Inequalities, Absolute Values, and Number Ranges in Data Sufficiency Practice Questions — GMAT — Page 2

Question 21Question →

If $x$ is a real number and $x \neq -1$ , is $\frac{|x - 3|}{x + 1} < 1$ ?

(1) $|2x - 1| > 3$
(2) $x(x - 1) > 0$

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 stem shows that

\frac{|x - 3|}{x + 1} < 1

is satisfied when

x < -1

or

x > 1

. Statement (1) solves to

x < -1

or

x > 2

. Since

(2, \infty)

is completely contained inside

(1, \infty)

, every value satisfying Statement (1) yields a definitive YES to the question. Statement (2) solves to

x < 0

or

x > 1

. Choosing

x = 2

gives a YES answer, while choosing

x = -0.5

gives a NO answer. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem by analyzing cases for the denominator

x + 1

.

Case 1 (

x > -1

):

x + 1 > 0

, so

|x - 3| < x + 1 \iff -(x + 1) < x - 3 < x + 1

. The right inequality

-3 < 1

is always true, and the left inequality

-x - 1 < x - 3

simplifies to

2x > 2 \iff x > 1

.
Case 2 (

x < -1

):

x + 1 < 0

. The numerator

|x - 3| > 0

while the denominator is negative, making the ratio negative, which is always

< 1

.
Combining both cases: The target inequality holds if and only if

x < -1

or

x > 1

.

Rephrasing the stem target simplifies complex absolute value expressions into clear number line intervals.

2

Evaluate Statement (1):

|2x - 1| > 3

.

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

, or

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

.
Range:

x \in (-\infty, -1) \cup (2, \infty)

. Every value in this range satisfies

x < -1

or

x > 1

, guaranteeing a definitive YES to the question.

Statement 1 specifies a subset of the valid target range, making it sufficient alone.

3

Evaluate Statement (2):

x(x - 1) > 0

.

Range:

x < 0

or

x > 1

. Testing values within this range:
- If

x = 2

:

2(1) = 2 > 0

, and

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

(YES).
- If

x = -0.5

:

-0.5(-1.5) = 0.75 > 0

, but

\frac{|-0.5 - 3|}{-0.5 + 1} = \frac{3.5}{0.5} = 7 \not< 1

(NO).
Since both YES and NO outcomes are possible, Statement (2) is insufficient.

Finding a counterexample within the statement's solution set proves insufficiency.

Key Concept

Data Sufficiency evaluation for rational expressions containing absolute values and rephrasing inequality ranges.

Estimated Time:2m 0s

Question 22Question →

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.

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

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

EITHER 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.

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.

Step-by-Step Solution

1

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

.

2

Evaluate Statement (1):

x^2 < y^2

Taking the principal square root of both sides gives

|x| < |y|

.

If

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.

3

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.

Key Concept

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

Estimated Time:2m 0s

Question 23Question →

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

(1) $|x + y| < |x - y|$
(2) $x > y$

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 correct option states that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Squaring both sides of Statement (1) gives

x^2 + 2xy + y^2 < x^2 - 2xy + y^2

, which simplifies directly to

4xy < 0

, meaning

xy < 0

. When two numbers have opposite signs, their quotient

\frac{x}{y}

must be negative, making it strictly less than 1. This provides a definitive 'Yes' answer. Statement (2) gives

x > y

; if

x = 3

and

y = 2

,

\frac{3}{2} > 1

(No), but if

x = 1

and

y = -2

,

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

(Yes). Hence Statement (2) is not sufficient.

Step-by-Step Solution

1

Rephrase the target question

The target question asks if

\frac{x}{y} < 1

.

Understanding the algebraic condition required for a fraction to be less than 1 helps evaluate given statements.

2

Evaluate Statement (1):

|x + y| < |x - y|

Squaring both sides yields

(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0

.

Since both sides of the inequality are non-negative distance expressions, squaring preserves the inequality sign.

3

Determine the sufficiency of Statement (1)

Since

xy < 0

,

x

and

y

must have opposite signs. Therefore,

\frac{x}{y}

is negative, which means

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

. This gives a definitive 'Yes' to the question.

Any negative number is strictly less than 1, so Statement (1) alone is sufficient.

4

Evaluate Statement (2):

x > y

If

x = 3

and

y = 2

, then

x > y

and

\frac{x}{y} = 1.5 > 1

(No). If

x = 1

and

y = -2

, then

x > y

and

\frac{x}{y} = -0.5 < 1

(Yes).

Testing cases with positive vs. negative denominators shows that Statement (2) leads to both 'Yes' and 'No' answers.

5

Determine the sufficiency of Statement (2) and select the overall answer choice

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

Only Statement (1) provides a definitive answer to the question stem.

Key Concept

Distance interpretation of absolute values and algebraic rephrasing of inequalities involving quotient sign analysis.

Estimated Time:2m 0s

Question 24Question →

If $x$ is a real number, is $|x + 2| + |x - 4| \le 8$ ?

(1) $|x - 1| \le 4$
(2) $x^2 - x - 6 \le 0$

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 question stem shows that

|x + 2| + |x - 4| \le 8

is equivalent to

-3 \le x \le 5

. Statement (1) gives

-3 \le x \le 5

, which directly yields a definitive 'Yes'. Statement (2) gives

-2 \le x \le 3

, which is a subset of

[-3, 5]

, so any value of

x

satisfying Statement (2) must also satisfy

-3 \le x \le 5

, also yielding a definitive 'Yes'. Thus, each statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the question stem target by analyzing critical points

x = -2

and

x = 4

for

|x + 2| + |x - 4| \le 8

.

The target inequality

|x + 2| + |x - 4| \le 8

simplifies to the range

-3 \le x \le 5

.

For

x \ge 4

:

(x + 2) + (x - 4) \le 8 \implies 2x - 2 \le 8 \implies x \le 5

, giving

[4, 5]

. For

-2 \le x < 4

:

(x + 2) + (4 - x) = 6 \le 8

, which is true for all

x \in [-2, 4)

. For

x < -2

:

(-x - 2) + (4 - x) \le 8 \implies 2 - 2x \le 8 \implies -2x \le 6 \implies x \ge -3

, giving

[-3, -2)

. Combining all intervals gives

-3 \le x \le 5

.

2

Evaluate Statement (1):

|x - 1| \le 4

.

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

.

Statement (1) states that

x

is precisely in the range

[-3, 5]

. This provides a definitive 'Yes' answer to the question 'Is

-3 \le x \le 5

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

3

Evaluate Statement (2):

x^2 - x - 6 \le 0

.

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

.

Since

[-2, 3]

is entirely contained within

[-3, 5]

, every value of

x

satisfying Statement (2) automatically satisfies

-3 \le x \le 5

. This yields a definitive 'Yes' to the question. Thus, Statement (2) alone is sufficient.

4

Combine evaluations of Statement (1) and Statement (2).

Each statement alone is sufficient.

Because both statements independently provide enough information to give a definitive 'Yes' answer, the correct answer choice is that each statement alone is sufficient.

Key Concept

Absolute Value Distance Interpretation and Subset Range Sufficiency

Question 25Question →

If $m$ and $n$ are real numbers, is $m^2 < n^2$ ?

(1) $|m| + n < 0$
(2) $m + n > 0$

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.

Rephrased, the question asks whether

|m| < |n|

. Evaluating the first condition gives

n < -|m|

, which shows that

n

must be negative and its absolute value

|n| = -n

must be strictly greater than

|m|

. Hence,

m^2 < n^2

is definitely true. Evaluating the second condition allows both

m=2, n=5

(where

m^2 < n^2

) and

m=5, n=2

(where

m^2 > n^2

), making it insufficient. Therefore, the first condition alone is sufficient while the second condition alone is not.

Step-by-Step Solution

1

Rephrase the target question stem

m^2 < n^2

.

The target condition

m^2 < n^2

is equivalent to

|m|^2 < |n|^2

, which is true if and only if

|m| < |n|

.

Since both

m^2

and

n^2

are non-negative, taking the square root of both sides preserves the inequality order for non-negative magnitudes.

2

Evaluate Statement (1):

|m| + n < 0

.

Rearranging gives

n < -|m|

. Since

|m| \ge 0

, this implies

n

is strictly negative. Taking absolute values of both sides of

n < -|m|

gives

|n| = -n > |m|

. Therefore,

|n| > |m|

, which means

n^2 > m^2

or

m^2 < n^2

.

Statement (1) yields a definitive 'Yes' to the target question. Thus, Statement (1) alone is sufficient.

3

Evaluate Statement (2):

m + n > 0

.

Case 1: Let

m = 2

and

n = 5

. Then

2 + 5 = 7 > 0

, and

2^2 = 4 < 25 = 5^2

(Yes). Case 2: Let

m = 5

and

n = 2

. Then

5 + 2 = 7 > 0

, but

5^2 = 25 > 4 = 2^2

(No).

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

Key Concept

Data Sufficiency evaluation of absolute values and algebraic inequalities

Question 26Question →

If $x$ and $y$ are real numbers such that $x \neq 0$ , is $\frac{|x - y|}{x} > 1$ ?

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

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.

Statement (1) tells us

x < 0

. Since

|x - y| \ge 0

for all real numbers, dividing a non-negative numerator by a negative denominator guarantees that

\frac{|x - y|}{x} \le 0

. Consequently,

\frac{|x - y|}{x}

can never be greater than

1

, giving a definitive 'No' answer to the question. Because Statement (1) provides a single definitive answer, it is sufficient. Statement (2) tells us

y < 0

, but leaves the sign and value of

x

unconstrained. Choosing

x = -1

with

y = -2

gives a value of

-1

(answering 'No'), while choosing

x = 1

with

y = -2

gives a value of

3

(answering 'Yes'). Since Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient. Thus, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct.

Step-by-Step Solution

1

Analyze the target question stem and algebraic constraints.

We are asked whether

\frac{|x - y|}{x} > 1

. Notice that for any real numbers

x

and

y

, the absolute value in the numerator

|x - y| \ge 0

. The sign of the denominator

x

determines the sign of the entire quotient.

Understanding the algebraic behavior of non-negative numerators over signed denominators simplifies statement evaluation.

2

Evaluate Statement (1):

x < 0

.

Since

|x - y| \ge 0

and

x < 0

, the ratio

\frac{|x - y|}{x}

is a non-negative number divided by a negative number. Thus,

\frac{|x - y|}{x} \le 0

. Since a number

\le 0

can never be greater than

1

, the answer to the question is a definitive 'No'. A definitive 'No' means the statement IS sufficient.

In Data Sufficiency, any statement that allows us to answer the question with a single, unambiguous 'Yes' or 'No' is sufficient.

3

Evaluate Statement (2):

y < 0

.

Test values for

x

while holding

y < 0

(e.g.,

y = -2

). If

x = -1

, then

\frac{|-1 - (-2)|}{-1} = \frac{1}{-1} = -1 \ngtr 1

(Answer: No). If

x = 1

, then

\frac{|1 - (-2)|}{1} = \frac{3}{1} = 3 > 1

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

Getting conflicting answers from permissible test cases proves a statement is insufficient.

Key Concept

Definitive Yes/No logic and sign properties of absolute value quotients in Data Sufficiency

Question 27Question →

If $m$ is a real number, is $|m^2 - 2m - 3| < 5$ ?

(1) $|m - 2| < 3$
(2) $|m + 1| < 4$

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 question stem requires solving

|m^2 - 2m - 3| < 5

. This breaks down into

-5 < m^2 - 2m - 3 < 5

. The left inequality

m^2 - 2m + 2 > 0

holds for all real numbers because its discriminant is negative and the quadratic opens upward. The right inequality

m^2 - 2m - 8 < 0

factors as

(m - 4)(m + 2) < 0

, giving the target range

-2 < m < 4

.

Evaluating the first statement gives

|m - 2| < 3 \implies -1 < m < 5

. This interval extends past 4 (e.g.,

m = 4.5

yields a 'No', while

m = 1

yields a 'Yes'), so the first statement alone is not sufficient.

Evaluating the second statement gives

|m + 1| < 4 \implies -5 < m < 3

. This interval extends past

-2

(e.g.,

m = -4

yields a 'No', while

m = 1

yields a 'Yes'), so the second statement alone is not sufficient.

Combining both statements requires taking the intersection of

-1 < m < 5

and

-5 < m < 3

, which gives

-1 < m < 3

. Since every number in the range

(-1, 3)

lies strictly between

-2

and

4

, the target condition is guaranteed to be true. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target inequality

|m^2 - 2m - 3| < 5

is equivalent to

-5 < m^2 - 2m - 3 < 5

. Solving the upper bound gives

m^2 - 2m - 8 < 0 \implies (m - 4)(m + 2) < 0 \implies -2 < m < 4

. The lower bound

m^2 - 2m - 3 > -5 \implies m^2 - 2m + 2 > 0 \implies (m - 1)^2 + 1 > 0

is true for all real

m

. Thus, the question simplifies to: 'Is

-2 < m < 4

?'

Simplifying the target stem establishes the exact numerical interval required to yield a definitive 'Yes' or 'No' response.

2

Evaluate Statement (1) independently.

Statement (1) states

|m - 2| < 3 \implies -3 < m - 2 < 3 \implies -1 < m < 5

. If

m = 1

,

-2 < 1 < 4

(Yes). However, if

m = 4.5

,

m

is not in

(-2, 4)

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

A statement is sufficient only if every value in its allowed range produces the same answer to the rephrased question.

3

Evaluate Statement (2) independently.

Statement (2) states

|m + 1| < 4 \implies -4 < m + 1 < 4 \implies -5 < m < 3

. If

m = 1

,

-2 < 1 < 4

(Yes). However, if

m = -4

,

m

is not in

(-2, 4)

(No). Thus, Statement (2) is NOT sufficient.

Checking boundary values reveals that Statement (2) allows values outside the required interval.

4

Evaluate Statement (1) and Statement (2) together.

Combining Statement (1) range

(-1 < m < 5)

and Statement (2) range

(-5 < m < 3)

requires taking their intersection:

-1 < m < 3

. Since every value in

(-1, 3)

satisfies

-2 < m < 4

, the answer to the question is a definitive 'Yes'.

The intersection of the two ranges falls strictly inside the target range, guaranteeing sufficiency.

Key Concept

Data Sufficiency range evaluation for quadratic absolute value inequalities

Estimated Time:2m 0s

Question 28Question →

If $x$ is a real number, is $|x - 5| < x + 1$ ?

(1) $x^2 - 6x + 8 < 0$
(2) $|x - 3| < 2$

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 question target

|x - 5| < x + 1

simplifies directly to

x > 2

. Statement (1) bounds

x

to

2 < x < 4

, which guarantees that

x > 2

is always true, providing a definitive 'Yes'. Statement (2) bounds

x

to

1 < x < 5

, allowing values such as

1.5

(where

x > 2

is false) and

3

(where

x > 2

is true), which is insufficient. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem by simplifying the absolute value inequality

|x - 5| < x + 1

.

The inequality holds if and only if

-(x + 1) < x - 5 < x + 1

. Solving

-x - 1 < x - 5

yields

2x > 4

, or

x > 2

. Solving

x - 5 < x + 1

yields

-5 < 1

, which is always true. Thus, the question rephrases to: 'Is

x > 2

?'

Simplifying absolute value conditions in the stem reduces complex expressions to a single target range.

2

Evaluate Statement (1):

x^2 - 6x + 8 < 0

.

Factoring gives

(x - 2)(x - 4) < 0

, which yields the range

2 < x < 4

. For any value of

x

in

(2, 4)

,

x

is strictly greater than

2

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

Since every value satisfying Statement (1) satisfies

x > 2

, Statement (1) alone is sufficient.

3

Evaluate Statement (2):

|x - 3| < 2

.

Unfolding the absolute value gives

-2 < x - 3 < 2

, or

1 < x < 5

. If

x = 1.5

, then

x > 2

is false ('No'). If

x = 3

, then

x > 2

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

Statement (2) permits values both above and below the threshold of 2.

Key Concept

Question Stem Rephrasing for Absolute Value Inequalities

Estimated Time:2m 0s

Question 29Question →

If $k$ is a real number such that $k \neq -1$ , is $\frac{|k - 3|}{k + 1} < 1$ ?

(1) $|k| > 1$
(2) $k^2 - 2k - 3 > 0$

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

\frac{|k - 3|}{k + 1} < 1

holds whenever

k < -1

or

k > 1

. Statement (1) gives

k < -1

or

k > 1

, yielding a definitive YES. Statement (2) gives

k < -1

or

k > 3

, which is a subset of

k < -1

or

k > 1

, also yielding a definitive YES. Thus, each statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the question stem target inequality

The target inequality

\frac{|k - 3|}{k + 1} < 1

is satisfied if and only if

k < -1

or

k > 1

.

Analyze by cases depending on the sign of the denominator

k + 1

.
- Case 1:

k + 1 < 0 \implies k < -1

. The denominator is negative and the numerator

|k - 3| \ge 0

, so

\frac{|k - 3|}{k + 1} \le 0 < 1

is always true for all

k < -1

.
- Case 2:

k + 1 > 0 \implies k > -1

. Multiplying by

k + 1 > 0

yields

|k - 3| < k + 1

, which expands to

-(k + 1) < k - 3 < k + 1

. The left boundary

-k - 1 < k - 3

reduces to

2k > 2 \implies k > 1

. The right boundary

k - 3 < k + 1 \implies -3 < 1

is universally true. Thus for

k > -1

, the condition holds when

k > 1

.
Combining both cases, the inequality holds whenever

k < -1

or

k > 1

.

2

Evaluate Statement (1)

Statement (1) alone is SUFFICIENT.

Statement (1) states

|k| > 1

, which unwraps to

k < -1

or

k > 1

. This matches the target condition exactly. For every value of

k

satisfying this statement, the answer to the stem question is a definitive YES.

3

Evaluate Statement (2)

Statement (2) alone is SUFFICIENT.

Statement (2) states

k^2 - 2k - 3 > 0

, which factors as

(k - 3)(k + 1) > 0

. Solving gives

k < -1

or

k > 3

.
- If

k < -1

, it falls into the left target region

k < -1

.
- If

k > 3

, it falls into the right target region

k > 1

.
In all cases allowed by Statement (2), the answer to the stem question is a definitive YES.

Key Concept

Inequalities with absolute values and variables in the denominator require case analysis based on the denominator's sign.

Question 30Question →

If $r$ is a real number, is $r^3 < r$ ?

(1) $|r - 0.5| < 0.5$
(2) $|r| < 1$

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.

Rephrasing

r^3 < r

shows that the inequality holds true if and only if

r < -1

or

0 < r < 1

. Statement (1) bounds

r

precisely within the interval

(0, 1)

, ensuring a definitive 'Yes' answer. Statement (2) defines the range

(-1, 1)

, which includes values where the inequality holds (such as

0.5

) and values where it fails (such as

-0.5

), making it insufficient.

Step-by-Step Solution

1

Rephrase the target inequality algebraically.

The inequality

r^3 < r

rearranges to

r^3 - r < 0

, which factors as

r(r - 1)(r + 1) < 0

.

Simplifying the stem isolates the specific intervals on the number line where the statement holds true.

2

Determine the valid number ranges for the rephrased question.

The expression

r(r - 1)(r + 1)

is strictly negative when

r < -1

or when

0 < r < 1

.

Testing sign changes across key boundary points (

r = -1

,

r = 0

,

r = 1

) identifies the target ranges.

3

Evaluate Statement (1):

|r - 0.5| < 0.5

.

This absolute value inequality expands to

-0.5 < r - 0.5 < 0.5

, which simplifies to

0 < r < 1

.

Since every value in the range

0 < r < 1

satisfies

r^3 < r

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

4

Evaluate Statement (2):

|r| < 1

.

This absolute value inequality simplifies to

-1 < r < 1

.

If

r = 0.5

, then

r^3 = 0.125 < 0.5

(Yes). However, if

r = -0.5

, then

r^3 = -0.125 > -0.5

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

Key Concept

Data Sufficiency evaluation of cubic inequalities and absolute value range constraints

Question 31Question →

If $x$ is a real number, is $|x^3 - x| > x^2 - 1$ ?

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

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 option is the one stating that Statement (2) alone is sufficient, but Statement (1) alone is not sufficient. Simplifying the stem target

|x^3 - x| > x^2 - 1

into

|x||x^2 - 1| > x^2 - 1

reveals that the inequality holds for every real number except

x = 1

and

x = -1

(where both sides equal zero). Statement (1) states

|x - 1| > 0

, meaning

x \neq 1

, but it permits

x = -1

, which yields a 'No' answer, making Statement (1) insufficient. Statement (2) states

|x + 2| < 1

, which expands to

-3 < x < -1

. Because this interval strictly excludes both

x = -1

and

x = 1

, every value of

x

in this range satisfies the stem inequality, yielding a definitive 'Yes'.

Step-by-Step Solution

1

Rephrase the question stem target algebraically.

The target inequality

|x^3 - x| > x^2 - 1

simplifies to

|x(x^2 - 1)| > x^2 - 1

, which is equivalent to

|x| \cdot |x^2 - 1| > x^2 - 1

.

Factoring allows analysis of the critical values where

x^2 - 1 = 0

.

2

Analyze the conditions under which

|x| \cdot |x^2 - 1| > x^2 - 1

holds true.

Case 1: If

x^2 - 1 = 0

(meaning

x = 1

or

x = -1

), LHS =

0

and RHS =

0

, giving

0 > 0

, which is FALSE. Case 2: If

x^2 - 1 < 0

(meaning

-1 < x < 1

), LHS is positive and RHS is negative, so non-negative > negative is TRUE. Case 3: If

x^2 - 1 > 0

(meaning

x > 1

or

x < -1

), dividing both sides by

x^2 - 1 > 0

yields

|x| > 1

, which is TRUE. Thus, the inequality holds for ALL real numbers EXCEPT

x = 1

and

x = -1

. The question simplifies to: Is

x \neq 1

and

x \neq -1

?

Simplifying the target reveals that the inequality is true everywhere except at the two boundary roots

x = 1

and

x = -1

.

3

Evaluate Statement (1):

|x - 1| > 0

.

This implies

x \neq 1

. However,

x

could equal

-1

. If

x = -1

,

|-1 - 1| = 2 > 0

is satisfied, but

|(-1)^3 - (-1)| = 0

and

(-1)^2 - 1 = 0

, giving

0 > 0

(NO). If

x = 2

,

|2 - 1| = 1 > 0

is satisfied, and

|8 - 2| = 6 > 3

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

Statement (1) rules out

x = 1

but permits

x = -1

.

4

Evaluate Statement (2):

|x + 2| < 1

.

Solving the absolute value inequality gives

-1 < x + 2 < 1

, which simplifies to

-3 < x < -1

. In this range,

x

cannot be

1

or

-1

because the upper boundary is strictly less than

-1

. Thus, for all

x \in (-3, -1)

,

x \neq 1

and

x \neq -1

is guaranteed, yielding a definitive 'YES'. Statement (2) is SUFFICIENT.

The open interval

(-3, -1)

excludes both

x = -1

and

x = 1

.

Key Concept

Question Stem Simplification with Absolute Value Inequalities

Estimated Time:2m 0s

Question 32Question →

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

(1) $|a + 2b| < |a - 2b|$
(2) $\frac{|a|}{b} + \frac{|b|}{a} < 0$

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 stem by squaring both sides gives

|a + b|^2 > |a - b|^2 \iff (a + b)^2 > (a - b)^2 \iff 4ab > 0 \iff ab > 0

. Statement (1) simplifies similarly by squaring both sides to

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

. Since

ab < 0

guarantees that

ab > 0

is false, Statement (1) produces a definitive 'No' and is therefore sufficient. Statement (2) is satisfied when both

a

and

b

are negative (giving

ab > 0

, a 'Yes' answer) as well as when

a

and

b

have opposite signs with

|a| > |b|

(giving

ab < 0

, a 'No' answer). Because Statement (2) permits both outcomes, it is insufficient.

Step-by-Step Solution

1

Rephrase the question stem target using algebraic properties of absolute values.

Squaring both non-negative sides of

|a + b| > |a - b|

yields

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

. The question asks: 'Is

ab > 0

?'

Simplifying the stem target into a fundamental sign constraint (

ab > 0

) prevents algebraic errors during statement evaluation.

2

Evaluate Statement (1):

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

.

Squaring both sides yields

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

.

Since

ab < 0

,

ab

cannot be greater than

0

. This yields a definitive 'No' to the question 'Is

ab > 0

?'. A definitive 'No' is sufficient.

3

Evaluate Statement (2):

\frac{|a|}{b} + \frac{|b|}{a} < 0

.

Combine the fractions:

\frac{a|a| + b|b|}{ab} < 0

. Test specific cases:
- Case A:

a = -2, b = -2 \implies \frac{|-2|}{-2} + \frac{|-2|}{-2} = -2 < 0

. Here

ab = 4 > 0

(Answer: YES).
- Case B:

a = 3, b = -1 \implies \frac{|3|}{-1} + \frac{|-1|}{3} = -3 + \frac{1}{3} = -\frac{8}{3} < 0

. Here

ab = -3 < 0

(Answer: NO).

Since Statement (2) allows both a 'Yes' and a 'No' response to whether

ab > 0

, Statement (2) alone is not sufficient.

Key Concept

Data Sufficiency Yes/No decision logic combined with distance and squaring properties of absolute value inequalities.

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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