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If a < b < 0, which of the following must be true?

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If a < b < 0, which of the following must be true?  [#permalink]

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New post 21 Apr 2015, 06:01
2
8
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

63% (01:48) correct 37% (01:48) wrong based on 246 sessions

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Re: If a < b < 0, which of the following must be true?  [#permalink]

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New post 21 Apr 2015, 08:21
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Bunuel wrote:
If a < b < 0, which of the following must be true?

A. a^2 < b^2
B. b − 10 < a
C. b + a > a
D. ab < b^2
E. ab < a^2

Kudos for a correct solution.


a) \(a^2<b^2\); \(a^2-b^2<0\) false because \(|a|>|b|\)
b) \(b − 10 < a\); \(b<a+10\) false if \(|a|-|b| > 10\)
c) \(b + a > a\); \(b > 0\) false
d) \(ab < b^2\) false because \(|b| < |a|\)
e) \(ab < a^2\) true because \(|a|>|b|\)

And answer is E
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Re: If a < b < 0, which of the following must be true?  [#permalink]

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New post 21 Apr 2015, 09:18
1
Bunuel wrote:
If a < b < 0, which of the following must be true?

A. a^2 < b^2
B. b − 10 < a
C. b + a > a
D. ab < b^2
E. ab < a^2

Kudos for a correct solution.


A. a^2 < b^2 --> (a+b)(a-b) < 0 --> (-ive)*(-ive)<0 ; NO
B. b − 10 < a --> b-a<10,let a=-1000 and b=-1; then b-a=999 . NO
C. b + a > a --> b>0 ; NO
D. ab < b^2 --> b*(a-b) <0; (-ive)*(-ive)<0 ; NO
E. ab < a^2 --> a*(b-a)<0; (-ive)*(+ive) <0 YESSSSSSSS!!. this is our answer.
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Re: If a < b < 0, which of the following must be true?  [#permalink]

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New post 22 Apr 2015, 00:04
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1
Bunuel wrote:
If a < b < 0, which of the following must be true?

A. a^2 < b^2
B. b − 10 < a
C. b + a > a
D. ab < b^2
E. ab < a^2

Kudos for a correct solution.


A. a^2<b^2. Put a=-2,b=-1, 4>1.Not true
B. b-10<a. Put a=-25,b=-3. -13>-25.Not true
C.b+a>a. Put b=-1,a=-2, -3<-2. Not true
D. ab<b^2. Put b=-1,a=-2, ab=2 and b^2=1. Not true
E.ab<a^2. Always true. Put b=-2,a=-5. ab=10, a^2=25. True
Answer E
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Re: If a < b < 0, which of the following must be true?  [#permalink]

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New post 22 Apr 2015, 01:16
1
picking no for a and b :
a b OpA
-2 -1 4<1 NT
-1.5 -.5 2.25<.25 NT
-20 -1 400<1 NT

similarly work out the table for other options.. Thus we see that putting these nos of a and b in A, B, C, D give Not True(NT) or maybe true ans. E gives true always. Thus E.
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Re: If a < b < 0, which of the following must be true?  [#permalink]

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New post 27 Apr 2015, 03:13
Bunuel wrote:
If a < b < 0, which of the following must be true?

A. a^2 < b^2
B. b − 10 < a
C. b + a > a
D. ab < b^2
E. ab < a^2

Kudos for a correct solution.


VERITAS PREP OFFICIAL SOLUTION:

E. In this problem you should see that a and b are both negative numbers, and that a has a greater absolute value (it's further from zero) than b does. So when you square a, you'll end up with a larger positive number than when you multiply a by a lesser-absolute-value b. So choice E must be true. Choosing numbers can help exemplify:

a = -10, b = -5

Would mean that a^2 = 100 and ab = 50, so a^2 is bigger.

Even if you try with fractions (always a good idea to consider), a = -3/4 and b = -1/2 yields:

a^2 = 9/16 (greater than 1/2)

ab = 3/8 (less than 1/2)

Proving E to be true. For the incorrect answers:

A) Since a has a larger absolute value, when it's squared it will be larger than b-squared.

B) Because you don't know the relative values of a and b, you can't determine that moving b 10 places to the left will reduce it past a.

C) Adding a negative number (b) to a will make it smaller, not larger, than the original a.

D) See demonstration above.
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Re: If a < b < 0, which of the following must be true?  [#permalink]

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Re: If a < b < 0, which of the following must be true?   [#permalink] 18 Mar 2019, 02:01
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