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If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y
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21 Jan 2012, 09:23
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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 \(0\).
If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y
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09 Jul 2013, 14:05
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.
(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 )
(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]
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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Re: If y >= 0, what is the value of x?
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16 May 2015, 11:43
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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
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).
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 \(y\geq{0}\)?
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).
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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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!
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!
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.. _________________
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 is greater than or equal to 0, what is the value of x?
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