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Is |a| + |b| > |a + b| ?

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Is |a| + |b| > |a + b| ?

by Gmat_mission » Sun May 13, 2018 10:20 am

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Is |a| + |b| > |a + b| ?

(1) a^2 > b^2
(2) |a|*b < 0

[spoiler]OA=E[/spoiler].

How can I solve this DS question? I'd really appreciate any help. Thanks in advance. <i class="em em-confounded"></i>
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by Vincen » Mon May 14, 2018 2:02 am

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Hello Gmat_mission.

Here, we have the strict triangular inequality.

The only way to be strict is when "a" and "b" have different signs.
(1) a^2 > b^2
This statement doesn't help us because it doesn't give us information about the sign of "a" and "b":

- If a=2 and b=1 then a^2=4 > 1=b^2 and

3 = 2 + 1 = |2| + |1| = |a| + |b| = |a + b| = |2+1| = |3| = 3.

This tells that the answer to the original question is NO.

But

- If a=-2 and b=1, then a^2 = 4 > 1 = b^2 and

3 = 2 + 1 = |-2| + |1| = |a| + |b| > |a + b| = |-2+1| = |-1|=1.

This tells that the answer to the original question is YES.

Since we got two different answers, we conclude that this statement is NOT SUFFICIENT.
(2) |a|*b < 0
This statement tells us that b<0, but we don't know anything about "a". Therefore, this statement also is NOT SUFFICIENT..

Using both statements together again we will obtain that b is negative, but we don't know anything about "a".

In conclusion, the correct answer for this DS question is the option E.

I hope it helps.
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by GMATGuruNY » Mon May 14, 2018 2:35 am

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Gmat_mission wrote:Is |a| + |b| > |a + b| ?

(1) a^2 > b^2
(2) |a|*b < 0
>
Is |a| + |b| > |a + b|?
Because an absolute value cannot be negative, both sides of the inequality above are NONNEGATIVE, allowing us to SQUARE the inequality

(|a| + |b|)² > (|a + b|)²
a² + 2|a||b| + b² > a² + 2ab + b²
2|a||b| > 2ab
|a||b| > ab.
The resulting inequality is valid only if a and b have DIFFERENT SIGNS.
Question stem, rephrased:
Do a and b have different signs?

Statement 1: a² > b²
Statement 2: |a|b < 0

Both statements are satisfied by the following cases:
Case 1: a=2 and b=-1
Case 2: a=-2 and b=-1

In Case 1, a and b have different signs, so the answer to the rephrased question stem is YES.
In Case 2, a and b have the same sign, so the answer to the rephrased question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.
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