GMAT Challenging Question 2 | Inequalities in Exponents

The concept tested in this GMAT hard math question is tested quite often in the GMAT - especially as the data sufficiency variant. The relation between two exponents vary depending on whether the numbers are negative or positive or greater than 1 or less than -1.

GMAT Data Sufficiency | Directions | Click Here ▼

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
3. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
4. EACH statement ALONE is sufficient to answer the question asked.
5. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Numbers

All numbers used are real numbers.

Figures

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Note

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 2: Is x³ > x² ?

Statement 1: x > 0
Statement 2: x < 1

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Video Explanation

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Explanatory Answer | GMAT Inequalities DS

Step 1: Decode the Question Stem and Get Clarity

Q1: What kind of an answer will the question fetch?
An "Is" question will fetch an "Yes" or a "No" as an answer.
The data provided in the statements will be considered sufficient if the question is answered with a conclusive Yes or a conclusive No.

Q2: When is the answer yes?
If x³ > x², the answer to the question is a conclusive Yes.

Q3: When is the answer yes?
If x³ ≤ x², the answer to the question is a conclusive No.
Note: When x³ = x², the answer is No.

It pays rich dividend to note down the answer to questions 2 and 3 mentioned above in your scratch paper while solving DS questions.

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Step 2: Key concepts of Statement 1:

x³ is greater than x² for certain values of x and will not be greater for other values of x.
There are 4 intervals to keep in mind while evaluating the relation between two different exponents of x.
Interval 1: -∞ < x < -1 x³ is negative in this interval and x² is positive in this interval. So, x³ < x².
Interval 2: -1 < x < 0 x³ is negative in this interval and x² is positive in this interval. So, x³ < x².
Interval 3: 0 < x < 1 x³ is positive and so is x².
Let us substitute a value and check. When x = \\frac{1}{2}), x³ = \\frac{1}{8}) and x² = \\frac{1}{4})
\\frac{1}{8}) < \\frac{1}{4}). So, x³ < x² in this interval as well
Interval 4: 1 < x < ∞ Both x³ and x² are positive in this interval.
Let us plug in a value and check. When x = 2, x³ = 8 and x² = 4. So, x³ > x².
In addition to these 4 intervals, we also need to check the relation between x³ and x² when x = -1, x = 0 and x = 1.

Step 2A: Evaluate Statement 1 ALONE

Statement 1: x > 0

We need to check if we get a conclusive Yes or No using this statement to determine whether statement 1 is sufficient

When x > 0, we need to evaluate the following conditions:

1. 0 < x < 1: In this inverval x³ < x². i.e, the answer is NO.
2. x = 1: x³ = x². i.e, the answer is NO.
3. 1 < x < ∞: x³ > x². In this interval, the answer to the question is YES.

The values that x takes based on statement 1 do not give a conclusive answer to the question.
Hence, statement 1 is NOT Sufficient
Eliminate answer options A and D.

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Step 3: Evaluate Statement 2 ALONE

Statement 2: x < 1

We need to check if we get a conclusive Yes or No using this statement to determine whether statement 2 alone is sufficient

When x < 0, we need to evaluate the following conditions:

1. 0 < x < 1: In this inverval x³ < x². i.e, the answer is NO.
2. x = 0: x³ = x². i.e, the answer is NO.
3. -1 < x < 0: x³ < x². In this interval, the answer to the question is NO.
4. -∞ < x < -1: x³ < x². In this interval too, the answer to the question is NO.

The values that x takes based on statement 2 give a conclusive 'NO' as an answer to the question.
Statement 2 alone is Sufficient.
Eliminate answer options C and E.

Choice B is the correct answer.

Please bear in mind that what is important is whether we get a uniform YES or a uniform NO as an answer. A NO does not mean that the data is not sufficient. Getting a conclusive NO means we have a definitive answer to the question.