Is |x| = y + z?

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.

The answer to this question is E, not C.

Consider below 2 cases:
\(x=-1\), \(y=1\) and \(z=0\) --> \(|x|=1\) and \(y+z=1\) --> answer YES;
\(x=-1\), \(y=2\) and \(z=1\) --> \(|x|=1\) and \(y+z=3\) --> answer NO.

I think you refer to the following question:

Is \(|x|=y-z\)?

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 --> \(-x=y-z\)
if \(x>0\) --> \(y-z\) is negative --> no good for us;
if \(x\leq{0}\) --> \(y-z\) is positive --> good.
Two possible answers not sufficient;

(2) \(x<0\)
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Answer: C.

Hope it's clear.

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.

(1) \(x = y-z\). Then \(|x|=|y-z|\) which is either \(y-z\) or \(z-y.\)
In order to have \(y+z=y-z\), necessarily \(z=0\) and also \(y\geq0.\)
In order to have \(y+z=z-y\), necessarily \(y=0\) and also \(z\geq0.\)
Obviously not sufficient.

(2) Clearly not sufficient.

(1) and (2)
We are in the case \(x=y-z<0\) so \(|x|=z-y\). For \(|x|=y+z\) as seen above we need \(y=0\) and \(z\geq0.\) Neither condition is guaranteed.
Not sufficient.

Answer E

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.

Hi KarishmaB . Please help. I can't understand any of the solutions here. Thank you.