3

Hope it's clear.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If y >= 0, what is the value of x?  [#permalink]

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1
Hi All,

When complex-looking questions show up on Test Day, there's almost always some type of built-in pattern involved. If you can't immediate spot the pattern, then you have to put in a bit of work to prove what the pattern actually is....TESTing VALUES can help you to prove that a pattern exists.....

Here, we're told that Y >= 0. We're asked for the value of X.

Fact 1: |X-3| >= Y

IF....
Y = 0
Then |X-3| >= 0, so X can be ANY number. As Y gets bigger, certain options are eliminated, but given this 'restriction', X has an infinite number of possibilities.
Fact 1 is INSUFFICIENT

Fact 2: |X-3| <= -Y

Here, we have to be CAREFUL with the details. Notice how there's a NEGATIVE sign in front of the Y.....

IF....
Y = 0
|X-3| <= 0

Absolute values CANNOT have negative results - the result is ALWAYS 0 or a positive, so this TEST has JUST ONE solution...
X = 3

IF....
Y = 1
|X-3| <= - 1 which is NOT POSSIBLE.

From the prompt, we know that Y >= 0, so choosing a positive value for Y will NOT fit the absolute value given in Fact 2. This means that the ONLY possible value for Y is 0. By extension, there is ONLY ONE possible value for X....X = 3.
Fact 2 is SUFFICIENT

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Re: If y>=0, What is the value of x?  [#permalink]

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Bunuel wrote:

There is a following problem with your solution:
If $$x<3$$ --> $$-(x-3)\geq{y}$$ --> $$3-y\geq{x}$$;
OR:
If $$x\geq{3}$$ --> $$(x-3)\geq{y}$$ --> $$x\geq{3+y}$$;

But you can not combine these inequalities and write: $$3+y\leq{x}\leq{3-y}$$ as they are OR scenarios not AND scenarios (meaning that depending on the value of x we'll have either the first one or the second one).

Also discussed here: if-y-geq-0-what-is-the-value-of-x-1-x-3-geq-y-91640.html

Hope it helps..

Hi Bunuel,

I understand the OR scenario you mentioned above. However, what if the statement reads $$|x-3|\leq{y}$$? Then we can have $$3-y\leq{x}\leq{3+y}$$, can't we? So here we have the AND scenario. Is that right?

And if I am correct, how can we solve the above inequalities, given that $$y\geq{0}$$?

Thank you very much!
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Re: If y>=0, What is the value of x?  [#permalink]

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truongynhi wrote:
Bunuel wrote:

There is a following problem with your solution:
If $$x<3$$ --> $$-(x-3)\geq{y}$$ --> $$3-y\geq{x}$$;
OR:
If $$x\geq{3}$$ --> $$(x-3)\geq{y}$$ --> $$x\geq{3+y}$$;

But you can not combine these inequalities and write: $$3+y\leq{x}\leq{3-y}$$ as they are OR scenarios not AND scenarios (meaning that depending on the value of x we'll have either the first one or the second one).

Also discussed here: if-y-geq-0-what-is-the-value-of-x-1-x-3-geq-y-91640.html

Hope it helps..

Hi Bunuel,

I understand the OR scenario you mentioned above. However, what if the statement reads $$|x-3|\leq{y}$$? Then we can have $$3-y\leq{x}\leq{3+y}$$, can't we? So here we have the AND scenario. Is that right?

And if I am correct, how can we solve the above inequalities, given that $$y\geq{0}$$?

Thank you very much!

Hi,
this too is OR scenario,
because $$3-y\leq{x}$$ and $${x}\leq{3+y}$$ are dependent on different set of values of x..
$$3-y\leq{x}$$ is when $$x<3$$..
and $${x}\leq{3+y}$$ is when $$x\geq{3}$$..
Example of AND is
when we have two eqs in x, and not dependent on each other..
say x<3.. and x+2>1..
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Re: If y>=0, What is the value of x?  [#permalink]

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Hi chetan2u,

Thank you for the prompt reply. But I still have a doubt.

Take the inequation $$x^2<4$$ for example. We then have $$|x|<2$$, which means $$-2<x<2$$. What I understand is that x must be greater than -2 AND less than 2 for the inequalitiy to hold. So I think this is an AND scenario.

If, however, $$x^2>4$$, then $$x<-2$$ OR $$x>2$$. This is clearly an OR scenario. Here OR makes sense to me.

My inequality skill is pretty rusty. Thank you for bearing with me. I very appreciate your help!
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Re: If y>=0, What is the value of x?  [#permalink]

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truongynhi wrote:
Hi chetan2u,

Thank you for the prompt reply. But I still have a doubt.

Take the inequation $$x^2<4$$ for example. We then have $$|x|<2$$, which means $$-2<x<2$$. What I understand is that x must be greater than -2 AND less than 2 for the inequalitiy to hold. So I think this is an AND scenario.

If, however, $$x^2>4$$, then $$x<-2$$ OR $$x>2$$. This is clearly an OR scenario. Here OR makes sense to me.

My inequality skill is pretty rusty. Thank you for bearing with me. I very appreciate your help!

Hi truongynhi,

I am happy to help you and clear a few doubts you have..

WHAt does OR and AND mean..

1) Take the inequation $$x^2<4$$ for example. We then have $$|x|<2$$, which means $$-2<x<2$$
So here too you had 2 inequalities, x<2 and x>-2..
x<2 can mean x is -3 so this is a solution when we are using OR since we are not looking at both together..
But here x<2 and x>-2 has a range which OVERLAPS, so this is the combined solution for two inequalities..
when you are choosing a value in this range, you are using AND, since taht value will satisfy both the inequalities..

2)If, however, $$x^2>4$$, then $$x<-2$$ OR $$x>2$$. This is clearly an OR scenario.
YES, this is OR situation, because there is no overlap and hence there is no possiblity of a combined solution..
any solution will satisfy just one inequality..

Now when you have two variable as in the case of Q mentioned here..
3+y≤x≤3−y.. you cannot take this as a solution..
WHY..
because the two inequalities you have combined have come in two different scenarios of OR while taking value of x..
there is no OVERLAP in values of x.. x<3 for one inequality and x>= 3 for other..
so don't combine the two..

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Re: If y>=0, What is the value of x?  [#permalink]

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Apex231 wrote:
If y >= 0, What is the value of x?

(1) |x-3| >= y
(2) |x-3| <= -y

Statement 1 is not sufficient, since if y=0, x can be anything.

In Statement 2, the left side is 0 or greater, since it is an absolute value. The right side is 0 or smaller, since it is the negative of y, which is at least zero. If the left side (which is > 0) is no bigger than the right side (which is < 0), the only possibility is that they are both exactly equal to zero. And that only happens if x=3.
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Re: If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

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ugimba wrote:
If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$

(2) $$|x - 3|\leq{-y}$$

If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$
$$|x - 3|\geq{y}\geq{0}$$
$$|x-3| \geq 0$$
NOT SUFFICIENT

(2) $$|x - 3|\leq{-y}$$
$$|x - 3|\leq{-y} \leq 0$$
$$|x-3| \leq 0$$
|x-3| = 0
x = 3
SUFFICIENT

IMO B
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Re: If y is greater than or equal to 0, what is the value of x?  [#permalink]

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