If m and n are real numbers such that mn \neq 0, is \frac{|m + n|}{m} > 1? (1) m < 0 (2) n… — GMAT

If $m$ and $n$ are real numbers such that $mn \neq 0$ , is $\frac{|m + n|}{m} > 1$ ?

(1) $m < 0$

(2) $n > 2|m|$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The option stating that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient is correct. For Statement (1), since

m < 0

and the absolute value

|m + n|

is always non-negative, the quotient

\frac{|m + n|}{m}

is non-positive (

\le 0

). Therefore, it can never be greater than 1, giving a definitive 'No' to the question stem, which makes Statement (1) sufficient. For Statement (2),

m

can be positive (yielding a 'Yes') or negative (yielding a 'No'), making Statement (2) insufficient.

Analyze the target question stem and constraints.

The absolute value

|m + n|

is always non-negative (

|m + n| \ge 0

). The target asks whether

\frac{|m + n|}{m} > 1

Establishing the properties of the numerator and denominator helps simplify the condition.

Evaluate Statement (1):

m < 0

Since

|m + n| \ge 0

and

m < 0

, dividing a non-negative number by a negative number yields a result that is less than or equal to 0 (

\frac{|m + n|}{m} \le 0

). Thus, the expression can never be greater than 1, yielding a definitive 'No'.

A definitive 'No' answer means Statement (1) alone IS SUFFICIENT.

Evaluate Statement (2):

n > 2|m|

Since

2|m| > 0

n

must be positive. Test cases for

m

:
- Case 2a: If

m = 1

, then

n > 2

. Let

n = 3

. Then

\frac{|1 + 3|}{1} = 4 > 1

(Answer: YES).
- Case 2b: If

m = -1

, then

n > 2(1) = 2

. Let

n = 3

. Then

\frac{|-1 + 3|}{-1} = \frac{2}{-1} = -2 \ngtr 1

(Answer: NO).
Since both 'Yes' and 'No' are possible, Statement (2) alone is NOT SUFFICIENT.

Testing cases with positive and negative values of

m

shows that Statement (2) does not yield a unique answer.

Data Sufficiency Yes/No Decision Logic with Signed Denominator and Absolute Value Properties

Estimated Time:2m 0s

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