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22 Jul 2008, 08:32
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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
I know that this can be solved by experimenting with different numbers, but I honestly hate this approach because my mind just turns nuts and even more confused. More importantly, experimenting will make you waste time by doing the same approach multiple times until you reach your desired answer. I prefer the algebraic approach because you have to do the approach only once and then there you have your result. So would anyone please show me how to approach this problem using strictly algebra?? Here is what I did so far:
b+c>a
- b>a
c>0, so what the question is really asking is whether C is positive
(1) we don't know whether both are positive or negative
(2) either all the variables are positive or only 2 of the variables are negative, which C can be one of them
(1&2) Together, when we look at the worst case scenario that 2 variables are negative for abc>0, and when taking b>a and c>a into consideration, since a is smaller than both b and c, we know that a must be at least negative. However, what algebraic approach can we do to determine whether c or b is positive?
(1) c>a
(2) abc>0
I know that this can be solved by experimenting with different numbers, but I honestly hate this approach because my mind just turns nuts and even more confused. More importantly, experimenting will make you waste time by doing the same approach multiple times until you reach your desired answer. I prefer the algebraic approach because you have to do the approach only once and then there you have your result. So would anyone please show me how to approach this problem using strictly algebra?? Here is what I did so far:
b+c>a
- b>a
c>0, so what the question is really asking is whether C is positive
(1) we don't know whether both are positive or negative
(2) either all the variables are positive or only 2 of the variables are negative, which C can be one of them
(1&2) Together, when we look at the worst case scenario that 2 variables are negative for abc>0, and when taking b>a and c>a into consideration, since a is smaller than both b and c, we know that a must be at least negative. However, what algebraic approach can we do to determine whether c or b is positive?
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22 Jul 2008, 08:44
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Rather than subtracting out c>a, I would add c>a to b>a
this gives you b+c>2a
#1 should be sufficient. We already know that b>a. If c>a, then 2 numbers larger than a must have a sum larger than a, even if c is negative. Try picking some numbers.
b = 3, a = -2 c = -1
3 + -1 > -2 True.
b = 1, a = -1 c = 0
1 + 0 > -1
I don't believe #2 is sufficient.
this gives you b+c>2a
#1 should be sufficient. We already know that b>a. If c>a, then 2 numbers larger than a must have a sum larger than a, even if c is negative. Try picking some numbers.
b = 3, a = -2 c = -1
3 + -1 > -2 True.
b = 1, a = -1 c = 0
1 + 0 > -1
I don't believe #2 is sufficient.
tarek99
22 Jul 2008, 08:57
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If you pick numbers b=c=-2 and a=-3 then although b>a and c>a but b+c<a
Combining statement 1) and 2)
b>a and c>a , b + c > 2a and abc>0
This means either a>0, b>0, c>0 or a<0, b<0 and c>0 or a<0, c<0 and b>0
If all are +ve and b + c > 2a then b + c > a is true as well
If a<0, b<0 and c>0 , b > a and b + c > 2a then b + c > a is true as well
If a<0, c<0 and b>0 , c > a and b + c > 2a then b + c > a is true as well
Therefore answer is C)
Combining statement 1) and 2)
b>a and c>a , b + c > 2a and abc>0
This means either a>0, b>0, c>0 or a<0, b<0 and c>0 or a<0, c<0 and b>0
If all are +ve and b + c > 2a then b + c > a is true as well
If a<0, b<0 and c>0 , b > a and b + c > 2a then b + c > a is true as well
If a<0, c<0 and b>0 , c > a and b + c > 2a then b + c > a is true as well
Therefore answer is C)
jallenmorris
. so you're required to check ONLY the scenario of b being positive 
