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VP  Joined: 04 May 2006
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If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a
(2) abc > 0

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Originally posted by sondenso on 26 May 2009, 02:46.
Last edited by Bunuel on 10 Aug 2012, 00:22, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58435
Re: If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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14
8
If a, b and c are integers such that b > a, is b+c > a ?

Question: is $$b+c > a$$ ? --> or: is $$b+c-a > 0$$?

(1) c > a. If $$a=1$$, $$b=2$$ and $$c=3$$, then the answer is clearly YES but if $$a=-3$$, $$b=-2$$ and $$c=-1$$, then the answer is NO. Not sufficient.

(2) abc > 0. Either all three unknowns are positive (answer YES) or two unknowns are negative and the third one is positive. Notice that in the second case one of the unknowns that is negative must be $$a$$ (because if $$a$$ is not negative, then $$b$$ is also not negative so we won't have two negative unknowns). To get a NO answer for the second case consider $$a=-3$$, $$b=1$$ and $$c=-4$$ and . Not sufficient.

(1)+(2) We have that $$b > a$$, $$c > a$$ ($$c-a>0$$) and that either all three unknowns are positive or $$a$$ and any from $$b$$ and $$c$$ is negative. Again for the first case the answer is obviously YES. As for the second case: say $$a$$ and $$c$$ are negative and $$b$$ is positive, then $$b+(c-a)=positive+positive>0$$ (you can apply the same reasoning if $$a$$ and $$b$$ are negative and $$c$$ is positive). So, we have that in both cases we have an YES answer. Sufficient.

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sondenso wrote:
If a, b and c are integers such that b > a, is b+c > a ?
(1) c > a
(2) abc > 0

E

a = -4 or 3
b = -3 or 4

The above satifies b>a, using - or + as examples.

(1) c>a
c = -2 or 4
Insuff
(2) abc>0
This means either 2 of the intergers are negative or all of the intergers are positive.
Insuff
Manager  Joined: 14 May 2009
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sondenso wrote:
If a, b and c are integers such that b > a, is b+c > a ?
(1) c > a
(2) abc > 0

b>a <==> 0>a-b <==> a-b is negative
Question:(b+c>a)?
Question:(c>a-b)?
Question:(c >= 0)?

(1) Insufficient, as a-b<0 could mean that a<0 or a>0, which means c<0 or c>=0.

(2) We have an even # of negatives, ie 0/2. If we have 0 negatives, then sufficient. Again if we have 2 insufficient.

I can see the pattern, the answer is E.

Here are the Yes/No cases:

Yes: 4 5 7
No: -9 3 -10

Final Answer, $$E$$.
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2
why cant it be C?

1. b>a and c>a and abc>0:

If a<0, b can be less than or greater than 0.

If a<0, b<0 then c>0 (a=-6, b=-4, c=2) => b+c >a
If a<0, b>0, c<0 (a=-3, b=5, c=-1) => b+c >a

If all are positive, then also b+c >a.

Hence, C.

Are there any assumptions which do not satisfy both conditions simultaneously?

What is the OA?
Manager  Affiliations: Beta Gamma Sigma
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1
2
The answer is C

statement 1 does work for positives, but not for negatives. a= -2 b= -1 c= -1.

statement 2 doesnt work on its own because c could be way less than a, negating b,
a= -3 b=2 c= -5

however, both together mean that there can only be two negatives, and c must be larger than A.
for any value A, negative or not, two values that are greater than it with one positive are going to together be greater than A.
if A is positive = a=2, c=3, b=3, a=1 b=2 c=2
if A is negative = a= -3, c= -2, b=1, a= -2 b=-1 c=1

There is no solution with (1) and (2) true that doesn't end up with B + C > A

gmatprep is right
Manager  Joined: 08 Jan 2009
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1
b>a
stmt 1 :

c>a so b+c>2a
we cannot tell whether b+c >a may or may not. So insufficient.

stmt 2 :
abc>0 and b >a
a = +ve b = +ve c =+ve so definetly b+c > a
a = -ve b = -ve c = +ve so definetly b+c > a. ( since c =+ve and b>a)
a = -ve b = +ve c = -ve ( here it is just the opposite u know b =+ve but do not know c>a).so u cant tell whether b+c > a

Combing : yes u know you have got what u wanted so C.

Thanks gmatprep09 and dk94588.
Manager  Joined: 14 May 2009
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1
Whoops the answer is C

If you look at both together,

we have that b>a & c>a.

from (1) we get b+c>2a and we're almost sufficient, but only if 2a>a.
Well if a<0, no, but if a>0, yes.

(2) tells us that 0 or 2 are negative. If all 3 are positive, sufficient.
Now if 2 are negative, then a has to be negative as well. Suppose a were positive-- then b/c would be negative, and b>a & c>a would be false. Hence a has to be positive.

(C)

Very tricky
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Re: If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

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If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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sondenso wrote:
If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a
(2) abc > 0

The question stem says that b > a. There are following possibilities for this
a. Both Positive Lets say $$b = 5 and a = 2$$

b. Both Negative, lets say $$b = -3 and a = -5$$

c. One positive one negative, lets say $$b = 3 and a = -1$$

d. Either of the two zero

1. c > a

C can be positive or negative and can fit in any of the above mentioned scenarios.

Insufficient

2. abc > 0

This means either all are positive or any two are negative. Since b > a, there are numerous possibilities for c.

So, Insufficient.

c > a and abc > 0

This would mean that either all are positive - b + c > a

or two negatives and one one positive - a has to be negative, either of b and c has to be negative and the other one positive. This would also mean that c + b > a

Ans - C
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Give KUDOS if the post helps you... Manager  Joined: 24 May 2013
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If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a
(2) abc > 0
Either two are negative and one positive or all are positive.

Individually both the statements are not sufficient.
Combining 1 and 2 Sufficient.

Hence C.
Attachments abc integer.png [ 6.31 KiB | Viewed 18433 times ]

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Re: If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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Bunuel

In this problem can we subtract the following inequalities?

b>a
b+c>a

and arrive at 'is c>0??'
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If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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.rr1990 wrote:
Bunuel

In this problem can we subtract the following inequalities?

b>a
b+c>a

and arrive at 'is c>0??'

Let me try to answer.

The answer is NO you can not.

b>a and b+c>a can mean both c<0 and c>0 by taking the following cases:

b=5, c=-3 and a=1 but the given conditions are also true for b=5, c=3 and a=1. So you have both the cases possible.

With,

b>a and b+c>a, you can not directly subtract the 2 inequalities as the sign for b>a MUST be reversed. You can only ADD in inequalities without thinking about the signs.

If I give you,

b>a and b+c>a ---> then the only MUST be true statement will be ---> b+b+c > 2a ---> 2b+c > 2a

BUT,

If you are given, b<a and b+c > a, as you can not add 2 inequalities with dissimilar inequality signs ---> you must do an additional operation (b<a ---> -b>-a) before you add the 2.

After the manipulation you get,

-b > -a and this added to b+c > a --> -b+b+c > 0 ---> c>0

Hope this helps.
Math Expert V
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Posts: 7977
Re: If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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.rr1990 wrote:
Bunuel

In this problem can we subtract the following inequalities?

b>a
b+c>a

and arrive at 'is c>0??'

Hi,

NO you cannot..
if
b>a...I
b+c>a..II

I-II
b-(b+c)>a-a
-c>0
or c<0

II-I
b+c-b>a-a
c>0

so you see, you have two opposite answer depending on the way you subtract..

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If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a

lets say b=-1 and a=-2. If c=-1 then -1+-1 = -2 ---> is b+c > a NO. If c=1 then -1+1>-2 YES.

Hence insufficient.

(2) abc > 0

Only tell us either a,b,c are all positive or 2 negative and 1 positive, besides this info we get nothing. Hence insufficient.

IF two combines. When b is positive a, c must be negative (all three are positive will be a straight answer. And all three negative will never happen because abc > 0) (abc>0, b>a, c>a). Either outcome makes b+c > a. (C) Correct
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Re: If a, b and c are integers such that b > a, is b+c > a  [#permalink]

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_________________ Re: If a, b and c are integers such that b > a, is b+c > a   [#permalink] 11 Oct 2018, 00:32
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