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

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 | > | b |?

- \\frac{1}{(a - b)}) > \\frac{1}{(b - a)})
- a + b < 0

#### 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".

#### When is the data sufficient?

We need to determine whether | a | is greater than | b |.

The data is sufficient if we get a conclusive Yes or a conclusive No as the answer to the question.

The answer to this question will be a conclusive 'yes' if | a | > | b |.

The answer will be a conclusive 'no' if | a | ≤ | b |

#### Statement 1: \\frac{1}{(a - b)}) > \\frac{1}{(b - a)})

We can rewrite the same inequality as \\frac{1}{(a - b)}) > \\frac{-1}{(a - b)}).

##### When will a number be greater than the negative of the number?

The number has to be a positive number.

So, we can conclude that a - b > 0 or a > b.

If a > b, | a | may or may not be greater than | b |.

Example: a = 4, b = 2. Then | a | > | b |. For positive "a" and "b", when a > b, | a | > | b |.

Counter example: a = 2 and b = -10. a > b. But | a | < | b |.

Hence, we cannot conclude from statement 1 whether | a | > | b |.

Statement 1 ALONE is NOT sufficient.

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

#### Statement 2: a + b < 0

Either both a and b are negative or one of a or b is negative.

If only one of the two numbers is negative, then the magnitude of the negative number is greater than the magnitude of the positive number.

Example: a = -3 and b = -4. a + b < 0, | a | < | b |

Counter example: a = -4 and b = -3. a + b < 0 and | a | > | b |.

Statement 2 ALONE is NOT sufficient.

Eliminate choice B. Choices narrow down to C or E.

#### Statements Together:

We know a > b from statement 1 and a + b < 0 from statement 2.

Possibility 1: **Both a and b are negative**. We know that a > b. So, | a | < | b |. Note in negative numbers, lesser the magnitude, larger the value.

Possibility 2: **One of 'a' or 'b' is negative**. Because a > b, a has to be positive and b has to be negative.

The sum, a + b < 0. Therefore, the magnitude of the positive number "a" has to be lesser than the magnitude of the negative number "b".

So, we can conclude that | a | < | b |.

Hence, by combining the two statements we can conclude that | a | is not greater than | b |.

Statements Together ARE sufficient.

Eliminate choice E.

Choice C is the answer.

#### Alternative Method

Here is an alternative explanation for the same.

From statement 1, we know a - b > 0.

From statement 2, we know a + b < 0.

So, (a - b) (a + b) < 0

Or a^{2} - b^{2} < 0 or a^{2} < b^{2}

If a^{2} < b^{2}, we can conclude that | a | < | b |.

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