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If r, s, and t are integers such that r^2 > s^2 > t^2, which of the

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Bunuel
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If r, s, and t are integers such that r^2 > s^2 > t^2, which of the [#permalink] New post   Updated on: 05 May 2017, 03:57
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If r, s, and t are integers such that r^2 > s^2 > t^2, which of the following must be true?

I. r > s
II. |r| > |s|
III. (t + 1)^2 ≥ r^2

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III
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A

Originally posted by Bunuel on 30 Jan 2017, 10:46.
Last edited by Bunuel on 05 May 2017, 03:57, edited 1 time in total.
Edited the question.
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ankujgupta
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Re: If r, s, and t are integers such that r^2 > s^2 > t^2, which of the [#permalink] New post   30 Jan 2017, 11:12
Bunuel wrote:
If r, s, and t are integers such that r^2 > s^2 > t^2, which of the following must be true?

I. r > s
II. | r | > | s |
III. (t + 1)2 ≥ r2

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III

Case 1 r=-5, s = -4, t=-3 r2> s2> t2 r<s so I is out
But II will always be true.
III is also negated with this example. So, only II is correct, thus A.
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Abhishek009
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Re: If r, s, and t are integers such that r^2 > s^2 > t^2, which of the [#permalink] New post   30 Jan 2017, 11:27
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Bunuel wrote:
If r, s, and t are integers such that r^2 > s^2 > t^2, which of the following must be true?

I. r > s
II. | r | > | s |
III. (t + 1)2 ≥ r2

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III


\(r^2 > s^2 > t^2\)

Let \(r = 5\) , \(s = 4\) & \(t = 3\)

So, \(r^2 > s^2 > t^2\) = 25 > 16 > 9

Again , \(r^2 > s^2 > t^2\)

Let \(r = -5\) , \(s = -4\) & \(t = -3\)

So, \(r^2 > s^2 > t^2\) = 25 > 16 > 9


Now, check the options -

I. \(r > s\) - Not true if \(r = -5\) & \(s = -4\)

II. | r | > | s | - True for both +ve and -ve values of r & s

III. \((t + 1)^2 ≥ r^2\) - Not true


Hence, answer will be (A) only II.

PS : Assuming \((t + 1)2 ≥ r2\) as \((t + 1)^2 ≥ r^2\)
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Re: If r, s, and t are integers such that r^2 > s^2 > t^2, which of the [#permalink] New post   30 Jan 2017, 12:48
What does the third case III. (t + 1)2 ≥ r2 mean?

2(t+1) >= 2r or (t+1)^2 >= r^2?
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Re: If r, s, and t are integers such that r^2 > s^2 > t^2, which of the [#permalink] New post   20 May 2017, 02:24
Abhishek009 wrote:
Bunuel wrote:
If r, s, and t are integers such that r^2 > s^2 > t^2, which of the following must be true?

I. r > s
II. | r | > | s |
III. (t + 1)2 ≥ r2

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III


\(r^2 > s^2 > t^2\)

Let \(r = 5\) , \(s = 4\) & \(t = 3\)

So, \(r^2 > s^2 > t^2\) = 25 > 16 > 9

Again , \(r^2 > s^2 > t^2\)

Let \(r = -5\) , \(s = -4\) & \(t = -3\)

So, \(r^2 > s^2 > t^2\) = 25 > 16 > 9


Now, check the options -

I. \(r > s\) - Not true if \(r = -5\) & \(s = -4\)

II. | r | > | s | - True for both +ve and -ve values of r & s

III. \((t + 1)^2 ≥ r^2\) - Not true


Hence, answer will be (A) only II.

PS : Assuming \((t + 1)2 ≥ r2\) as \((t + 1)^2 ≥ r^2\)



I understood the solution.Can you please help me with my below dought.

I really find it difficult to figure out when to apply the mod properties? Like in this case when we are asked that |r|>|s|, then it holds true only if r and s are positive. In case they are negative then |r| equals -r and |s| equals -s, so -r<-s. Is this not the case?

I understand by mod we mean only positive values. But i get confuse whenever i see |x| equals -x,x<0. I do not understand when to consider only positive values and when to consider |x| equals -x,x<0 in case of mod.

It will be really helpful if you can help :? :?
PS: I have read the theory also.
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Re: If r, s, and t are integers such that r^2 > s^2 > t^2, which of the [#permalink]
20 May 2017, 02:24
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