Absolute Values
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Source: Beat The GMAT — Data Sufficiency |
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jayanti
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Hey, the right answer is E... Ive the solution but, didn't understand, Im posting it anyways, let me know if you get it.
Thanks for the effort,
Let us start be examining the conditions necessary for |a|b > 0. Since |a| cannot be negative,
both |a| and b must be positive. However, since |a| is positive whether a is negative or positive,
the only condition for a is that it must be non-zero.
Hence, the question can be restated in terms of the necessary conditions for it to be answered
"yes":
"Do both of the following conditions exist: a is non-zero AND b is positive?"
(1) INSUFFICIENT: In order for a = 0, |ab| would have to equal 0 since 0 raised to any power is
always 0. Therefore (1) implies that a is non-zero. However, given that a is non-zero, b can be
anything for |ab| > 0 so we cannot determine the sign of b.
(2) INSUFFICIENT: If a = 0, |a| = 0, and |a|b = 0 for any b. Hence, a must be non-zero and the
first condition (a is not equal to 0) of the restated question is met. We now need to test whether
the second condition is met. (Note: If a had been zero, we would have been able to conclude
right away that (2) is sufficient because we would answer "no" to the question is |a|b > 0?)
Given that a is non-zero, |a| must be positive integer. At first glance, it seems that b must be
positive because a positive integer raised to a negative integer is typically fractional (e.g., a-2 =
1/a2). Hence, it appears that b cannot be negative. However, there is a special case where this is
not so:
If |a| = 1, then b could be anything (positive, negative, or zero) since |1|b is always equal to 1,
which is a non-zero integer . In addition, there is also the possibility that b = 0. If |b| = 0,
then |a|0 is always 1, which is a non-zero integer.
Hence, based on (2) alone, we cannot determine whether b is positive and we cannot answer the
question.
An alternative way to analyze this (or to confirm the above) is to create a chart using simple
numbers as follows:
a B Is |a|b a non-zero integer? Is |a|b > 0?
1 2 Yes Yes
1 -2 Yes No
2 1 Yes Yes
2 0 Yes No
We can quickly confirm that (2) alone does not provide enough information to answer the
question.
(1) AND (2) INSUFFICIENT: The analysis for (2) shows that (2) is insufficient because, while we
can conclude that a is non-zero, we cannot determine whether b is positive. (1) also implies that
a is non-zero, but does not provide any information about b other than that it could be
anything. Consequently, (1) does not add any information to (2) regarding b to help answer the
question and (1) and (2) together are still insufficient. (Note: As a quick check, the above chart
can also be used to analyze (1) and (2) together since all of the values in column 1 are also
consistent with (1)).
Thanks for the effort,
Let us start be examining the conditions necessary for |a|b > 0. Since |a| cannot be negative,
both |a| and b must be positive. However, since |a| is positive whether a is negative or positive,
the only condition for a is that it must be non-zero.
Hence, the question can be restated in terms of the necessary conditions for it to be answered
"yes":
"Do both of the following conditions exist: a is non-zero AND b is positive?"
(1) INSUFFICIENT: In order for a = 0, |ab| would have to equal 0 since 0 raised to any power is
always 0. Therefore (1) implies that a is non-zero. However, given that a is non-zero, b can be
anything for |ab| > 0 so we cannot determine the sign of b.
(2) INSUFFICIENT: If a = 0, |a| = 0, and |a|b = 0 for any b. Hence, a must be non-zero and the
first condition (a is not equal to 0) of the restated question is met. We now need to test whether
the second condition is met. (Note: If a had been zero, we would have been able to conclude
right away that (2) is sufficient because we would answer "no" to the question is |a|b > 0?)
Given that a is non-zero, |a| must be positive integer. At first glance, it seems that b must be
positive because a positive integer raised to a negative integer is typically fractional (e.g., a-2 =
1/a2). Hence, it appears that b cannot be negative. However, there is a special case where this is
not so:
If |a| = 1, then b could be anything (positive, negative, or zero) since |1|b is always equal to 1,
which is a non-zero integer . In addition, there is also the possibility that b = 0. If |b| = 0,
then |a|0 is always 1, which is a non-zero integer.
Hence, based on (2) alone, we cannot determine whether b is positive and we cannot answer the
question.
An alternative way to analyze this (or to confirm the above) is to create a chart using simple
numbers as follows:
a B Is |a|b a non-zero integer? Is |a|b > 0?
1 2 Yes Yes
1 -2 Yes No
2 1 Yes Yes
2 0 Yes No
We can quickly confirm that (2) alone does not provide enough information to answer the
question.
(1) AND (2) INSUFFICIENT: The analysis for (2) shows that (2) is insufficient because, while we
can conclude that a is non-zero, we cannot determine whether b is positive. (1) also implies that
a is non-zero, but does not provide any information about b other than that it could be
anything. Consequently, (1) does not add any information to (2) regarding b to help answer the
question and (1) and (2) together are still insufficient. (Note: As a quick check, the above chart
can also be used to analyze (1) and (2) together since all of the values in column 1 are also
consistent with (1)).
