GMAT Math : Inequalities

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Inequalities

Data sufficiency question- do not actually solve the question

Is $\small xy< 12$ ?

1. $\small x^2=64; 1\leq y\leq 1.5$

2. $\small x+y=6$

Possible Answers:

Statements 1 and 2 together are not sufficient, and additional information is needed to answer the question

Statement 1 alone is sufficient, but statement 2 along is not sufficient to answer the question

Both statements taken together are sufficient to answer the question but neither statement alone is sufficient

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Each statement alone is sufficient

Correct answer:

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Explanation:

From statement 1, we can conclude that $\small xy\leq 12$ but not $\small xy< 12$ . From the second statement, we can conclude that the greatest product will result from $\small 3+3=6$ or 9, which is less than 12.

Report an Error

Example Question #1 : Inequalities

How many solutions does the equation $\left | x - A \right | = B$ have?

Statement 1: A = 8

Statement 2: B = -8

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.

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.

Correct answer:

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

Explanation:

If we only know that A = 8 , then the above statement becomes $\left | x - 8 \right | = B$ , and it can have zero, one, or two solutions depending on the value of . For example:

If B = -5 , the equation is $\left | x - 8 \right | = -5$ , which has no solution, as an absolute value cannot be negative.

If B = 0 , the equation is $\left | x - 8 \right | = 0$ , which requires that x - 8=0 , or x = 8 , since only 0 has absolute value 0; this means the equation has one solution.

If we only know that B = -8 , then the equation becomes $\left | x - A \right | = -8$ , which has no solution regardless of the value of ; this is because, as stated before, an absolute value cannot be negative.

Report an Error

Example Question #3 : Inequalities

True or false: is a positive number.

Statement 1: $N = \left | N \right |$

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

Possible Answers:

EITHER 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.

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:

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

Explanation:

If is negative, then |N| = -N and $N ^{2} > 0 > N$ . Therefore, either Statement 1 or Statement 2 alone proves nonnegative. However, if N = 0 , then $N = \left | N \right |$ , but $N > N^{2}$ is false.

Therefore, Statement 2 proves positive, but Statement 1 only proves nonnegative.

Report an Error

Example Question #4 : Inequalities

True or false: x < 5

Statement 1: |x| > 5

Statement 2: $x^{2}> 25$

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

x= 6 makes both statements true, since |6|= 6 > 5 and $6^{2}= 36 > 25$ .

x= - 6 makes both statements true, since |-6|= 6 > 5 and $(-6)^{2}= 36 > 25$ .

One of the two values is less than 5, and one is greater than 5. The statements together provide insufficient information.

Report an Error

Example Question #2 : Inequalities

is a whole number.

True or false: is odd.

Statement 1: $2^{x} > 0$

Statement 2: $(-2)^{x} > 0$

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Statement 1 alone is a superfluous statement, since a positive number raised to any power must yield a positive result.

Statement 2 alone answers the question, since a negative number raised to a whole number exponent yields a positive result if and only if the exponent is even. Since Statement 2 states that $(-2)^{x}$ is positive, is even, not odd.

Report an Error

Example Question #6 : Inequalities

True or false: x < 9

Statement 1: |x| < 9

Statement 2: $x^{2}< 81$

Possible Answers:

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. |x| < 9 can be rewritten as -9 < x < 9 .

Assume Statement 2 alone. It can be rewritten as

$x^{2}< 81$

the solution set of which is -9 < x < 9

From either statement alone, it follows that x < 9 .

Report an Error

Example Question #4 : Dsq: Solving Inequalities

True or false: x > 0

Statement 1: $2^{x}<4^{x}$

Statement 2: $x^{3}> 0$

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.

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Since $2^{x}$ and $4^{x}$ are both positive, we can divide both sides by $2^{x}$ to yield the statement

$2^{x}<4^{x}$

$4^{x} > 2^{x}$

$\frac{ 4^{x}}{2^{x}} >\frac{ 2^{x}}{2^{x}}$

$\left (\frac{ 4}{2} \right )^{x} >1$

$2^{x}> 1$

Since $f(x) = 2^{x}$ increases as increases, and since $2^{0}= 1$ , it follows that x > 0 .

Assume Statement 2 alone. Since the cube root of a number assumes the same sign as the number itself, $x^{3}> 0$ implies that x>0 .

From either statement alone it follows that x>0 .

Report an Error

Example Question #5 : Dsq: Solving Inequalities

True or false: x < 11

Statement 1: $x^{2} > 121$

Statement 2: $x^{3}> 1,331$

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

Explanation:

Assume Statement 1 only. Both and 12 make the statement true, since $12^{2}= (-12)^{2}= 144 > 121$ . But one is less than 11 and one is not.

Assume Statement 2 only. Then, since an odd (third) root of a number assumes the sign of that number, and an odd root function is an increasing function, we can simply take the cube root of each side:

$x^{3}> 1,331$

$\sqrt[3]{x^{3}}>\sqrt[3]{ 1,331}$

x > 11 .

Report an Error

Example Question #5 : Dsq: Solving Inequalities

True or false: x > 0

Statement 1: $x^{5}< 0$

Statement 2: $3 ^{x}> 6^{x}$

Possible Answers:

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.

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. Since the fifth (odd) power of a number assumes the same sign as the number itself, and $x^{5}$ have the same sign, and $x^{5}< 0$ implies that x<0 .

Assume Statement 2 alone. Since $3^{x}$ and $6^{x}$ are both positive, we can divide both sides by $3^{x}$ to yield the statement

$3^{x}>6^{x}$

$6^{x} < 3^{x}$

$\frac{6^{x}}{3^{x}} <\frac{ 3^{x}}{3^{x}}$

$\left (\frac{6}{3} \right )^{x} <1$

$2^{x}< 1$

Since $f(x) = 2^{x}$ increases as does, and since $2^{0}= 1$ , it follows that x < 0 .

Report an Error

Example Question #6 : Dsq: Solving Inequalities

True or false: x >- 2

Statement 1: $x^{8} >256$

Statement 2: $x^{10} >1,024$

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements together provide insufficient information. For example,

If x = 3 , then:

$x^{8}= 3^{8}= 6,561 > 256$

$x^{10}= 3^{10}= 59,049 > 1,024$

If x = -3 , then

$x^{8}= (-3)^{8}= 6,561 > 256$

$x^{10}= \left (-3 \right )^{10}= 59,049 > 1,024$

Both values fit the conditions of both statements, but only one is greater than . The question is not answered.

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Boston, GRE Tutors in Dallas Fort Worth, English Tutors in San Diego, Statistics Tutors in San Francisco-Bay Area, Biology Tutors in New York City, Reading Tutors in Philadelphia, Spanish Tutors in San Diego, Math Tutors in Dallas Fort Worth, ISEE Tutors in Dallas Fort Worth, MCAT Tutors in Houston

Popular Courses & Classes

ISEE Courses & Classes in New York City, ACT Courses & Classes in Washington DC, SSAT Courses & Classes in Washington DC, LSAT Courses & Classes in Denver, SSAT Courses & Classes in Atlanta, LSAT Courses & Classes in San Diego, GMAT Courses & Classes in Denver, GMAT Courses & Classes in Houston, ACT Courses & Classes in Phoenix, MCAT Courses & Classes in Miami

Popular Test Prep

GRE Test Prep in New York City, GMAT Test Prep in Houston, GRE Test Prep in Seattle, ACT Test Prep in Washington DC, GMAT Test Prep in Denver, GMAT Test Prep in Washington DC, ISEE Test Prep in Boston, GMAT Test Prep in Boston, ACT Test Prep in New York City, ISEE Test Prep in Washington DC

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : Inequalities

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Inequalities

Example Question #1 : Inequalities

Example Question #3 : Inequalities

Example Question #4 : Inequalities

Example Question #2 : Inequalities

Example Question #6 : Inequalities

Example Question #4 : Dsq: Solving Inequalities

Example Question #5 : Dsq: Solving Inequalities

Example Question #5 : Dsq: Solving Inequalities

Example Question #6 : Dsq: Solving Inequalities

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details