If x and y are non-zero real numbers, is \frac{|x|}{y} < 1? (1) x^2 < y^2 (2) x + y < 0 — GMAT

If $x$ and $y$ are non-zero real numbers, is $\frac{|x|}{y} < 1$ ?

(1) $x^2 < y^2$
(2) $x + y < 0$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
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
EITHER statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

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

Statement (1) gives

x^2 < y^2

, which means

|x| < |y|

. If

y > 0

|y| = y

, so

|x| < y \implies \frac{|x|}{y} < 1

. If

y < 0

|x| > 0

implies

\frac{|x|}{y} < 0 < 1

. Thus, Statement (1) alone yields a definitive 'Yes' and is sufficient. Statement (2) allows

x = -3, y = 1

(yielding a ratio of 3, which is not less than 1) and

x = -1, y = -2

(yielding a ratio of -0.5, which is less than 1), so Statement (2) alone is insufficient.

Adım Adım Çözüm

Rephrase the target question

The target question asks whether

\frac{|x|}{y} < 1

for non-zero real numbers

x

and

y

Since

|x| > 0

for any non-zero real number

x

, if

y < 0

, the ratio

\frac{|x|}{y}

is strictly negative, which is always less than 1. If

y > 0

\frac{|x|}{y} < 1

is equivalent to

|x| < y

Evaluate Statement (1):

x^2 < y^2

Taking the principal square root of both sides gives

|x| < |y|

y > 0

|y| = y

, so

|x| < y

, which implies

\frac{|x|}{y} < 1

(YES). If

y < 0

, then

y

is negative and

|x|

is positive, so

\frac{|x|}{y} < 0 < 1

(YES). Since Statement (1) yields a definitive YES in all cases, Statement (1) ALONE is sufficient.

Evaluate Statement (2):

x + y < 0

Test suitable numbers.

Case A: Let

x = -3

and

y = 1

. Then

x + y = -2 < 0

, but

\frac{|-3|}{1} = 3 \not< 1

(NO). Case B: Let

x = -1

and

y = -2

. Then

x + y = -3 < 0

, and

\frac{|-1|}{-2} = -0.5 < 1

(YES). Because Statement (2) can yield both YES and NO, Statement (2) ALONE is not sufficient.

Anahtar Kavram

Data Sufficiency evaluation of absolute values and algebraic inequalities with unknown signs

Tahmini Süre:2m 0s

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