GMAT Data Sufficiency Practice : Inequalities & Number Properties


Is a + b > 0?
1. a - b > 0
2. |a| < |b|

Correct Answer is Choice C. Both statements together are sufficient to answer the question.

Explanatory Answer

An "IS" question is answered when you can provide a definite YES or a definite NO as an answer to the question using the data.

We need to answer if a + b > 0.

Statement 1: a - b > 0.
We can infer that a > b.
If both a and b are negative and a > b, say  a = -2 and b = -10, the sum of a and b, a + b < 0
On the contrary if both a and b are positive, the sum will be positive.
We cannot answer the question based on the data in statement 1.

Statement 1 is INSUFFICIENT.

Statement 2: |a| < |b|
The magnitude of a is less that of b.
a and b could both be negative. In that scenario a + b will be negative.
Both the numbers a and b could be positive. In that case a + b will be positive.
 We cannot determine whether a + b is positive with this statement either.

Statement 2 is INSUFFICIENT.

Let us combine the data in the two statements.
a > b and  |a| < |b|
If a and b are both positive, then if a > b, |a| also has to be greater than |b|.
i.e., for positive numbers larger the magnitude, larger the number.

So, we can infer that both a and b cannot be positive.
Either both a and b are negative or one is negative and the other is positive.

If both a and b are negative if a > b, |a| will be less than |b|. The sum of a and b, a + b < 0

If one of the two numbers is positive, a has to be positive as a > b.
If |a| is less than |b| as given in statement 2, then the magnitude of the positive number is lesser than the magnitude of the negative number.
So, the sum of a and b, a + b will be negative.

Hence, using the data in the two statements we can determine that a + b < 0.

So, the correct answer is choice C.

Here is an alternative method to determine this when combining the two statements.

Statement 1 : a - b > 0

Statement 2: |a| < |b|.
If |a| < |b|, we can conclude that a^2 < b^2.
So, we can determine that a^2 - b^2 < 0

a + b = (a^2 - b^2) / (a - b).
If a^2 - b^2 is negative and a - b is positive, a + b has to be negative.

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