If x is a nonzero real number, is \frac{x^2 - 4x}{x} > 0? (1) x > 5 (2) x^2 > 16 — GMAT

If $x$ is a nonzero real number, is $\frac{x^2 - 4x}{x} > 0$ ?

(1) $x > 5$
(2) $x^2 > 16$

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.

Simplifying the target expression

\frac{x^2 - 4x}{x}

for

x \neq 0

yields

x - 4

. Thus, the question target rephrases to 'Is

x > 4

?'. Statement (1) states

x > 5

, which guarantees

x > 4

, yielding a definitive 'Yes' answer. Statement (2) allows

x > 4

x < -4

, yielding both 'Yes' and 'No' outcomes. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Simplify the question stem expression

Since

x \neq 0

, factor out

x

from the numerator:

\frac{x(x - 4)}{x} = x - 4

Simplifying the expression reveals the underlying condition required by the question stem.

Rephrase the target question

The original question 'Is

\frac{x^2 - 4x}{x} > 0

?' simplifies to 'Is

x - 4 > 0

?', which is equivalent to 'Is

x > 4

?'.

Target rephrasing turns a fraction inequality into a simple comparison.

Evaluate Statement (1):

x > 5

x > 5

, then

x

is strictly greater than 4. The answer to 'Is

x > 4

?' is a definitive YES.

Since Statement (1) provides a conclusive 'Yes', Statement (1) alone is sufficient.

Evaluate Statement (2):

x^2 > 16

Taking the square root gives

|x| > 4

, meaning

x > 4

x < -4

.
- If

x = 5

, then

x > 4

(YES).
- If

x = -5

, then

x < 4

(NO).

Because Statement (2) allows both 'Yes' and 'No' answers, it is not sufficient.

Question Stem Simplification and Target Rephrasing

Question