# GMAT Math Questions | GMAT Inequalities Q5

#### Inequalities in Absolute Values | Data Sufficiency | GMAT Sample Questions

A GMAT DS question in Inequalities. Tests your understanding of finding solution set to inequalities of absolute values (modulus of an expression). A GMAT 650+ quantitative reasoning sample question in 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.

Question 5: Is | a | > | b |?

Statement 1: $$frac{1}{$a - b$}) > $$frac{1}{$b - a$})
Statement 2: a + b < 0

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### Explanatory Answer | GMAT Inequalities DS

#### Step 1: Decode the Question Stem

Q1. What kind of an answer will the question fetch?
The question is an "IS" question. For "is" questions, the answer is "YES" or "NO".

Q2. When is the data sufficient?
The data is sufficient if we are able to get a CONCLUSIVE YES or a CONCLUSIVE NO from the information in the statements.
If using the information in the statement(s), we arrive at an answer that is sometimes yes and sometimes no, the data is not sufficient.

Q3. When is the answer yes?
The answer to this question will be a conclusive 'yes' if | a | > | b |.

Q4. When is the answer no?
The answer will be a conclusive 'no' if | a | ≤ | b |

#### Step 2: Evaluate Statement 1 ALONE

Statement 1: $$frac{1}{$a - b$}) > $$frac{1}{$b - a$})

Approach: Counter Example
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 answer options A and D.

#### Step 3: Evaluate Statement 2 ALONE

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, because the sum is negative, 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 |.

We are not able to get a conclusive answer using statement 2.
Hence, statement 2 is not sufficient.
Eliminate answer option B. Choices narrow down to C or E.

#### Step 4: Evaluate Statements TOGETHER

Statements: $$frac{1}{$a - b$}) > $$frac{1}{$b - a$}) and a + b < 0

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.

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