Data Sufficiency

If $x$ is a real number, is $x > 0$?

(1) $x^3 - x = 0$ and $x \neq 0$
(2) $|x + 2| = -x - 2$

Which of the following choices correctly identifies the sufficiency of the statements?

In a GMAT Data Sufficiency question framed as a 'Yes/No' question (such as 'Is $x > 0$?'), a statement that conclusively proves that the condition is always false (e.g., $x < 0$) is classified as insufficient because it results in a negative response.

If $x$ and $y$ are non-zero real numbers, is $\frac{x}{y} + \frac{y}{x} < 2$?

(1) $(x + y)^2 > x^2 + y^2$
(2) $x^3 y^2 < 0$

If $a$ and $b$ are real numbers, is $a + b > 0$?

(1) $a < -2$ and $b < -3$
(2) $ab > 0$

Which of the following choices correctly describes the sufficiency of the statements to answer the question?

If $k$ is a real number, is $|k| < 4$?

(1) $k > -3$
(2) $k < 3$

Which of the following describes the sufficiency of the statements?

If $x$ is a positive integer, is $x^2 - 1$ divisible by 8?

(1) $x$ is an odd integer.
(2) $x = 2k$ for some integer $k$.

In GMAT Data Sufficiency, if a statement evaluated in a Yes/No question type yields a definitive 'No' for every possible case satisfying the statement, that statement must be classified as insufficient because the answer to the question stem is negative.

If $k$ is a real number, is $k > 0$?

(1) $k^2 + 5k + 6 = 0$
(2) $|k| = 2$

In a GMAT Data Sufficiency 'Yes/No' question, if a statement allows multiple possible numerical values for a variable, but every permitted value yields a consistent 'Yes' answer to the question stem, that statement must be declared insufficient because it fails to determine a single unique value for the variable.

In a GMAT Data Sufficiency 'Value' question asking for the numerical value of an expression $E$, if a statement allows multiple distinct numerical values for a variable $x$, but every permitted value of $x$ yields the exact same unique numerical value for $E$, the statement is sufficient to answer the question.

If $x$ and $y$ are real numbers such that $x \neq 0$, is $|x| < y$?

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

Which of the following correctly describes the sufficiency of the statements?

If $k$ is a real number, is $k^3 > k$?

(1) $k^2 - k < 0$
(2) $k^2 > 1$

If $x$ is a real number, is $|x - 2| < 3$?

(1) $x^2 - 4x + 3 < 0$
(2) $|x| < 4$

Which of the options correctly describes the sufficiency of the statements?

In a GMAT Data Sufficiency 'Yes/No' question, if a statement allows us to establish conclusively that the answer to the question stem is 'No' under all permissible conditions, that statement is deemed insufficient because the outcome is negative.

If $a$ and $b$ are non-zero real numbers, is $|a + b| < |a| + |b|$?

(1) $a^3 b^2 < 0$
(2) $|a - b| > |a + b|$

Which of the following describes the sufficiency of the statements?

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

(1) $m < 0$

(2) $n > 2|m|$

If $x$ and $y$ are non-zero real numbers, is $\frac{x}{y} > 0$?

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

In a GMAT Data Sufficiency question asking whether the real number $x$ is positive, a statement that uniquely restricts $x$ to the set $\{-3, -1\}$ is sufficient to answer the question.