If y=0, What is the value of x? (1) |x-3|=y (2) |x-3|

Question

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

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

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

Bunuel · Accepted Answer

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 . Answer: B. Hope it helps..

gurpreetsingh · Answer

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

Observer · Answer

https://i.imgur.com/q5wkB1a.jpg 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.

WholeLottaLove · Answer

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 :lol:

)

Thanks!

) Thanks!

pqhai · Answer

Hi WholeLottaLove The important property concerning absolute value inequalities is: |a| <= b <--> -b <= a <=b 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.

EMPOWERgmatRichC · Answer

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

Final Answer:

Show Spoiler

B

GMAT assassins aren't born, they're made,
Rich

truongynhi · Answer

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!

chetan2u · Answer

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..

truongynhi · Answer

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!

chetan2u · Answer

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 -2-2.. 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..

IanStewart · Answer

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.

Kinshook · Answer

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

TheBipedalHorse · Answer

If y≥0, what is the value of x?

(1) |x−3|≥y
(2) |x−3|≤−y

As per Statement 1 
if y ≥0, then |x-3| ≥ 0, which is true for all real numbers. Once you mod an expression, it will always be non negative (i.e ≥ 0), so all real numbers satisfy this equation, and we do not have a specific value of x. Therefore statement 1 is insufficient

As per statement 2
if y≥0, then −y≤0, then |x-3|≤0, Once you mod an expression, it will always be non negative (either 0 or positive), which throws out any possibility of |x-3|<0, 
So all we need to find is the value of x for which x-3=0, which points towards a unique solution, therefore statement 2 by itself is sufficient

KarishmaB · Answer

Given : y is non negative i.e. either it is 0 or positive. Asked : What is x? (1) |x - 3|\geq{y} An absolute value is always non negative. If y is 0, |x - 3| >= 0 means all real values of x are acceptable. So we can't get a unique value of x. Let's move on. We don't even need to think about what happens when y is a positive number. Not sufficient alone. (2) |x - 3|\leq{-y} Since y is non-negative, -y becomes non positive i.e. it can be 0 or negative. An absolute value cannot be less than any negative value or 0. It can only be equal to 0. So only possibility is |x - 3| = 0 Hence x must be 3. Sufficient Answer (B) Here is a post on absolute values that explains these concepts in detail: https://anaprep.com/algebra-the-why-beh ... questions/

manan01 · Answer

In my first try I got confused, but after thinking that GMAT is a more logic based rather than algebraic exam I was able to do this question under 2 mins.

We're given y>=0, y is 0 or a positive number, Now S1 states |x-3| >= ( positive number ), obviously a mod number would always be positive so x can take multipe values, depending on y's value. hence  Insufficient

In S2) this is where logic helps, |x-3|<= ( a negative number or 0 ), how can a mod number be less than -ve number, it has to be 0 or a positive number!! Hence y is 0 and basis that clearly x has to be 3 for this condition to satisfy.  Sufficient

If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y

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