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If x and y are integers such that \(x<0<y\),and z is non negative integer then which of the following must be true?

A) \(x^2<y^2\)

B) \(x+y=0\)

C) \(xz<yz\)

D)\(xz=yz\)

E) \(\frac{x}{y}<z\)

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

Given: \(x<0<y\) and \({0}\leq{z}\).

Evaluate each option:

A. \(x^2<y^2\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

B. \(x+y=0\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

C. \(xz<yz\) --> not necessarily true, if \(z=0\) then \(xz=yz=0\);

D. \(xz=yz\) --> not necessarily true, it's true only for \(z=0\);

E. \(\frac{x}{y}<z\) --> as \(x<0<y\) then \(\frac{x}{y}=\frac{negative}{positive}=negative<0\) and as \({0}\leq{z}\) then \(\frac{x}{y}<0\leq{z}\) --> always true.

Re: PS-which of the following must be true [#permalink]

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22 May 2012, 01:19

Bunuel wrote:

LM wrote:

If x and y are integers such that \(x<0<y\),and z is non negative integer then which of the following must be true?

A) \(x^2<y^2\)

B) \(x+y=0\)

C) \(xz<yz\)

D)\(xz=yz\)

E) \(\frac{x}{y}<z\)

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

Given: \(x<0<y\) and \({0}\leq{z}\).

Evaluate each option: A) \(x^2<y^2\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

B) \(x+y=0\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

C) \(xz<yz\) --> not necessarily true, if \(z=0\) then \(xy=yz=0\);

D)\(xz=yz\) --> not necessarily true, it's true only for \(z=0\);

E) \(\frac{x}{y}<z\) --> as \(x<0<y\) then \(\frac{x}{y}=\frac{negative}{positive}=negative<0\) and as \({0}\leq{z}\) then \(\frac{x}{y}<0\leq{z}\) --> always true.

Answer: E.

amazing ! couldn't figure out how option 3 was not necessarily true , forgot that non negative could mean that 0 is possible ,folks : non negative does not mean only positive integers , it could be 0 as well

Hypothetically speaking, Bunuel so if a question says, non positive numbers can we consider 0 as well , rather than only negative numbers.

Set of Non positive numbers { 0,-1,-5,-9 } Set of Non negative numbers { 0,1, 4, 7, }

If x and y are integers such that \(x<0<y\),and z is non negative integer then which of the following must be true?

A) \(x^2<y^2\)

B) \(x+y=0\)

C) \(xz<yz\)

D)\(xz=yz\)

E) \(\frac{x}{y}<z\)

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

Given: \(x<0<y\) and \({0}\leq{z}\).

Evaluate each option: A) \(x^2<y^2\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

B) \(x+y=0\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

C) \(xz<yz\) --> not necessarily true, if \(z=0\) then \(xy=yz=0\);

D)\(xz=yz\) --> not necessarily true, it's true only for \(z=0\);

E) \(\frac{x}{y}<z\) --> as \(x<0<y\) then \(\frac{x}{y}=\frac{negative}{positive}=negative<0\) and as \({0}\leq{z}\) then \(\frac{x}{y}<0\leq{z}\) --> always true.

Answer: E.

amazing ! couldn't figure out how option 3 was not necessarily true , forgot that non negative could mean that 0 is possible ,folks : non negative does not mean only positive integers , it could be 0 as well

Hypothetically speaking, Bunuel so if a question says, non positive numbers can we consider 0 as well , rather than only negative numbers.

Set of Non positive numbers { 0,-1,-5,-9 } Set of Non negative numbers { 0,1, 4, 7, }

is this correct?

Yes, a set of non-positive numbers consists of zero and negative numbers.
_________________

Re: PS-which of the following must be true [#permalink]

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24 May 2012, 02:08

Bunuel wrote:

Yes, a set of non-positive numbers consists of zero and negative numbers.

Isn't that one of the first few things one gets to learn when trying to read the number system. Thank you bunnel for reminding everyone about it
_________________

Re: If x and y are integers such that x<0<y, and z is non [#permalink]

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25 May 2012, 22:34

So 0 "Zero" is even and can be part of both a NON Postive set and a NON negative set Set of Non positive numbers { 0,-1,-5,-9 } Set of Non negative numbers { 0,1, 4, 7, }

Re: If x and y are integers such that x<0<y, and z is non [#permalink]

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08 Apr 2015, 07:37

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