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If x, y are non-zero integers and x/y < |x/y| < y, which of the

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mathhelp001
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If x, y are non-zero integers and x/y < |x/y| < y, which of the [#permalink] New post   Updated on: 26 Apr 2023, 23:56
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If x, y are non-zero integers and x/y < |x/y| < y, which of the following must be true?

I. x is negative
II. y is positive
III. x/y is negative

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II and III
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E

Originally posted by mathhelp001 on 26 Apr 2023, 20:23.
Last edited by Bunuel on 26 Apr 2023, 23:56, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: If x, y are non-zero integers and x/y < |x/y| < y, which of the [#permalink] New post   26 Apr 2023, 20:49
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This seems like a perfectly fine problem-solving question so I’m going to go ahead and move it to the PS sub forum 😇

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Re: If x, y are non-zero integers and x/y < |x/y| < y, which of the [#permalink] New post   27 Apr 2023, 00:12
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mathhelp001 wrote:
If x, y are non-zero integers and x/y < |x/y| < y, which of the following must be true?

I. x is negative
II. y is positive
III. x/y is negative

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II and III


When a < 0, then a < |a|. Hence, x/y < |x/y| implies that x/y must be negative. III must be true.

Since the absolute value of nonnegative number is positive, the |x/y| < y, means 0 < |x/y| < y, which implies that y is positive. II must be true.

If x/y is negative and y is positive, means x must be negative. I must be true.

Answer: E.
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Re: If x, y are non-zero integers and x/y < |x/y| < y, which of the [#permalink] New post   27 Apr 2023, 02:05
Asked: If x, y are non-zero integers and x/y < |x/y| < y, which of the following must be true?

x/y < |x/y|
One of x & y is negative and other positive

x/y < y
x/y - y < 0
(x- y^2)/y < 0

If x<0
x- y^2 < 0
y > 0
Feasible
x = -2 ; y = 2
x/y = -1
|x/y| = 1
y = 2
x/y < |x/y| < y

But if x>0
x - y^2 may be positive or negative
Case 1: x -y^2 > 0
x > y^2
x/y > y^2/y = y
Not feasible
Case 2: x -y^2 < 0
y > 0; Not feasible since x/y < |x/y|

x<0 & y>0 is only feasible case.

I. x is negative
II. y is positive
III. x/y is negative

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II and III

IMO E
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Re: If x, y are non-zero integers and x/y < |x/y| < y, which of the [#permalink] New post   27 Apr 2023, 03:35
when |x/y| < y ( a positive value can only be less then a positive value , therefore y is positive)

Also x/y< | x/y|
means | x/y| cannot be equal to x/y


Lastly, in x/y
we know y is positive so x can either be negative or positive
but if we take x to be positive ,it will make x/y = |x/y| ,which makes x negative.

therefore, Y is positive
X is negative
x/y is negative.
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Re: If x, y are non-zero integers and x/y < |x/y| < y, which of the [#permalink]
27 Apr 2023, 03:35
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