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

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Math Expert
Joined: 02 Sep 2009
Posts: 53063
If a < b < 0, which of the following must be true?  [#permalink]

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21 Apr 2015, 05:01
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Question Stats:

65% (01:49) correct 35% (01:54) wrong based on 287 sessions

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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.

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

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21 Apr 2015, 07:21
1
2
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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21 Apr 2015, 08:18
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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21 Apr 2015, 23:04
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
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Re: If a < b < 0, which of the following must be true?  [#permalink]

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22 Apr 2015, 00:16
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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Kudos!!

Math Expert
Joined: 02 Sep 2009
Posts: 53063
Re: If a < b < 0, which of the following must be true?  [#permalink]

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27 Apr 2015, 02:13
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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21 Dec 2017, 19:17
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Re: If a < b < 0, which of the following must be true?   [#permalink] 21 Dec 2017, 19:17
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# If a < b < 0, which of the following must be true?

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