Is |x-z| + |x| = |z|?

Question

(1) zy < xy < 0

(2) y > 0

Is there a faster way other than trying out different positive / negative cases for 1?

Bunuel · Answer

Is |x-z| + |x| = |z|? (1) zy < xy < 0. Consider two cases: (i) If y > 0, then we'd get: z < x < 0. z < x means that x - z > 0, thus |x - z| = x - z; x < 0 means that |x| = -x; z < 0 means that |z| = -z. Therefore, LHS = |x - z| + |x| = (x - z) - x = -z and RHS = |z| = -z --> -z = -z. (ii) If y < 0, then we'd get: z > x > 0 (flip the sign when reducing by negative value). z > x means that x - z < 0, thus |x - z| = -(x - z); x > 0 means that |x| = x; z > 0 means that |z| = z. Therefore, LHS = |x - z| + |x| = -(x - z) + x = z and RHS = |z| = z --> z = z. Since in both case we got an YES answer to the question, then this statement is sufficient. (2) y > 0. Clearly insufficient. Answer: A. Hope it's clear.

GMATinsight · Answer

|x-z| + |x| = |z| will always be true when either z < x < 0 or z > x > 0 or when x=0 Statement 1 : zy < xy < 0 This will be true only when y is positive and z < x < 0 or This will be true only when y is Negative and z > x > 0 Hence, SUFFICIENT Statement 2 : y > 0 Nothing Can be predicted about x and z hence NOT SUFFICIENT Answer: option A

GMATinsight · Answer

Bunuel : This question is available elsewhere on GMAT CLUB Forum

https://gmatclub.com/forum/if-zy-xy-0-is-x-z-x-z-123830.html

Bunuel · Answer

The question you are referring to is similar but not the same as this one. That question is flawed old GMAT Prep question: if-zy-xy-0-is-x-z-x-z-101210.html

arnabs · Answer

bunuel,
in case the first statement had been zy<xy onlly, then A would have been clearly insufficient, correct??

Bunuel · Answer

Yes, in this case the statement would not be sufficient.

sahil7389 · Answer

For Absolute/ mod questions i try to go by number line method...
So from question |x-z| + |x| = |z|.. Distance between x and z + distance between x and zero = distance between z and zero
that is only possible if on number line.....z...x...0 or 0...x....z....
From statement 1--->zy<xy<0 if y >0 the z<x<0 ok as per negative side... For Y<0 z>x>0 reversing sign of inequality as we are multiplying with negative no-----for positive side of number line.-- So A is sufficient.
From statement 2-> y>0 doesn't give any relation between x and z.
So ans is A. Bunuel M I Correct?

rohit8865 · Answer

i m having a doubt that for a YES as answer z & x have same sign and z should be greater than x.
but if Y>0 then z is < x.....

plz clear me ....

thanks

ENGRTOMBA2018 · Answer

Not necessarily.

If x=2 and z=3, you do get a YES and also when you have z=-3 and x=-2

Thus, for this question, you will get a yes for z<x<0 or 0<x<z.

Hope this helps.

rohit8865 · Answer

so, should i mean that i have to search only for xy>0 ?????
Becoz squaring both sides Ix-zI=IzI-IxI
i got IxzI-xz=0 soit means x and y have same sign..

Plz clear ...
Thanks

chetan2u · Answer

Hi,
you are correct 
 In the given equation zy<xy<0.. 
there are two cases..
1)as you say when y>0.. zand x -ive and z<x..
2)if y<0, z and x are +ive and zhas to be greater to make zy<xy

ENGRTOMBA2018 · Answer

You could have gotten to your final conclusion in a more structured and easier manner when you realize that from zy < xy < 0 ---> xy<0 --> x and y are of opposite signs and from yz<0 --> y and z are of opposite signs ---> x and z are of same sign. Ix-zI=IzI-IxI only means that you have to figure out if the distance of x from z = distance of z from 0 + distance of x from 0, when can we have this happen? When z

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