GMAT Math : Inequalities

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Example Questions

Example Question #21 : Algebra

The variables A,B,C,D,P,Q,R,S stand for integer quantities.

Order from least to greatest:

$A^{P}, B^{Q}, C^{R}, D^{S}$

Statement 1: 1< A < B < C < D

Statement 2: P = Q= R = S >1

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 only gives us the order of the bases; we cannot order the powers without any information about the exponents. Similarly, Statement 2 alone only tells us the common exponent; without knowing the order of the bases, we cannot order the powers.

Assume both statements are true. Let's look at $A^{P}$ and $B^{Q}$ .

Since and are both positive, and B>A , we can apply the multiplication property of equality:

$B\cdot B>A \cdot A$

$B^{2}> A^{2}$

Similarly, $B^{3}> A^{3}, B^{4}> A^{4},$ etc.

So, if P > 1 and B>A ,

$A^{P}< B^{P}$ .

Since A < B < C < D from Statement 1, it follows that

$A^{P}< B^{P}< C^{P}< D^{P}$ ,

which, from Statement 2, can be rewritten as

$A^{P}< B^{Q}< C^{R}< D^{S}$ .

The order has been determined.

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Example Question #12 : Dsq: Solving Inequalities

True or false: x > 10

Statement 1: $x^{5} >100,000$

Statement 2: $x^{7} >10,000,000$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Then, since an odd (fifth) root of a number assumes the sign of that number, and an odd root function is an increasing function, we can simply take the fifth root of each side:

$x^{5} >100,000$ ,

$\sqrt[5]{x^{5}} >\sqrt[5]{100,000}$

x > 10

Assume Statement 2 alone.By a similar argument,

$x^{7} >10,000,000$ ,

$\sqrt[7]{x^{7}} >\sqrt[7]{10,000,000}$

x > 10

From either statement alone, it follows that x > 10 is true.

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Example Question #21 : Algebra

The variables A,B,C,D,P,Q,R,S stand for integer quantities.

Order from least to greatest:

$A^{P}, B^{Q}, C^{R}, D^{S}$

Statement 1: A < B < C < D

Statement 2: P = Q= R = S

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show the staements together provide insufficient information by examining two situations.

Suppose A=1, B= 2 , C= 3, D=4 .

If P =Q=R=S=2 , the four expressions become:

$A^{P}= 1^{2}= 1$

$B^{Q}= 2^{2} = 4$

$C^{R}=3^{2}= 9$

$D^{S}= 4^{2}= 16$

and $A^{P} < B^{Q}<C^{R}< D^{S}$

If P =Q=R=S=-2 , the four expressions become:

$A^{P}= 1^{-2}= 1$

$B^{Q}= 2^{-2} = \frac{1}{4}$

$C^{R}=3^{-2}= \frac{1}{9}$

$D^{S}= 4^{-2}= \frac{1}{16}$

In ascending order, the expressions are $D^{S}<C^{R}< B^{Q}<A^{P}$ .

Since the orderings are different in the two cases, the two statements together give insufficient information as to their correct ordering.

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Example Question #12 : Dsq: Solving Inequalities

The variables A,B,C,D,P,Q,R,S stand for integer quantities.

Order from least to greatest:

$A^{P}, B^{Q}, C^{R}, D^{S}$

Statement 1: A= B = C = D > 1

Statement 2: P <Q< R < S

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone only tells us the common base; without knowing the order of the exponents, we cannot order the powers. Similarly, Statement 2 only gives us the order of the exponents; we cannot order the powers without any information about the bases.

Assume both statements are true. Since N > 1 , it holds that

$N^{2}= N \cdot N > N$

$N^{3}= N^{2} \cdot N > N^{2}$

et cetera, and

$N^{0} = 1 < N^{1}$

$N^{-1} = N^{0} \div N < N^{1} \div N = N^{0}$

and so forth in both directions. That is,

$...<N^{-2} <N^{-1}< N^{0} <N ^{1}< N^{2} < ...$

Therefore, since P <Q< R < S , it follows that

$N^{P}< N^ {Q} <N ^{R}< N^{S}$ ,

and the ordering is determined.

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Example Question #15 : Dsq: Solving Inequalities

True or false: x < 7

Statement 1: $x^{2} > 49$

Statement 2: $x^{3}$ is positive.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. From $x^{2} > 49$ , it can be determined that either x > 7 or x < -7 , but nothing more; the statement alone provides insufficient information. Statement 2 taken alone provides insufficient information as well, since the positive numbers include numbers both less than and greater than 7.

Assume both statements are true. From Statement 1, either x > 7 or x < -7 , but Statement 2 gives us that x > 0 . Therefore, x > 7 , and the question is answered.

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Example Question #21 : Algebra

True or false: |x| < 12

Statement 1: $x^{2}< 144$

Statement 2: $x^{3}< 1,728$

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The absolute value of a number is its unsigned value - that is, if the number is nonnegative, it is the number itself, and if the number is negative, it is the corresponding positive.

From Statement 1 alone, since $x^{2}< 144$ , then it follows that -12 < x < 12 - this is equivalent to saying |x| < 12 .

Statement 2 alone provides insufficient information. For example:

If x= 11 , then $x^{3}= 11^{3} = 1,331 < 1,728$

if x= -13 , then $x^{3}= (-13)^{3} =- 2,197 <- 1,728$

Both numbers fit the condition of Statement 2, but

|11| = 11 < 12

and

|-13| = 13 > 12

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Example Question #17 : Dsq: Solving Inequalities

The variables A,B,C,D,P,Q,R,S stand for integer quantities.

Order from least to greatest:

$A^{P}, B^{Q}, C^{R}, D^{S}$

Statement 1: A= B = C = D

Statement 2: P <Q< R < S

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show the two statements provide insufficient information by assuming both statements to be true and showing that the ordering is different depending on the common exponent. In both cases, we let P =1, Q=2, R= 3, R= 4 .

If A= B = C = D = 2 , then the expressions become

$A^{P}= 2^{1}= 2$

$B^{Q}= 2^{2} = 4$

$C^{R}= 2^{3}= 8$

$D^{S}= 2^{4}= 16$

The correct ordering is $A^{P}<B^{Q}<C^{R}< D^{S}$ .

If A= B = C = D = -2 , then the expressions become

$A^{P}= \left ( -2\right )^{1}= -2$

$B^{Q}= \left ( -2\right )^{2} = 4$

$C^{R}= \left ( -2\right )^{3}=- 8$

$D^{S}= \left ( -2\right )^{4}= 16$

The correct ordering is $C^{R}< A^{P}< B^{Q}<D^{S}$ .

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Example Question #18 : Dsq: Solving Inequalities

is a real number. True or false: is positive.

Statement 1: $N^{2}+ 10N + 24 < 0$

Statement 2: |N + 5| < 1

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. If is positive, then $N^{2} > 0$ and 10 N > 0 ; since $N^{2}+ 10N + 24$ is the sum of three positive numbers, then $N^{2}+ 10N + 24 > 0$ , and $N^{2}+ 10N + 24 < 0$ is a false statement. Therefore, cannot be positive.

Assume Statement 2. If is positive, then so is N+5 , and the inequality can be rewritten as

N+5 < 1

Consequently,

N+5 - 5 < 1 - 5

N < -4 ,

a contradiction since is positive. Therefore, is not positive.

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Example Question #19 : Dsq: Solving Inequalities

is a real number. True or false: is positive.

Statement 1: $2^{N} > 8$

Statement 2: $N^{2} > 9$

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. $2^{N} > 8$ , or $2^{N} > 2^{3}$ . For any base a > 1 , if $a^{b}> a^{c}$ , then b > c , so it follows that N > 3 . is therefore positive.

Assume Statement 2 alone. Both N = -4 and N = 4 are solutions:

$4^{2} = 16 > 9$

and

$(-4)^{2} = 16 > 9$

The sign of cannot be determined.

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GMAT Math : Inequalities

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #21 : Algebra

Example Question #12 : Dsq: Solving Inequalities

Example Question #21 : Algebra

Example Question #12 : Dsq: Solving Inequalities

Example Question #15 : Dsq: Solving Inequalities

Example Question #21 : Algebra

Example Question #17 : Dsq: Solving Inequalities

Example Question #18 : Dsq: Solving Inequalities

Example Question #19 : Dsq: Solving Inequalities

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