# GMAT 700+ Math Question #15 | Number Properties

#### GMAT Sample Questions | Data Sufficiency Practice

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: x2 + y2 = 4
Statement 2: x3 + y2 = 0

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### Explanatory Answer | GMAT Number Properties DS

#### Step 1: Decode the Question Stem and Simplify it

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?

#### Step 2: Evaluate Statement 1 ALONE

Statement 1: x2 + y2 = 4
Squares of real numbers are non-negative.
So, both x2 and y2 are non negative.
So, x2 could be 0 or positive.

If x2 = 0, x is 0.
Answer to the question "Is x < 0?" is NO.

If x2 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.

#### Step 3: Evaluate Statement 2 ALONE

Statement 2: x3 + y2 = 0
Squares of real numbers are non-negative. So, y2 is definitely not a negative number.

Two possibilities exist for x3 and y2
Possibility 1: Both x3 and y2 are 0.
If x3 = 0, the value of x = 0.
The answer to the question "Is x < 0?" is NO.

Possibility 2: x3 is negative and y2 is positive.
If x3 < 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.

#### Step 4: Evaluate Statements TOGETHER

Statement 1: x2 + y2 = 4
Statement 2: x3 + y2 = 0

From Statement 1, if x = 0, y2 = 4
And from statement 2, if x = 0, y2 = 0.
So, if x = 0, the statements contradict each other.
So, x cannot be 0.

Therefore, y2 has to be positive and x3 has to be negative to satisfy both statements.
If x3 < 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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