Is |x| = y + z?

Question:
Is |x| = y + z?
We don't know what to do when we have absolute values in the equation so we should remove it.
The question is this:
If x >= 0, is x = y + z?
and if x < 0, is -x = y + z?

(1) x + y = z
We don't know whether x is positive/zero or negative so for now we don't know what we have to answer actually (which equation we have to consider).
Not sufficient.

(2) x < 0
So our question is: Is -x = y + z?
But this statement alone doesn't even mention y and z. Not sufficient.

Using both, x is negative and we need to answer whether -x = y + z i.e. whether x = -y - z?
We know that x + y = z (statement 1) so we know that x = -y + z.
But is x = -y - z? We cannot say. If z = 0, then x could be equal to -y - z. Else not.

So even after using both statements we cannot answer our question.
Not sufficient.

Answer (E)

|x| = y + x

(A) x = y + z AND/IF x > 0
(B) x = - ( y + z ) AND/IF x <0

I don't get Statement (1)

Shouldn't it be x = - y - z ??

But this is insufficient because we have to have the condition that x > 0

Statement (2)

x < 0 <-- insufficient (this is the condition we are looking for in statement (1))

Clearly, Statement (1) and Statement (2) = (C)

Statement 1, x=-y-z is same as -x=y+z. and this implies x<0 and hence the statement should be sufficient. This is where I got lost.

Yeah I also got this wrong the first time because I thought Statement (1) was already sufficient. However, we need Statement (2).

Did you copy this problem down correctly? -x=y+z =/=> x+y=z.

This is where you're incorrect. -x=y+z alone does not imply that x<0 without the condition that y+z >0.

Hi ,

Though it looks reasonable , I am not sure on what is wrong with this logic.



1. -x = y +z ==> x+y=z (This is 1. and satisfies the question with different values of y and z for a negative x)
x=y+z ==> is not the same as given equation.
So A is sufficient.

given that x = y+z or -x=Y+z

so 1. x+Y = z..... not sufficient
2. it does not tell everything either... so not sufficient..

together.... -x+y = z

-x = -y+z, hence not same as given in the question, hence the answer to the main question is NO and with C option we are answering it.

Answer is E. Plugged in a bunch of numbers.

1) x=3, y=4, z=7 gives us the answer no, but x= -5, y=3, z=2 gives us yes. Insufficient

2) x<0 insufficient since y and z are unknown

Put them together, x=-5, y=3, z=2 gives us yes, but x=-5, y=-7, z=-12 gives us no. Insufficient.

Hi Bunuel!
I have a doubt.
|x|= y+z gives us two equations..
X= y+ z or -x= y+z
Statement one says x+y=z .. It is not possible to get the above mentioned statements with this equation.
Statement two says x<0 ..this should be sufficient to answer right?

Similarly. In the question |x|= y-z.
We can x= y-z and -x= y-z
Statement one by substituting we get -x= y-z so this should be suffice to right?

Not sure I understand your logic there...

Number plugging proving that E is the answer: is-x-y-z-133977.html#p1094912
Algebraic approach proving the same: is-x-y-z-133977.html#p1114409

Hope this helps.

In the question |x|= y-z.
We get two equations I.e.(removing the modulus . x= y-z and -x= y-z
Statement one says x+z =y
Therefore by substituting we get -x= y-z. so this should be suffice to answer right?
As the question stem also has the same equation.

Posted from my mobile device

Since the correct answer is E, then this is obviously not right.

If \(x\geq{0}\), the questions asks: is \(x=y+z\)?
If \(x<{0}\), the questions asks: is \(-x=y+z\)?

When we combine the statements, since it's given that x<0, the questions becomes: is \(-x=y+z\). From (1) we have that \(-x=y-z\), which is not sufficient to get whether \(-x=y+z\).

Consider the examples given here: https://gmatclub.com/forum/is-x-y-z-133977.html#p1094912

(I think the question could have been better stated.)

My answer is 'E'.

statement-1: Insufficient.
rewrite x+y=z as x=z-y.
If (z=0 and y=0, thus z+y=0 and x=z-y=0) then |x|=z+y. But, if (z=2 and y=1, thus z+y=3 and x=z-y=1) then |x|\(\neq\)z+y.

statement-2: Insufficient as it doesn't give any info on y and z. Example, x=-3, z=-1, y=-2, thus |x|\(\neq\)z+y OR x=-3, z=1, y=2, thus |x|=z+y.

1+2: Insufficient.
Here, we know that, x=z-y and x<0,
when z=1 and y=2 then x = z-y=-1, and |x|\(\neq\)=y+z
but when z=0, y=1 then x=z-y=-1, and |x| = y+z

The way I approached this question was this:

since abs(x)>=0, the question is asking is y+z>=0

Statement 1 says that Y=Z-X. This means that Y+Z=2Z-X. We don't have specific values for Z or X so not sufficient

Statement 2 says x>0. This alone means nothing. Insufficient.

Together x>0 and the question is whether 2Z-X>0. We have no indication of the direction of Z, so insufficient together.

For statement (1) we derive that

x+y = z

-x = y - z

When x is negative, y-z>0 and when x is positive, y-z<0 .. Knowing that y-z would be greater or less than 0 is insufficient to give us a similar indication for y+z ..

My question is whether this is a correct understanding of the information and can it be universally applied for two variables getting subtracted and added?

Hi TheMastermind ,

Yeah, this is another way to approach such questions.

Since you don't have any information about the sign of (y-z), you cannot conclude anything for y+z.

This is universally accepted concept and is not specific to this question.