A GMAT DS practice question in Inequalities. Tests your understanding of absolute values and inequalities.

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 -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- 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 | a | > a?

- a
^{2}< a - \(\frac{a}{2})) > \(\frac{2}{a}))

#### Explanatory Answer

Video explanation will be added soon#### What kind of an answer will the question fetch?

The question is an "IS" question. For "is" questions, the answer is "YES" or "NO".

#### How can we rewrite the question?

The magnitude of 'a' will be greater than 'a' only if 'a' is a negative number. For positive numbers the magnitude of 'a' will be equal to 'a'.

So, what we have to determine using the two statements is whether 'a' is negative.

#### Statement 1: a^{2} < a

This inequality holds good only when 0 < a < 1.

i.e., we can conclude that a is positive.

Hence, we can answer the question, "whether 'a' is negative" with a definite NO.

Statement 1 ALONE is sufficient.

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

#### Statement 2: \(\frac{a}{2})) > \(\frac{2}{a}))

'a' is an unknown and can therefore, take both positive and negative values.

We have to evaluate two possibilities from the information given in statement (2).

Possibility 1: a > 0

If a > 0, then \(\frac{a}{2})) > \(\frac{2}{a})) = a^{2} > 4

i.e., a^{2} - 4 > 0 and a > 0

or (a + 2) (a - 2) > 0 and a > 0

Rule: If (x - a)(x - b) > 0, x will not lie between 'a' and 'b'.

So, 'a' will not lie between -2 and 2. i.e., {a > 2 or a < -2} and a > 0

In possibility 1, only positive values of 'a' are possible.

∴ the values of 'a' that will satisfy the inequality are a > 0 and a > 2

a > 0 and a > 2 implies that a > 2

Possibility 1 therefore, answers the question with a NO.

Possibility 2: a < 0

If a < 0, then \(\frac{a}{2})) > \(\frac{2}{a})) = a^{2} < 4 (the sign of the inequality changes when multiplied with a negative number on both sides)

i.e., a^{2} - 4 < 0 and a < 0

or (a + 2) (a - 2) < 0 and a < 0

Rule: If (x - a)(x - b) < 0, x will in between 'a' and 'b'.

or -2 < a < 2 and a < 0

In possibility 2, only negative values of 'a' possible

∴ the range of values that 'a' can take narrows down to -2 < a < 0.

Possibility 2 therefore, points to the result that 'a' is negative and answers the question with Yes.

Statement (2) leaves us with both the possibilities: 'a' could be positive or 'a' could be negative.

Statement 2 ALONE is NOT sufficient.

Eliminate choice D. Choice A is the answer.

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