Bunuel
If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?
(1) \(a > b > c\)
(2) \(a + b > 0\)
Given information: \(|a| > |b| > |c|\)
To determine if: \(a*b^3*c^3 > a*b^4*c^2\)?
(1) \(a > b > c\)So, a>b>c as well as |a|>|b|>|c|. Based on these 2 conditions, we can infer the following:
1) a cannot be negative (because if a=-1, and a>b>c, then b=-2 and c=-3 for example but then their absolute value inequality condition will not satisfy 1<2<3, not the other way around)
2) b cannot be negative (same reason as above)
3) c can be negative
4) a,b,c can all be positive fractions (negative fractions will not satisfy the original inequality condition of the question stem)
Now that we have established certain possibilities regarding values, we can test values:
a b c \(a*b^3*c^3\) \(a*b^4*c^2\) \(a*b^3*c^3 > a*b^4*c^2\)
3 2 1 24 48 NO
1/2 1/3 1/4 1/3456 1/2592 NO
We can test other similar patterns but result will be the same and we have tested both integers as well as fractions
Negative a,b are not possible and having c as negative will only make \(a*b^3*c^3\) as NEGATIVE and the other one as POSITIVE so again it will be a NO as to if greater or not, hence no point checking that with values
Negative fractions will not satisfy original inequality condition
SUFFICIENT2) \(a + b > 0\)So, a+b>0 as well as |a|>|b|>|c|. Based on these 2 conditions, we can infer the following:
1) a cannot be negative (because if a is negative, then b will have to be greater than a for a+b>0 to hold true and that will go against the original inequality condition of the question)
2) b and c can be negative as long as they satisfy original inequality condition
Let us test using some values:
a b c \(a*b^3*c^3\) \(a*b^4*c^2\) \(a*b^3*c^3 > a*b^4*c^2\)
3 2 1 24 48 NO
1/2 1/3 1/4 1/3456 1/2592 NO
5 -3 -2 1080 1620 NO
1 -1/2 -1/3 1/216 1/144 NO
Only b or c as negative is one case not tested with values because it will lead to \(a*b^3*c^3\) as negative and then again NO to the question
SUFFICIENTAnswer - D