Q) Y >= 0,
What is the value of x?

1) |x-3| >= y

2) |x-3| <= -y


What is the best way to approach this problem?

Thank,
Champ

what is the OA?

OA is B

This is an annoying question. At first glance, I guessed the answer might be C because of the similarity between the statements ("hmm, maybe they overlap in agreement that y = 0...," I thought).

This is how I did it:

|x - 3| can be thought of as "x's distance from 3 on the number line." (Try a few x values, both above and below 3, to see why.)

So (1) tells us that x's distance from 3 on the number line is at least 0. Well, that just implies that x could be 3 or any number any distance away from 3. x = anything, so (1) is clearly insufficient (eliminate A and D). But since (1) tells us nothing, if (2) were insufficient and needed "help," we wouldn't get any help from (1). The answer can't be C.

So to decide between B and E, we look at (2).

The number line approach broke down for me (in a good way) when I realized that -y could be negative, but x's distance from 3 on the number line can't be negative.

|x-3|<=-y
-|x-3|>=y (from (2)) and y>=0 (from the question constraint)
-|x-3| can't be positive, but it can be 0.
Therefore, |x-3| = 0, or x = 3.

_________________
Emily Sledge
Instructor
ManhattanGMAT

Thanks Emily for clear explanation.

Hi Emily, this number line thought process has blown me away.
When i approched this problem it was like a modulus problem in the lines of the below -

for a.
|x-3| >= y
so we have 2 results -
x-3>=y and x-3<=-y and so on but got nowhere as there were two variables here.

could you also explain the method to get this done with the above though process ... or could you also explain the number line approach in more detail (maybe another another example) if thats the only way out.

This is how I approached and is very similar to Emily's but didn't consider the number line.

I think Stmt#1 is easy and everyone got the part that Stmt#1 is InSUFF
Moving onto Stmt #2

It's given that y>=0, i.e y is either 0 or a positive number
looking at Stmt#2, Mod of something <= -y
Mod of something is always positive or zero but never negative. If y is > 0, then mod of something can't be negative. ( b/c -y will be a negative number if y > 0)
Hence Y has to be 0. Mod of something =0, hence unique solution

OA:B

GOOOOODDDDD :)

sanidhya510, I'm not sure I quite grasped your question. If I understood it correctly, then kramacha1979's answer was exactly what you wanted. Post again otherwise.