Inequalities, Absolute Values, and Number Ranges in Data Sufficiency Practice Questions — GMAT

Question 1Question →

If $x$ is a real number, is $|x^2 - 4| < 5$ ?

(1) $|x - 1| < 2$
(2) $|x + 1| < 3$

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

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

Rephrasing the question target

|x^2 - 4| < 5

gives

-5 < x^2 - 4 < 5 \implies -1 < x^2 < 9

. Since

x^2 \ge 0

for all real numbers, the condition reduces to

x^2 < 9

, or

-3 < x < 3

.

Statement (1) states

|x - 1| < 2

, which translates to

-2 < x - 1 < 2 \implies -1 < x < 3

. Any number

x

in

(-1, 3)

automatically satisfies

-3 < x < 3

, giving a definitive 'Yes'. Hence, Statement (1) alone is sufficient.

Statement (2) states

|x + 1| < 3

, which translates to

-3 < x + 1 < 3 \implies -4 < x < 2

. If

x = 0

,

x

is in

(-3, 3)

('Yes'); if

x = -3.5

,

x

is not in

(-3, 3)

('No'). Because both 'Yes' and 'No' are possible, Statement (2) alone is not sufficient.

Therefore, the statement indicating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient, is the correct choice.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target inequality

|x^2 - 4| < 5

is equivalent to

-5 < x^2 - 4 < 5

. Adding 4 yields

-1 < x^2 < 9

. Since

x^2 \ge 0

for all real numbers,

-1 < x^2

is always true. Thus, the target question simplifies to whether

x^2 < 9

, which means

-3 < x < 3

.

Simplifying the question stem simplifies evaluating the statement ranges.

2

Evaluate Statement (1):

|x - 1| < 2

.

Expanding the absolute value gives

-2 < x - 1 < 2

, which simplifies to

-1 < x < 3

. Every value of

x

in the interval

(-1, 3)

is strictly inside

(-3, 3)

. Thus, Statement (1) guarantees a definitive 'Yes'.

Since Statement (1) provides a single definitive answer ('Yes'), Statement (1) alone is sufficient.

3

Evaluate Statement (2):

|x + 1| < 3

.

Expanding the absolute value gives

-3 < x + 1 < 3

, which simplifies to

-4 < x < 2

. If

x = 0

, then

-3 < 0 < 3

is true ('Yes'). If

x = -3.5

, then

|(-3.5)^2 - 4| = |12.25 - 4| = 8.25 < 5

is false ('No').

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

Key Concept

Rephrasing absolute value inequality bounds in Data Sufficiency questions

Question 2Question →

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

(1) $|x| = 5$
(2) $x + 3 > 5$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The choice stating that Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient, is correct because Statement (1) yields two possible values for

x

(

5

and

-5

), resulting in both 'Yes' and 'No' answers. In contrast, Statement (2) simplifies to

x > 2

, which guarantees that

x > 0

with a definitive 'Yes'.

Step-by-Step Solution

1

Rephrase the target question

The target asks whether

x

lies to the right of

0

on the real number line (a Yes/No question).

Establishing clear sufficiency criteria requires identifying what yields a definitive 'Yes' or definitive 'No'.

2

Evaluate Statement (1):

|x| = 5

x

can be

5

or

-5

. If

x = 5

, the answer is Yes (

5 > 0

). If

x = -5

, the answer is No (

-5 \ngtr 0

).

Since Statement (1) allows both a 'Yes' and a 'No' response, it is NOT sufficient.

3

Evaluate Statement (2):

x + 3 > 5

Subtracting

3

from both sides gives

x > 2

. Any number strictly greater than

2

is automatically greater than

0

.

Statement (2) provides a definitive 'Yes' answer to the question

x > 0

, so Statement (2) ALONE is sufficient.

Key Concept

Absolute Value Definition and Single-Variable Inequality Simplification in Data Sufficiency

Question 3Question →

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

(1) $|x| = -x$
(2) $x + |x| > 0$

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

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

The option stating that each statement alone is sufficient is correct. Statement (1) shows that

x \le 0

, which gives a definitive 'No' to whether

x > 0

. Statement (2) shows that

x > 0

, which gives a definitive 'Yes' to whether

x > 0

. Because both statements yield a clear, conclusive answer independently, each statement alone is sufficient.

Step-by-Step Solution

1

Analyze Statement (1):

|x| = -x

By definition of absolute value,

|x| = -x

holds true if and only if

x \le 0

.

If

x \le 0

, then

x

is not strictly positive, so the answer to 'Is

x > 0

?' is a definitive 'No'.

2

Evaluate sufficiency of Statement (1)

Statement (1) provides a definitive 'No' answer to the question stem.

In Data Sufficiency, a statement that consistently yields a definitive 'No' is sufficient.

3

Analyze Statement (2):

x + |x| > 0

If

x \le 0

, then

|x| = -x

, so

x + |x| = x - x = 0

, which is not greater than

0

. If

x > 0

, then

|x| = x

, so

x + |x| = 2x > 0

.

The inequality

x + |x| > 0

is satisfied if and only if

x > 0

.

4

Evaluate sufficiency of Statement (2)

Statement (2) provides a definitive 'Yes' answer to the question stem.

Since Statement (2) guarantees

x > 0

, it is sufficient on its own.

5

Combine evaluations

Each statement alone provides a definitive answer to the question stem.

Therefore, EACH statement ALONE is sufficient.

Key Concept

Absolute Value Definition and Yes/No Data Sufficiency Decision Logic

Estimated Time:1m 0s

Question 4Question →

If $x$ and $y$ are real numbers, is $|x - y| < |x + y|$ ?

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

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

Rephrasing the stem by squaring both non-negative sides reveals that

|x - y| < |x + y|

is strictly equivalent to

xy > 0

. Statement (1) specifies

x^3 y > 0

. Because

x^3

and

x

always have the same sign,

x^3 y > 0

means

x

and

y

share the same sign, which guarantees

xy > 0

. Thus, Statement (1) alone provides a definitive 'Yes'. Statement (2) states

|x| < y

, which forces

y > 0

but allows

x

to be either positive or negative, so

xy

can be either positive or negative. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem by squaring both sides of the inequality.

Since both

|x - y|

and

|x + y|

are non-negative,

|x - y| < |x + y| \iff (x - y)^2 < (x + y)^2 \iff x^2 - 2xy + y^2 < x^2 + 2xy + y^2 \iff 0 < 4xy \iff xy > 0

. The question asks whether

x

and

y

have the same sign (and neither is zero).

Simplifying absolute value inequalities before analyzing statements prevents misinterpretation.

2

Evaluate Statement (1):

x^3 y > 0

.

Since

x^3

always has the exact same algebraic sign as

x

for all real numbers,

x^3 y > 0 \implies xy > 0

. This yields a definitive 'Yes' to the rephrased question.

Statement (1) alone provides enough information to answer the target question conclusively.

3

Evaluate Statement (2):

|x| < y

.

Since

|x| \ge 0

, this implies

y > 0

. However,

x

can be positive (e.g.,

x=2, y=3 \implies xy=6 > 0

, Yes) or negative (e.g.,

x=-2, y=3 \implies xy=-6 < 0

, No).

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

Key Concept

Rephrasing absolute value inequalities and analyzing product sign constraints in Data Sufficiency

Question 5Question →

If $x$ is a real number, is $|x - 3| < 5$ ?

(1) $1 < x < 6$
(2) $x > 0$

Which of the following describes the sufficiency of the statements?

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The correct choice is the option stating that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Rephrasing

|x - 3| < 5

yields

-2 < x < 8

. Statement (1) restricts

x

to

(1, 6)

, which is completely inside

(-2, 8)

, guaranteeing a 'Yes' answer. Statement (2) allows values like

x = 2

('Yes') and

x = 10

('No'), so it is not sufficient.

Step-by-Step Solution

1

Rephrase the target inequality from the stem.

The absolute value inequality

|x - 3| < 5

is equivalent to

-5 < x - 3 < 5

. Adding 3 to all parts gives

-2 < x < 8

. The question asks: 'Is

x

strictly between

-2

and

8

?'

Simplifying the stem into a clear range for

x

makes evaluating sufficiency straightforward.

2

Evaluate Statement (1):

1 < x < 6

.

Every number in the interval

(1, 6)

is strictly between

-2

and

8

. Thus, the answer to 'Is

-2 < x < 8

?' is a definitive 'Yes'. Statement (1) alone is sufficient.

Since the entire set of allowed values under statement (1) is a subset of

(-2, 8)

, statement (1) provides a conclusive answer.

3

Evaluate Statement (2):

x > 0

.

If

x = 2

, then

-2 < 2 < 8

is True (Yes). However, if

x = 10

, then

-2 < 10 < 8

is False (No). Because both 'Yes' and 'No' are possible, Statement (2) alone is not sufficient.

A statement is insufficient if it permits values that produce conflicting answers to the target question.

Key Concept

Absolute Value Inequalities and Range Containment in Data Sufficiency

Question 6Question →

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?

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Rephrasing

|k| < 4

gives

-4 < k < 4

. Neither statement alone provides both an upper and lower bound. However, combining Statement (1) (

k > -3

) and Statement (2) (

k < 3

) establishes

-3 < k < 3

. Because the interval

(-3, 3)

is entirely contained inside

(-4, 4)

, any value of

k

satisfying both statements guarantees

|k| < 4

. Thus, both statements together are sufficient.

Step-by-Step Solution

1

Rephrase the question stem target

The target inequality

|k| < 4

is equivalent to

-4 < k < 4

. The question asks whether

k

lies strictly inside the range

(-4, 4)

.

Simplifying an absolute value inequality into a compound inequality clarifies the exact target range.

2

Evaluate Statement (1) independently

Statement (1) specifies

k > -3

. If

k = 0

, then

|0| = 0 < 4

(Yes). If

k = 5

, then

|5| = 5 \not< 4

(No).

Since Statement (1) yields both 'Yes' and 'No' answers, it is insufficient.

3

Evaluate Statement (2) independently

Statement (2) specifies

k < 3

. If

k = 0

, then

|0| = 0 < 4

(Yes). If

k = -5

, then

|-5| = 5 \not< 4

(No).

Since Statement (2) yields both 'Yes' and 'No' answers, it is insufficient.

4

Evaluate Statements (1) and (2) combined

Combining

k > -3

and

k < 3

gives

-3 < k < 3

. Any value of

k

in

(-3, 3)

automatically falls within

(-4, 4)

, yielding a definitive 'Yes' to

|k| < 4

.

The interval

(-3, 3)

is a complete subset of

(-4, 4)

, making the combined information sufficient.

Key Concept

Absolute Value Range Simplification and Subset Verification in Data Sufficiency

Question 7Question →

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?

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Combining both statements is sufficient. Statement (1) establishes that

|x| < |y|

because taking the square root of

x^2 < y^2

yields absolute values. Statement (2) specifies

y > 0

, meaning

|y| = y

. Substituting

y

for

|y|

gives

|x| < y

, which conclusively answers the question stem with 'Yes'.

Step-by-Step Solution

1

Rephrase the target question

The target question asks whether

|x| < y

. Since

|x| \ge 0

for all real numbers

x

, a necessary condition for

|x| < y

to be true is

y > 0

.

Simplifying absolute value conditions clarifies what values of variables are required.

2

Evaluate Statement (1):

x^2 < y^2

Taking the principal square root of both sides gives

\sqrt{x^2} < \sqrt{y^2}

, which simplifies to

|x| < |y|

.

The square root of a squared real number is its absolute value.

3

Test sufficiency of Statement (1) alone

If

x = 1

and

y = 2

, then

1^2 < 2^2

is true, and

|1| < 2

(Yes). If

x = 1

and

y = -2

, then

1^2 < (-2)^2

is true, but

|1| < -2

is false (No). Since both Yes and No are possible, Statement (1) alone is NOT sufficient.

Knowing

|x| < |y|

does not determine the sign of

y

.

4

Evaluate Statement (2):

y > 0

Statement (2) gives no information about

x

. For example, if

y = 5

and

x = 2

,

|2| < 5

(Yes). If

y = 5

and

x = 10

,

|10| < 5

is false (No). Thus Statement (2) alone is NOT sufficient.

No bound on

x

is provided.

5

Evaluate Statements (1) and (2) together

From Statement (1),

|x| < |y|

. From Statement (2),

y > 0

, which implies

|y| = y

. Substituting

|y| = y

into

|x| < |y|

gives

|x| < y

. This definitively answers 'Yes' to the question stem.

Combining the magnitude inequality with the sign constraint produces a unique, definitive Yes answer.

Key Concept

Absolute Value Inequalities and Number Ranges in Data Sufficiency

Question 8Question →

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?

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The correct answer specifies that Statement (1) alone is sufficient while Statement (2) alone is not. Rephrasing the question stem shows that

|x - 2| < 3

is equivalent to

-1 < x < 5

. Statement (1) yields

1 < x < 3

, which lies entirely inside

(-1, 5)

, guaranteeing a 'Yes' answer. Statement (2) yields

-4 < x < 4

, which allows values outside

(-1, 5)

(such as

x = -2

, giving 'No') as well as values inside (such as

x = 0

, giving 'Yes'), so it is not sufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The inequality

|x - 2| < 3

is equivalent to

-3 < x - 2 < 3

, which simplifies to

-1 < x < 5

. The question asks: Is

x

strictly between

-1

and

5

?

Simplifying the target inequality establishes the exact boundary values needed for evaluation.

2

Evaluate Statement (1):

x^2 - 4x + 3 < 0

.

Factoring the quadratic gives

(x - 1)(x - 3) < 0

, which holds when

1 < x < 3

.

Since the interval

(1, 3)

is entirely contained within the target range

(-1, 5)

, every value of

x

satisfying Statement (1) results in a definitive 'Yes' to the question. Thus, Statement (1) is sufficient.

3

Evaluate Statement (2):

|x| < 4

.

This absolute value inequality simplifies to

-4 < x < 4

.

If

x = 0

, then

-1 < 0 < 5

is True ('Yes'). If

x = -2

, then

-1 < -2 < 5

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

Key Concept

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

Question 9Question →

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?

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The correct response identifies that statement (2) alone provides enough information to answer the question, whereas statement (1) does not. Rephrasing the stem

|a + b| < |a| + |b|

reveals it is true if and only if

a

and

b

have opposite signs (

ab < 0

). Statement (1) only confirms that

a < 0

, leaving the sign of

b

unknown. Statement (2) can be squared to yield

ab < 0

, which conclusively answers the question with a 'Yes'.

Step-by-Step Solution

1

Rephrase the question stem

The inequality

|a + b| < |a| + |b|

holds if and only if

a

and

b

have opposite signs (i.e.,

ab < 0

). If

a

and

b

have the same sign or if either is zero,

|a + b| = |a| + |b|

. Thus, the target question is: 'Is

ab < 0

?'

Simplifying absolute value properties reduces the stem to a straightforward sign comparison.

2

Evaluate Statement (1):

a^3 b^2 < 0

Since

b \neq 0

,

b^2 > 0

for all real numbers. Thus,

a^3 b^2 < 0

implies

a^3 < 0

, which means

a < 0

. However,

b

can be either positive or negative. If

b > 0

, then

ab < 0

(YES). If

b < 0

, then

ab > 0

(NO).

Since statement (1) allows both a YES and a NO answer, Statement (1) ALONE is NOT sufficient.

3

Evaluate Statement (2):

|a - b| > |a + b|

Square both sides (valid since both sides are non-negative):

(a - b)^2 > (a + b)^2 \implies a^2 - 2ab + b^2 > a^2 + 2ab + b^2 \implies -2ab > 2ab \implies 4ab < 0 \implies ab < 0

.

Statement (2) directly proves that

ab < 0

, yielding a definitive YES to the question stem. Thus, Statement (2) ALONE is SUFFICIENT.

Key Concept

Absolute Value Distance and Sign Properties

Question 10Question →

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.

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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.

Step-by-Step Solution

1

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.

2

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.

3

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.

Key Concept

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

Estimated Time:2m 0s

Question 11Question →

If $r$ and $s$ are real numbers, is $r|s| < s|r|$ ?

(1) $rs < 0$
(2) $r + s > |r - s|$

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

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The choice stating that Statement (2) ALONE is sufficient while Statement (1) alone is not sufficient is correct. Rephrased,

r|s| < s|r|

is true only when

r < 0

and

s > 0

. Statement (1) allows

r

to be positive while

s

is negative, yielding a 'No', as well as

r

negative while

s

is positive, yielding a 'Yes', so it is insufficient. Statement (2) simplifies algebraically to

rs > 0

and

r + s > 0

, which guarantees both numbers are positive. When both numbers are positive,

r|s| < s|r|

reduces to

rs < rs

, which is false. Because Statement (2) yields a definitive 'No', it is sufficient on its own.

Step-by-Step Solution

1

Rephrase the question stem target

The inequality

r|s| < s|r|

holds if and only if

r < 0

and

s > 0

.

If

r

and

s

have the same sign or either is zero,

r|s| = s|r|

, making the strict inequality false. If

r > 0

and

s < 0

,

r|s| = -rs

and

s|r| = -rs

, making both sides equal. Only when

r < 0

and

s > 0

does

r|s| = rs < 0

and

s|r| = -rs > 0

, which makes the inequality true.

2

Evaluate Statement (1):

rs < 0

Statement (1) is NOT sufficient.

This condition means

r

and

s

have opposite signs. Case 1: If

r = -2

and

s = 3

, then

r < 0

and

s > 0

, so the answer is YES. Case 2: If

r = 3

and

s = -2

, then

r > 0

and

s < 0

, so the answer is NO. Since both YES and NO are possible, Statement (1) is insufficient.

3

Evaluate Statement (2):

r + s > |r - s|

Statement (2) ALONE is sufficient.

Since

|r - s| \ge 0

, we have

r + s > 0

. Squaring both positive sides gives

(r + s)^2 > (r - s)^2 \implies r^2 + 2rs + s^2 > r^2 - 2rs + s^2 \implies 4rs > 0 \implies rs > 0

. Since

rs > 0

and

r + s > 0

, both

r

and

s

must be positive. If

r > 0

and

s > 0

, then

r|s| = rs

and

s|r| = rs

, so

r|s| < s|r|

simplifies to

rs < rs

, which is FALSE. This gives a definitive NO to the question, making Statement (2) sufficient.

Key Concept

Data Sufficiency Yes/No Decision Logic with Absolute Value Inequalities

Question 12Question →

If $p$ and $q$ are non-zero real numbers, is $|p + q| > |p - q|$ ?

(1) $|p + 2q| < |p - 2q|$
(2) $p - q > 0$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The correct response identifies Statement (1) as sufficient and Statement (2) as insufficient. Rephrasing the question stem by squaring both sides of

|p + q| > |p - q|

yields

4pq > 0

, or

pq > 0

. Statement (1) expands and simplifies to

8pq < 0

, meaning

pq < 0

. This provides a definitive 'No' answer to the question 'Is

pq > 0

?', which constitutes full sufficiency. Statement (2) merely tells us

p > q

, which permits

p

and

q

to both be positive (

pq > 0

) or to have opposite signs (

pq < 0

), rendering Statement (2) insufficient.

Step-by-Step Solution

1

Rephrase the question stem target geometrically or algebraically.

Squaring both non-negative sides of

|p + q| > |p - q|

gives

(p + q)^2 > (p - q)^2 \implies p^2 + 2pq + q^2 > p^2 - 2pq + q^2 \implies 4pq > 0 \implies pq > 0

. The question asks: Is

pq > 0

?

Simplifying the question stem simplifies evaluation of both statements by focusing on whether

p

and

q

have the same sign.

2

Evaluate Statement (1):

|p + 2q| < |p - 2q|

.

Squaring both non-negative sides gives

(p + 2q)^2 < (p - 2q)^2 \implies p^2 + 4pq + 4q^2 < p^2 - 4pq + 4q^2 \implies 8pq < 0 \implies pq < 0

.

Since

pq < 0

, we can answer the rephrased target question ('Is

pq > 0

?') with a definitive 'NO'. In Data Sufficiency, a definitive 'No' means the statement is sufficient.

3

Evaluate Statement (2):

p - q > 0

.

If

p = 3

and

q = 1

, then

p - q = 2 > 0

and

pq = 3 > 0

(Yes). If

p = 1

and

q = -2

, then

p - q = 3 > 0

and

pq = -2 < 0

(No).

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

Key Concept

Rephrasing absolute value distance inequalities and recognizing that a definitive 'No' response in Yes/No Data Sufficiency constitutes sufficiency.

Question 13Question →

If $k$ is a real number, is $|k^2 - 4| < 3k$ ?

(1) $k^2 - 5k + 4 < 0$
(2) $|k - 1| < 2$

Which statement(s) provide sufficient information to determine whether the inequality holds?

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The stem inequality

|k^2 - 4| < 3k

simplifies to

1 < k < 4

. Statement (1) directly yields

1 < k < 4

, providing a definitive Yes answer. Statement (2) yields

-1 < k < 3

, which includes values inside

(1, 4)

such as

k = 2

(Yes) and values outside

(1, 4)

such as

k = 0.5

(No), making Statement (2) insufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target inequality

|k^2 - 4| < 3k

requires

3k > 0

(so

k > 0

) and translates to

-3k < k^2 - 4 < 3k

. Solving

k^2 - 4 < 3k

yields

(k - 4)(k + 1) < 0 \implies -1 < k < 4

. Solving

-3k < k^2 - 4

for

k > 0

yields

k^2 + 3k - 4 > 0 \implies (k + 4)(k - 1) > 0 \implies k > 1

. Combining these gives the rephrased question: 'Is

1 < k < 4

?'

Simplifying the question stem before evaluating the statements prevents misinterpretation of absolute value boundary conditions.

2

Evaluate Statement (1):

k^2 - 5k + 4 < 0

.

Factoring the quadratic gives

(k - 1)(k - 4) < 0

, which means

1 < k < 4

.

Since Statement (1) guarantees that

k

lies strictly between

1

and

4

, the answer to the rephrased question is a definitive Yes. Statement (1) ALONE is sufficient.

3

Evaluate Statement (2):

|k - 1| < 2

.

Removing the absolute value gives

-2 < k - 1 < 2

, or

-1 < k < 3

.

If

k = 2

, then

1 < 2 < 4

, giving a Yes answer. If

k = 0.5

, then

0.5

is in

(-1, 3)

but not in

(1, 4)

, giving a No answer (since

|0.5^2 - 4| = 3.75 > 3(0.5) = 1.5

). Because Statement (2) allows both Yes and No outcomes, it is NOT sufficient.

Key Concept

Absolute Value Inequalities and Question Stem Rephrasing

Alternative Method

Instead of solving the quadratic inequality

-3k < k^2 - 4 < 3k

algebraically, test the boundary points

k = 1

and

k = 4

in

|k^2 - 4| = 3k

: at

k = 1

,

|1 - 4| = 3 = 3(1)

; at

k = 4

,

|16 - 4| = 12 = 3(4)

. Since

3k

grows linearly while

k^2 - 4

grows quadratically, the inequality holds strictly between these two positive roots

1 < k < 4

.

Estimated Time:2m 0s

Question 14Question →

If $x$ is a real number, is $|x - 3| + |x + 2| \le 7$ ?

(1) $x^2 - 2x - 8 \le 0$
(2) $x(x - 5) \le 0$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

Rephrasing the question target

|x - 3| + |x + 2| \le 7

shows that it is satisfied if and only if

-3 \le x \le 4

. Statement (1) yields the interval

[-2, 4]

, which is completely contained within

[-3, 4]

, guaranteeing a definitive 'Yes'. Statement (2) yields

[0, 5]

, which includes values like

4.5

that result in a 'No'. Thus, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem inequality

|x - 3| + |x + 2| \le 7

.

The target inequality simplifies to

-3 \le x \le 4

.

Interpret

|x - 3| + |x + 2|

as the sum of distances from

x

to

3

and

-2

on the real number line. The distance between

-2

and

3

is

5

. For

-2 \le x \le 3

, the sum of distances is constantly

5 \le 7

. For

x > 3

, the sum is

2x - 1 \le 7 \implies x \le 4

. For

x < -2

, the sum is

1 - 2x \le 7 \implies x \ge -3

. Thus, the inequality holds if and only if

x \in [-3, 4]

.

2

Evaluate Statement (1):

x^2 - 2x - 8 \le 0

.

Statement (1) gives

-2 \le x \le 4

.

Factoring gives

(x - 4)(x + 2) \le 0

, which holds when

x \in [-2, 4]

. Since

[-2, 4] \subseteq [-3, 4]

, every value of

x

satisfying Statement (1) guarantees that

|x - 3| + |x + 2| \le 7

is true. This yields a definitive 'Yes'. Statement (1) alone is sufficient.

3

Evaluate Statement (2):

x(x - 5) \le 0

.

Statement (2) gives

0 \le x \le 5

.

The solution set is

x \in [0, 5]

. If

x = 1

,

-3 \le 1 \le 4

is true (Yes). If

x = 4.5

,

|4.5 - 3| + |4.5 + 2| = 1.5 + 6.5 = 8 \le 7

is false (No). Since Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) alone is not sufficient.

Key Concept

Question stem rephrasing for distance-based absolute values and set-inclusion sufficiency evaluation

Question 15Question →

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

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

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

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

Rephrasing the stem

|x - 3| < 2x

shows it is equivalent to asking whether

x > 1

. Statement (2) restricts

x

to the range

1 < x < 3

, where every value of

x

satisfies

x > 1

, providing a definitive YES answer. Statement (1) permits

x < 1

(giving a NO) and

x > 3

(giving a YES), making Statement (1) insufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target inequality is

|x - 3| < 2x

. Since absolute values are non-negative (

|x - 3| \ge 0

),

2x

must be strictly positive, so

x > 0

.
- If

x \ge 3

,

|x - 3| = x - 3 < 2x \implies x > -3

, which holds for all

x \ge 3

.
- If

0 < x < 3

,

|x - 3| = 3 - x < 2x \implies 3x > 3 \implies x > 1

.
Thus, the inequality

|x - 3| < 2x

holds if and only if

x > 1

. The target question simplifies to: 'Is

x > 1

?'

Simplifying the question stem before analyzing statements prevents misinterpreting the range of valid solutions.

2

Evaluate Statement (1):

x^2 - 4x + 3 > 0

.

Factoring the quadratic yields

(x - 1)(x - 3) > 0

, so

x < 1

or

x > 3

.
- If

x = 0

(which is

< 1

),

|0 - 3| = 3

and

2(0) = 0

. Is

3 < 0

? NO.
- If

x = 4

(which is

> 3

),

|4 - 3| = 1

and

2(4) = 8

. Is

1 < 8

? YES.
Since Statement (1) allows for both YES and NO outcomes, it is NOT sufficient.

Testing specific values from the disjoint solution set demonstrates that a single definitive answer cannot be reached.

3

Evaluate Statement (2):

|x - 2| < 1

.

Solving the absolute value inequality gives

-1 < x - 2 < 1 \implies 1 < x < 3

.
For every value of

x

in the interval

(1, 3)

,

x

is strictly greater than

1

. Therefore, the answer to 'Is

x > 1

?' is a definitive YES for all possible values of

x

in this statement.

A statement is sufficient if every allowed value of

x

yields a consistent YES answer to the rephrased question stem.

Key Concept

Question Stem Rephrasing for Absolute Value Inequalities

Estimated Time:2m 0s

Question 16Question →

If $x$ is a real number, is $x^2 < 4$ ?

(1) $|x - 1| < 2$
(2) $|x + 1| < 3$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Rephrasing the target question

x^2 < 4

shows that we need to determine if

-2 < x < 2

. Statement (1) gives

-1 < x < 3

, which allows values outside

(-2, 2)

like

x = 2.5

, so it is insufficient alone. Statement (2) gives

-4 < x < 2

, which allows values outside

(-2, 2)

like

x = -3

, so it is insufficient alone. Taking the intersection of both statements gives

-1 < x < 2

. Since every number in

(-1, 2)

satisfies

-2 < x < 2

, the combined statements definitively answer YES.

Step-by-Step Solution

1

Rephrase the target question

The target question asking whether

x^2 < 4

is equivalent to asking whether

-2 < x < 2

.

Taking the square root of both sides of

x^2 < 4

yields

|x| < 2

, which expands to

-2 < x < 2

.

2

Evaluate Statement (1) independently

Statement (1) states

|x - 1| < 2

, which expands to

-2 < x - 1 < 2

, or

-1 < x < 3

.

If

x = 0

, then

-1 < 0 < 3

is true and

0^2 < 4

(Yes). If

x = 2.5

, then

-1 < 2.5 < 3

is true, but

2.5^2 = 6.25 \not< 4

(No). Because we get both Yes and No answers, Statement (1) alone is insufficient.

3

Evaluate Statement (2) independently

Statement (2) states

|x + 1| < 3

, which expands to

-3 < x + 1 < 3

, or

-4 < x < 2

.

If

x = 0

, then

-4 < 0 < 2

is true and

0^2 < 4

(Yes). If

x = -3

, then

-4 < -3 < 2

is true, but

(-3)^2 = 9 \not< 4

(No). Because we get both Yes and No answers, Statement (2) alone is insufficient.

4

Evaluate Statements (1) and (2) together

Combining

-1 < x < 3

and

-4 < x < 2

requires

x

to satisfy both inequalities simultaneously, yielding the intersection

-1 < x < 2

.

Every value of

x

in the interval

(-1, 2)

lies strictly inside the required target interval

(-2, 2)

. Thus,

x^2 < 4

is definitively YES. Both statements together are sufficient.

Key Concept

Inequality range intersection and absolute value distance expansion in Data Sufficiency

Estimated Time:2m 0s

Question 17Question →

If $m$ and $n$ are non-zero real numbers, is $\frac{|m + n|}{m} > 1$ ?

(1) $m < -|n|$
(2) $mn > 0$ and $|m + n| = m + n$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

The correct choice is that EACH statement ALONE is sufficient. Statement (1) restricts

m

to negative values, guaranteeing that the numerator is non-negative while the denominator is negative, so the ratio can never exceed 1 (yielding a definitive 'No'). Statement (2) forces both

m

and

n

to be strictly positive, which simplifies the ratio to

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

(yielding a definitive 'Yes'). Since each statement provides a definitive answer independently, each statement alone is sufficient.

Step-by-Step Solution

1

Analyze the question stem target

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

.

Since absolute value

|m + n|

is always non-negative (

|m + n| \ge 0

), if

m < 0

, the ratio

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

, which can never be greater than 1. Thus, if

m < 0

, the answer is a definitive 'No'. If

m > 0

, the expression simplifies to

|m + n| > m

.

Establishing the behavior of the expression based on the sign of

m

simplifies statement evaluation.

2

Evaluate Statement (1):

m < -|n|

.

Since

n \neq 0

,

|n| > 0

, which implies

-|n| < 0

. Therefore,

m < 0

. Because

m < 0

and

|m + n| \ge 0

, the fraction

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

is non-positive (

\le 0

). Thus,

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

cannot be greater than 1, yielding a definitive 'No'. Statement (1) ALONE is sufficient.

A statement that yields a definitive 'No' to a Yes/No Data Sufficiency question is sufficient.

3

Evaluate Statement (2):

mn > 0

and

|m + n| = m + n

.

From

mn > 0

,

m

and

n

must have the same sign (both positive or both negative). The condition

|m + n| = m + n

requires

m + n \ge 0

. If both

m

and

n

were negative,

m + n

would be negative, contradicting

|m + n| \ge 0

. Hence, both

m

and

n

must be positive (

m > 0

and

n > 0

). Substituting into the expression:

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

. Since

n > 0

and

m > 0

,

\frac{n}{m} > 0

, so

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

. This yields a definitive 'Yes'. Statement (2) ALONE is sufficient.

Determining that

m

and

n

must both be positive allows exact evaluation of the inequality.

Key Concept

Data Sufficiency logic for Yes/No questions involving absolute value properties and variable sign analysis.

Question 18Question →

If $x$ is a real number, is $|2x - 3| \le 5$ ?

(1) $|x - 1| \le 2$
(2) $x^2 - 3x - 4 \le 0$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

Rephrasing the target inequality

|2x - 3| \le 5

gives

-1 \le x \le 4

. Statement (1) yields

-1 \le x \le 3

, which is completely contained within

[-1, 4]

and therefore guarantees a definitive 'Yes'. Statement (2) yields

-1 \le x \le 4

, which matches the condition directly and also gives a definitive 'Yes'. Hence, each statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the question stem target inequality algebraically.

-5 \le 2x - 3 \le 5 \implies -2 \le 2x \le 8 \implies -1 \le x \le 4

. Target Question: Is

-1 \le x \le 4

?

Simplifying the question stem converts an absolute value inequality into a clear number range for

x

.

2

Evaluate Statement (1):

|x - 1| \le 2

.

-2 \le x - 1 \le 2 \implies -1 \le x \le 3

. Since every value in

[-1, 3]

is also in

[-1, 4]

, the answer to 'Is

-1 \le x \le 4

?' is a definitive YES.

If a statement's allowed range is a subset of the target range, it guarantees a definitive 'Yes' answer.

3

Evaluate Statement (2):

x^2 - 3x - 4 \le 0

.

Factor the quadratic:

(x - 4)(x + 1) \le 0 \implies -1 \le x \le 4

. This matches the target range exactly, yielding a definitive YES.

Matching the target inequality range provides a definitive 'Yes' answer.

Key Concept

Data Sufficiency Yes/No decision logic for absolute value range constraints and subset range implications.

Estimated Time:1m 30s

Question 19Question →

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

(1) $|a + 2b| < |a - 2b|$
(2) $|a - 1| < |b - 1|$

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

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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. Squaring both sides of the question stem inequality

|a - b| > |a + b|

yields

a^2 - 2ab + b^2 > a^2 + 2ab + b^2

, which simplifies directly to

ab < 0

. Statement (1) states

|a + 2b| < |a - 2b|

. Squaring both sides yields

a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2

, which simplifies to

8ab < 0

, or

ab < 0

. This gives a definitive YES to the question, so Statement (1) is sufficient. Statement (2) compares the distances of

a

and

b

from 1 on the number line. Choosing

a = 0.5

and

b = -2

satisfies

|0.5 - 1| < |-2 - 1|

and gives

ab = -1 < 0

(YES), whereas choosing

a = 0.5

and

b = 3

satisfies

|0.5 - 1| < |3 - 1|

and gives

ab = 1.5 > 0

(NO). Thus, Statement (2) is insufficient.

Step-by-Step Solution

1

Rephrase the question stem target using absolute value properties.

Since both sides of

|a - b| > |a + b|

are non-negative, square both sides:

(a - b)^2 > (a + b)^2 \implies a^2 - 2ab + b^2 > a^2 + 2ab + b^2 \implies -2ab > 2ab \implies 4ab < 0 \implies ab < 0

. Thus, the question asks: 'Is

ab < 0

?' (Do

a

and

b

have opposite signs?)

Simplifying the target inequality converts an absolute value relationship into a clear sign constraint on the product

ab

.

2

Evaluate Statement (1):

|a + 2b| < |a - 2b|

.

Square both sides:

(a + 2b)^2 < (a - 2b)^2 \implies a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2 \implies 4ab < -4ab \implies 8ab < 0 \implies ab < 0

. This provides a definitive YES to the question 'Is

ab < 0

?'

Statement (1) uniquely guarantees that

a

and

b

have opposite signs, making Statement (1) ALONE sufficient.

3

Evaluate Statement (2):

|a - 1| < |b - 1|

.

Test numbers: Case 1: If

a = 0.5

and

b = -2

, then

|0.5 - 1| = 0.5

and

|-2 - 1| = 3

. Since

0.5 < 3

, Statement (2) holds, and

ab = (0.5)(-2) = -1 < 0

(YES). Case 2: If

a = 0.5

and

b = 3

, then

|0.5 - 1| = 0.5

and

|3 - 1| = 2

. Since

0.5 < 2

, Statement (2) holds, and

ab = (0.5)(3) = 1.5 > 0

(NO).

Because Statement (2) permits both

ab < 0

and

ab > 0

, it cannot determine whether

ab < 0

. Statement (2) ALONE is insufficient.

Key Concept

Properties of Absolute Value Inequalities and Rephrased Data Sufficiency Targets

Estimated Time:2m 0s

Question 20Question →

If $x$ is a real number, is $|x + 4| > 2x$ ?

(1) $|x - 1| < 3$
(2) $x^2 - x - 6 < 0$

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

Rephrasing the question stem shows that

|x + 4| > 2x

is equivalent to

x < 4

. Statement (1) establishes that

-2 < x < 4

, which guarantees

x < 4

(definitive Yes). Statement (2) establishes that

-2 < x < 3

, which also guarantees

x < 4

(definitive Yes). Therefore, each statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The inequality

|x + 4| > 2x

simplifies to

x < 4

.

If

x \ge -4

, then

x + 4 > 2x \implies x < 4

. If

x < -4

,

|x + 4| = -(x + 4) > 2x \implies 3x < -4 \implies x < -4/3

, which holds for all

x < -4

. Thus,

|x + 4| > 2x

is true if and only if

x < 4

.

2

Evaluate Statement (1):

|x - 1| < 3

.

Statement (1) yields the range

-2 < x < 4

.

Unpacking

|x - 1| < 3

gives

-3 < x - 1 < 3 \implies -2 < x < 4

. Since every value in

(-2, 4)

is strictly less than

4

, the answer to 'Is

x < 4

?' is a definitive 'Yes'. Thus, Statement (1) alone is sufficient.

3

Evaluate Statement (2):

x^2 - x - 6 < 0

.

Statement (2) yields the range

-2 < x < 3

.

Factoring gives

(x - 3)(x + 2) < 0 \implies -2 < x < 3

. Since every value in

(-2, 3)

is strictly less than

4

, the answer to 'Is

x < 4

?' is a definitive 'Yes'. Thus, Statement (2) alone is sufficient.

Key Concept

Rephrasing absolute value inequalities in Data Sufficiency Yes/No questions

Estimated Time:1m 30s

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