The given question is GMAT 700 level data question combining concepts in number properties and absolute values. A very interesting GMAT practice question to understand the nuances of what modulus of a number means and properties of squares of numbers.

Question 15: Is |x| > x?

**Statement 1**: x^{2} + y^{2} = 4

**Statement 2**: x^{3} + y^{2} = 0

@ INR

What does **"Is |x| > x?|** mean?

The modulus of a number gives the magnitude of that number.

Substitute a **positive value for x**: |x| = x

Substitute **zero for x**: |x| = x

Substitute a **negative value for x**: |x| > x

So, the question ultimately boils down to Is x < 0?

**Statement 1**: x^{2} + y^{2} = 4

Squares of real numbers are non-negative.

So, both x^{2} and y^{2} are non negative.

So, x^{2} could be 0 or positive.

If x^{2} = 0, x is 0.

Answer to the question "Is x < 0?" is **NO**.

If x^{2} is positive, x may be positive or negative.

Answer to the question "Is x < 0?" is **NO or YES**

We are not able to find a conclusive answer to the question using statement 1 ALONE.

Statement 1 alone is NOT Sufficient.

__Eliminate answer options A and D__.

**Statement 2**: x^{3} + y^{2} = 0

Squares of real numbers are non-negative. So, y^{2} is definitely not a negative number.

Two possibilities exist for x^{3} and y^{2}

**Possibility 1**: Both x^{3} and y^{2} are 0.

If x^{3} = 0, the value of x = 0.

The answer to the question "Is x < 0?" is NO.

**Possibility 2**: x^{3} is negative and y^{2} is positive.

If x^{3} < 0, x < 0

The answer to the question "Is x < 0?" is YES.

We are not able to find a conclusive answer to the question using statement 2 ALONE.

Statement 2 alone is NOT Sufficient.

__Eliminate answer option B__.

**Statement 1**: x^{2} + y^{2} = 4

**Statement 2**: x^{3} + y^{2} = 0

From Statement 1, if x = 0, y^{2} = 4

And from statement 2, if x = 0, y^{2} = 0.

So, if x = 0, the statements contradict each other.

So, x cannot be 0.

Therefore, y^{2} has to be positive and x^{3} has to be negative to satisfy both statements.

If x^{3} < 0, we can deduce that x < 0.

Answer to the question "Is x < 0?" is a __conclusive YES__.

We are able to answer the question by combining the two statements.

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