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If which x < y < 0, of the following inequalities must be tr

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If which x < y < 0, of the following inequalities must be tr [#permalink] New post 24 Jan 2016, 15:25
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If which \(x < y < 0\), of the following inequalities must be true?

A \(y + 1 < x\)

B \(y - 1 < x\)

C \(xy^2 < x\)

D \(xy < y^2\)

E \(xy < x^2\)

Practice Questions
Question: 24
Page: 463
Difficulty: medium/hard
[Reveal] Spoiler: OA

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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 24 Jan 2016, 15:35
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Solution

The question could become complex to evaluate theoretically. So the best way is to pick numbers

The stem can be tricky: it does not say that the numbers \(MUST\) be integers. As such, you need to take in account both options:\(X= -2\) and \(Y=-1\) OR \(X=\frac{-3}{2}\) and \(Y=\frac{-1}{2}\)

Testing all the answer choices we will end up that E will be \(-3<4\) and it is true;\(\frac{3}{4}<\frac{9}{4}\) and this is also true

The correct answer is \(E\)
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 19 Jul 2018, 11:39
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Carcass wrote:
If which \(x < y < 0\), of the following inequalities must be true?

 A \(y + 1 < x\)
 B \(y - 1 < x\)
 C \(xy^2 < x\)
 D \(xy < y^2\)
 E \(xy < x^2\)


We can see that both x and y are negative, and y is greater than x. So A is not true. B is not true, either, since we don’t know how much y is greater than x. If we divide both sides by x in the inequality in C (and switch the inequality sign since x is negative), we have:

y^2 > 1

However, since we don’t know the value of y, we can’t determine whether its square is in fact greater than 1. So C might not be true. If we divide both sides by y in the inequality in D (and switch the inequality sign since y is negative), we have:

x > y

This is not true since we know y > x. So the correct answer must be E. However, let’s verify it by dividing both sides of the inequality by x:

y > x

This is true since we know y is indeed greater than x.

Answer: E
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 19 Jul 2018, 12:05
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Carcass wrote:
If which \(x < y < 0\), of the following inequalities must be true?

 A \(y + 1 < x\)
 B \(y - 1 < x\)
 C \(xy^2 < x\)
 D \(xy < y^2\)
 E \(xy < x^2\)

Practice Questions
Question: 24
Page: 463
Difficulty: medium/hard



Given

\(x < y < 0\)

Here we understand that x is the most negative between x and y. So, whenever we square it , we get the maximum value which is greater than y.

Thus the best answer is E.
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 30 Sep 2018, 06:17
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Please correct the OA here. It is "E"
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 27 Nov 2018, 21:38
is there an explanation
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 28 Nov 2018, 09:46
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Dear Sir,

I think above your question there are 3 explanations of the same question. Please refer to them .

Regards
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 10 Feb 2019, 11:39
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Carcass wrote:
If which \(x < y < 0\), of the following inequalities must be true?

A \(y + 1 < x\)

B \(y - 1 < x\)

C \(xy^2 < x\)

D \(xy < y^2\)

E \(xy < x^2\)

Practice Questions
Question: 24
Page: 463
Difficulty: medium/hard


The question asks, "What MUST be true?"

So if we can find a case in which an answer choice is NOT true, then we can ELIMINATE that answer choice.

GIVEN: x < y < 0

A) y + 1 < x
If x < y < 0, then it's possible that x = -3 and y = -1
When plug these values into the above inequality, we get: -1 + 1 < -3
Simplify to get: 0 < -3, which is NOT true
ELIMINATE A

B) y - 1 < x
If x < y < 0, then it's possible that x = -3 and y = -1
When plug these values into the above inequality, we get: -1 - 1 < -3
Simplify to get: -2 < -3, which is NOT true
ELIMINATE B

C) xy < x
If x < y < 0, then it's possible that x = -3 and y = -1
When plug these values into the above inequality, we get: (-3)(-1) < -3
Simplify to get: 3 < -3, which is NOT true
ELIMINATE C


D) xy < y²
If x < y < 0, then it's possible that x = -3 and y = -1
When plug these values into the above inequality, we get: (-3)(-1) < (-1)²
Simplify to get: 3 < 1, which is NOT true
ELIMINATE D

By the process of elimination, the correct answer must be E

Cheers,
Brent
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Re: If which x < y < 0, of the following inequalities must be tr [#permalink] New post 06 Dec 2019, 11:07
[quote="Carcass"]If which \(x < y < 0\), of the following inequalities must be true?

A \(y + 1 < x\)

B \(y - 1 < x\)

C \(xy^2 < x\)

D \(xy < y^2\)

E \(xy < x^2\)

How about this approach after separating x < y from the given equation:

multiply both the sides with x to get x^2 > xy. -- sign changes because x is negative.
Re: If which x < y < 0, of the following inequalities must be tr   [#permalink] 06 Dec 2019, 11:07
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