# Inequalities in exponents : Data Sufficiency

The concept tested in this 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. Getting a clear understanding of this behavior will help crack questions of this kind.

## Directions

• DS
• 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 July or the meaning of the word counterclockwise), you must indicate whether:

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

## Question

Is x3 > x2?
Statement 1: x > 0
Statement 2: x < 1

Choice B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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## Detailed Solution

#### 1. 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.

#### 2. When is the answer an "Yes"?

If x3 > x2, the answer to the question is a conclusive Yes.

#### 3. When is the answer a "No"?

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.

Now let us evaluate each statement independently.

Some concepts before taking a look at 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 = ½, x3 = ⅛ and x2 = ¼
⅛ < ¼. 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.

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.

Therefore, we can eliminate choices A and D. We can narrow down the choices to this DS question to B, C, or E.

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.

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.

Hence, statement 2 alone is Sufficient.