Is |x| = y - z ? (1) x + y = z (2) x

Question

Is |x| = y - z ?

(1) x + y = z 
(2) x < 0

Bunuel · Accepted Answer

Is |x| = y - z ?

For a start, take into account that for the equation |x| = y - z to hold true, y - z must be greater than or equal to 0. This is due to the fact that it's equated to the absolute value of a number, |x|, which cannot be negative. In essence, the question is asking whether y - z is non-negative and if their difference equals |x|.

(1) -x = y - z.

If x > 0, -x would be negative, which would imply that y - z is also negative. This doesn't meet our initial criteria. For instance, if x = 1 and y - z = -1, then (|x| = 1) is not equal to (y - z = -1).

If x ≤ 0, -x would be non-negative, which would imply y - z is also non-negative, a valid scenario. For instance, if x = -1 and y - z = 1, then (|x| = 1) is equal to (y - z = 1).

This gives us two possible outcomes and does not definitively answer our question. Not sufficient.

(2)  x<0 .

This condition alone is not sufficient because it only gives information about x. We still need to know whether the value of y - z equals |x|.

(1) + (2) Given that x < 0, |x| = -x, and so our original question becomes: is -x = y - z? The second statement gives a direct YES answer to this question. Sufficient.

Answer: C.

sudhir18n · Answer

|X|= y-z means; X>0; X= Y-Z X<0; -X= Y-Z ; X= Z-Y St1: X= Z-Y We dont know if X>0 or <0 ............ insufficient St2: X<0 ; when dont know about Y and Z ......... insufficient 1&2 : we know X<0 and X= Z-Y hence sufficient

DenisSh · Answer

Brilliant, thank you! :^)

dispatch3d · Answer

is |x|=y-z?

given:
(1) x+y=z
(2) x<0

solving (1) first:

y=z-x
|x|=(z-x)-z
|x|=-x

take x=1, z=2, y=1
1=1-2 (no)
take x=-1, z=2, y=3
|x|=y-z?
|-1|=3-2=1 YES

so what solving for |x|=-x meant was that x MUST be negative for the equation to be true, if it is positive then it is not true (since in that case, |x| would not equal -x).

hence the answer is C.

EvaJager · Answer

(1) Can be rewritten as X = -Y + Z, so |X| = |-Y + Z|, which would be equal to Y - Z, if and only if  -Y+Z\leq0 . Obviously, we don't know that, so (1) insufficient.
(2) Cannot be sufficient, it doesn't say anything about Y and Z.
(1) and (2) together: X = -Y + Z < 0, therefore |X| = Y - Z, sufficient.

Answer: C

fameatop · Answer

Hi bunuel, 
I am not able to understand the solution for this problem. Can you kindly explain the highlighted areas.
 Note that y-z must be \geq{0} , because absolute value (in our case |x|) can not be negative.

Generally question asks whether y-z\geq{0} and whether the difference between them equals to |x|.

(1) -x=y-z
if x>0 -->  y-z is negative  --> no good for us;
if  x\leq{0} --> y-z is positive  --> good.

Waiting for reply.

Bunuel · Answer

Look at  |x|=y-z : the left hand side is absolute value (|x|), which cannot be negative, hence the right hand side (y-z) also cannot be negative. Therefore must be true that  y-z\geq{0} .

Next, for (1) given that  -x=y-z . Now, if  x>0 , or if  x  is positive, then we'll have that  -positive =y-z  -->  negative=y-z . But as we concluded above  y-z  cannot be negative, hence this scenario is not good.

Hope it's clear.

moulayabdesslam · Answer

Answer is C. From the question, if x>0, x = y-z if x<0, x = -(y-z) = z-y Let's start with the easy statement - Statement 2. (2): Insufficient x<0 doesn't give us any information about y and z. Eliminate B and D (1): Insufficient Statement 1 says that x = z - y If x<0 , then the answer is yes since |x| will be equal to y-z if x>0, then the answer is no since |x| = z-y We have a Yes and a No. Eliminate A. (1) and (2) together: Sufficient We have x<0 and x = z - y. hence, |x| = y-z -- Need 3 Kudos

bebs · Answer

|x| = y-z? ==> y-z>0? (since |x| > 0) ==> y>z?

St 1: x + y = z ==> Not sufficient since we do not know the signs

St 2: x<0 ==> Not sufficient since we are not told anything about z and y

Sts 1 and 2 together ==> From St 1 we see that x= z-y and from St 2 x<0 ==> z-y<0 ==> z<y ==> sufficient since it answers the question. Therefore the answer is C

romen2017 · Answer

|x| is always positive so the question asks us is the (y-z) a positive magnitude of X.
1. Now with stm-1 y-z=-x , this could be positive value of x or negative value of x depending on the value of x. Yes and No, so Not sufficient.
2. Stm -2 X< 0, what about the remaining terms - not sufficient.

1+2, X<0 so -X>0, so (y-z)= positive x = |x|, which is always positive.

Ans- C

RenB · Answer

Bunuel   KarishmaB 
I solved it this way. Let me know if there are any conceptual gaps in this method:
Qs: |x| = y-z
Thus qs is : Is -x=z-y or x=y-z

St 1: 
x+y=z
x=z-y
This is neither of the equations: -x=z-y or x=y-z
Not sufficient

St 2:
Clearly Insufficient

St: 1+2
x is negative. Substituting x=-x in x=z-y
-x=z-y
One of the derived question equations
Thus sufficient

I am a bit unsure about the substitution. Looking forward to your response for more clarity.
Thanks!

KarishmaB · Answer

Not Correct. 
Question: Is |x| = y - z
means
Is x = y - z if x is positive or is -x = y - z if x is negative.

We use the definition of absolute values to replace |x| by some expression in x. There is no multiplication by -1 on both sides. 
I suggest you to check out the absolute values section of some test prep company content.

shardsingh1134527 · Answer

We have :
Is |x| = y - z ?

(1) x + y = z
(2) x < 0

if x >0 then inequality will be x= y-z ; if x < 0 then inequality will be x= z-y 
1. x= z-y we can not say for sure if the condition hold
2. x <0 then again we can not sure for sure
combining both x < 0 then x = z-y which is what we are getting above in the inequality.
Hence C is the correct option.

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Is |x| = y - z ? (1) x + y = z (2) x < 0

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