Number Line & Exponents : GMAT DS

Concept: Comparison of 2 powers in different intervals

This GMAT DS question given below is from the topic Number Properties. The question is presented to elucidate the concept that comparison of values of x3 and x3 will yield different results for different range of values.

Directions for Data Sufficiency

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: Is x3 > x2?

  1. x > 0
  2. x < 1

Explanatory Answer

Video explanation will be added soon

What kind of an answer will the question fetch?

The question is an "Is" question. Answer to an "is" questions is either YES or NO.

When is the data sufficient?

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

If from the statements we get an answer that x3 > x2 in some instances and it is otherwise in other instances, the data is NOT sufficient.

Statement 1: x > 0

We know that x is a positive number.

Interval 1: If 0 < x < 1, then x3 < x2.

For example, (0.5)3 = 0.125, which is lesser than (0.5)2 = 0.25

The answer to the question is NO.

Interval 2: If x > 1, then x3 > x2

For example, 23 = 8 which is greater than 22 = 4

The answer to the question is YES.

We do NOT have a DEFINITE answer using statement 1.

  Statement 1 ALONE is NOT sufficient.

Eliminate choices A and D. Choices narrow down to B, C or E.

Statement 2: x < 1

Interval 1: For positive values of x, i.e., 0 < x < 1, we know x3 < x2.

The answer to the question is NO.

Interval 2:For negative values of x, x3 will be a negative number and x2 will be a positive number.

Hence, x3 < x2

The answer to the question is NO.

Lastly, what is the answer if x = 0?

When x = 0, x3 = x2.

The answer to the question is NO.

Hence, if we know that x < 1, we can conclude that x3 is NOT GREATER THAN x2.

We have a DEFNITE answer, even if it is NO.

  Statement 2 ALONE is sufficient.

Eliminate choices C and E. Choice B is the answer.

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