GMAT 700+ Math Question #15 | Number Properties

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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Video Explanation

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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?

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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.

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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.

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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.

Choice C is the correct answer.