What is the distance between x and y on the number line?

Question

(1) |x| – |y| = 5
(2) |x| + |y| = 11

Bunuel · Accepted Answer

What is the distance between x and y on the number line?

Question: |x-y|=?

(1) |x| – |y| = 5. Not sufficient: consider x=10, y=5 and x=10, y=-5.
(2) |x| + |y| = 11. Not sufficient: consider x=10, y=1 and x=10, y=-1.

(1)+(2) Solve the system of equation for |x| and |y|: sum two equations to get 2|x|=16 --> |x|=8 --> |y|=3. Still not sufficient to get the single numerical value of |x-y|, for example consider: x=8, y=3 and x=8, y=-3. Not sufficient.

Answer: E.

Abhii46 · Answer

Solving the two equations will give x as 8 and y as 3. But since mod sign is there, x and y can take any value, either positive or negative. Hence, both the statements are insufficient.

PrashantPonde · Answer

Bunuel can you clarify what can be wrong below approach?

Answer choice can be (C)
-------------------------------------
By multiplying statements 1 & 2
(1) |x| – |y| = 5
(2) |x| + |y| = 11

X^2-Y^2=55

i.e.  (x+y)(x-y)=55 = 11 * 5 = - 11 * - 5  (i.e. both factors are either positive or negative)

Hence only two possible solutions for this – i.e. either (x=8 & y=3) OR (x= -8 & y= -3)
In both cases the distance between them is 5.

-> Hence Answer is C.

Bunuel · Answer

(x+y)(x-y)=55 does not mean that either (x=8 & y=3) OR (x= -8 & y= -3). There are more integer solutions (for example x=+/-28 and y=+/-27) and infinitely many non-integer solutions.

PrashantPonde · Answer

Oh I see...That explains it! Thanks for your reply Bunuel!

Bunuel · Answer

From 100 hardest questions 
Bumping for review and further discussion.

WholeLottaLove · Answer

What is the distance between x and y on the number line?

(1) |x| – |y| = 5

|11|-|6|=5
Distance is five

|11|-|-6|=5
Distance is seventeen
INSUFFICIENT

(2) |x| + |y| = 11

|5|+|6| = 11
Distance is one

|5|+|-6| = 11
Distance is negative eleven

INSUFFICIENT

This problem, to me, seems much easier than a 700 level question. a and b provide us with multiple valid values for x and y, none of which entirely (i.e. are the same) Can someone tell me if I am oversimplifying this problem? Thanks!

mvictor · Answer

i picked 8 and 3
1. x=8, y=3 -> 5 or x=-8, y=3 -> distance is 11. 1 alone is insufficient.
2. x=8, y=3 -> 5 or x=-8, y=3 -> distance is 11. 2 alone is insufficient.

1+2
same info from 1 and 2. C is out, and the answer must be E.

enasni · Answer

Hi... I have a question when we solve both the eqs tog we get two values for |y|=3 and -13. Right?

Bunuel · Answer

|y| is an absolute value of y, so it cannot be negative. When we solve for |y|, we get that |y| = 3, so y = 3, or y = -3.

Hope it's clear.

mbaapp1234 · Answer

Picking numbers works well here. The trick is to realize that you can simply make either x or y negative to install a new case to test for sufficiency.

GOAL: What is the distance between x and y? Must be one discrete value, not multiple.

Statement 1: Pick 11 and 6. 11-6 = 5, so this case matches the given information and the distance is 5. Now you can make either 11 or 6 negative, so set x = -11. Now abs(-11) - 6 = 5, but the distance is 17 units. So this case is not sufficient because we have multiple possible distances on the number line.

Statement 2: Here abs(x) + abs(y) = 11. Pick 5 and 6, the most obvious choices. The distance between them is 1. Now turn 5 into -5, and the statement is still valid (abs(-5) + abs(6) =11, but their distance is now 11. Not sufficient.

Combined they are not sufficient. You can test via 8 = x, 3 = y and then -8 = x, 3=y.

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Condition (1)
In the case that  x = 6 ,  y = 1 , the distance  x  and  y ,  | x - y | = 5 .
In the case that  x = 6 ,  y = -1 , the distance  x  and  y ,  | x - y | = 7 .
Thus we don't have a unique solution.

Condition (2)
In the case that  x = 6 ,  y = 5 , the distance  x  and  y ,  | x - y | = 1 .
In the case that  x = 6 ,  y = -5 , the distance  x  and  y ,  | x - y | = 11 .

Thus we don't have a unique solution.

Condition (1) & (2)
If we add two equation, we have  2|x| =16  or  |x| = 8 . Thus  x = \pm 8 .
If we subtract the first equation from the second one, we have  2|y| =6  or  |y| = 3 . Thus  y = {\pm}3 .

In the case that  x = 8  and  y = 3 , the distance  x  and  y ,  | x - y | = 5 .
In the case that  x = 8  and  y = -3 , the distance  x  and  y ,  | x - y | = 11 .
Thus we don't have a unique solution.

Therefore the answer is E.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using 1) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

BrentGMATPrepNow · Answer

Target question :    What is the distance between x and y on the number line?

When I scan the two statements, they both feel insufficient, AND I’m pretty sure I can identify some cases with conflicting answers to the  target question . So, I’m going to head straight to……

Statements 1 and 2 combined    
Statement 1 tells us that |x| – |y| = 5
Statement 2 tells us that |x| + |y| = 11
If we ADD the two given equations, we get: 2|x| = 16, which means  |x| = 8 
Similarly, if we SUBTRACT but the bottom equation from the top equation we get: -2|y| = -6, which means  |y| = 3

At this point, we can see that there are several possible pairs of values for x & y
Consider these two conflicting cases: 
 Case a : x = 8 and y = 3. In this case, the answer to the target question is   the distance between x and y is 5 
 Case b : x = 8 and y = -3. In this case, the answer to the target question is   the distance between x and y is 11 
Since we can’t answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer: E

tmx426 · Answer

In |x|-|y|=5 where x=8, does it matter if Y= 3 or -3 because due to the mod sign, shouldn't it be interpreted as (10-5), irrespective of the sign??

Sorry if the question is basic, i am very new to this.

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What is the distance between x and y on the number line?

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