GMAT DS Practice : Inequalities - Exponents

Concept: Comparison of two numbers in different intervals

A GMAT Inequality DS question. Tests your understanding of how exponents of a number compare at different intervals.

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.
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.

What is the data NOT sufficient?

If the information in the statements provide an ambiguous answer - Yes for certain values and No for others that satisfy the information in the statements, the data is NOT sufficient.

What is the answer Yes and when No?

If a3 > a2, the answer is YES
If a3 ≤ a2, the answer is YES.

Statement 1: $$frac{1}{a}$ > a For positive values of 'a' if $\frac{1}{a}$ > a, a has to lie in the interval 0 < a < 1. In this interval a3 < a2 For negative values of 'a' a3 < a2 because odd powers of negative numbers are negative and even powers of negative numbers are positive. Hence, from statement$1) we can conclude that a3 is not greater than a2

Statement 1 ALONE is sufficient.

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

Statement 2: a5 > a3

For positive values of 'a' if a5 < a3 a has to be greater than 1. In this interval a3 > a2.

For negative values of 'a' a3 < a2

We get both possibilities from the data in statement 2. Hence, we will not be able determine whether a3 > a2

Statement 2 ALONE is NOT sufficient.

Eliminate choice D. Choice (A) is the answer.

Note

In questions of this type, look out for the following intervals. -∞ to -1; -1 to 0; 0 to 1 and 1 to ∞. Numbers, especially powers of numbers behave differently - ascending order descending order rules - in each of these intervals. Many an inequality question tests whether you have understood this difference in pattern and have evaluated the condition in these four intervals properly.

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