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MGMAT: Abs question
Source: Beat The GMAT — Data Sufficiency |
C for me
1) substitue following values for a, b
-3, 2, true
-2, 1, false
Not suff
2) substitue following values for a, b
-3, -2, true
-2, -1, true
2, 1, false
Not suff
taken together C
Second equation also test the cases when a, b are taken
1) substitue following values for a, b
-3, 2, true
-2, 1, false
Not suff
2) substitue following values for a, b
-3, -2, true
-2, -1, true
2, 1, false
Not suff
taken together C
Second equation also test the cases when a, b are taken
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The answer is E, but only because in Statement 2, the inequality is not strict. Notice in Statement 2, it's possible for b to be equal to zero. The question asks:
is a*|b| < a - b ?
If b is zero, this becomes:
is
0 < a ?
We know from Statement 1 that a is negative, so the answer is no. However, for any other valid values of a and b, the inequality will be true. Hence, E.
The question is much more interesting if the second statement is made into a strict inequality (i.e. remove the possibility that ab = 0). Then the answer is indeed C (though I'd note that it's vitally important that a and b are integers here). I think it's been demonstrated above that neither statement is sufficient on its own. Taken together, if we make the 2nd statement a strict inequality:
from 1: a is negative
from 2: a and b have the same sign, so b is also negative.
We also know that |a| > |b|, or in other words, a is further from zero than b is. If both a and b are negative, this means a < b.
So looking at the question- is a*|b| < a - b ?
Well, b is an integer, and we're now assuming it's not zero, so a*|b| can't be larger than a. It might be 2a, 3a, etc, but notice that since a is negative, we're getting smaller and smaller values the larger we make |b|. So the left side of that inequality is less than or equal to a. The right side is a - b, and b is negative: subtracting a negative is equivalent to adding a positive, and a-b must therefore be larger than a. So the left side is less than or equal to a, the right side is greater than a, and the inequality must be true. C.
is a*|b| < a - b ?
If b is zero, this becomes:
is
0 < a ?
We know from Statement 1 that a is negative, so the answer is no. However, for any other valid values of a and b, the inequality will be true. Hence, E.
The question is much more interesting if the second statement is made into a strict inequality (i.e. remove the possibility that ab = 0). Then the answer is indeed C (though I'd note that it's vitally important that a and b are integers here). I think it's been demonstrated above that neither statement is sufficient on its own. Taken together, if we make the 2nd statement a strict inequality:
from 1: a is negative
from 2: a and b have the same sign, so b is also negative.
We also know that |a| > |b|, or in other words, a is further from zero than b is. If both a and b are negative, this means a < b.
So looking at the question- is a*|b| < a - b ?
Well, b is an integer, and we're now assuming it's not zero, so a*|b| can't be larger than a. It might be 2a, 3a, etc, but notice that since a is negative, we're getting smaller and smaller values the larger we make |b|. So the left side of that inequality is less than or equal to a. The right side is a - b, and b is negative: subtracting a negative is equivalent to adding a positive, and a-b must therefore be larger than a. So the left side is less than or equal to a, the right side is greater than a, and the inequality must be true. C.
