GMAT Math : Solving 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 : Solving Inequalities

How many integers $\dpi{100} \small (x)$ can complete this inequality?

$7< 2x-3 <15$

Possible Answers:

$\dpi{100} \small 3$

$\dpi{100} \small 9$

$\dpi{100} \small 4$

$\dpi{100} \small 5$

$\dpi{100} \small 6$

Correct answer:

$\dpi{100} \small 3$

Explanation:

$7< 2x-3 <15$

3 is added to each side to isolate the $\dpi{100} \small x$ term:

$10< 2x <18$

Then each side is divided by 2 to find the range of $\dpi{100} \small x$ :

$5< x <9$

The only integers that are between 5 and 9 are 6, 7, and 8.

The answer is 3 integers.

Report an Error

Example Question #2 : Solving Inequalities

Solve $5 < 3x + 10 \leq 16$ .

Possible Answers:

$\left ( -\frac{5}{3}, 2 \right )$

$\left [ -\frac{5}{3}, 2 \right ]$

$\left [ \frac{5}{3}, 2\right )$

$\left ( \frac{5}{3}, 2 \right ]$

$\left ( -\frac{5}{3}, 2 \right ]$

Correct answer:

$\left ( -\frac{5}{3}, 2 \right ]$

Explanation:

$5 < 3x + 10 \leq 16$

Subtract 10: $-5 < 3x \leq 6$

Divide by 3: $-5/3 < x \leq 2$

We must carefully check the endpoints. is greater than $-\frac{5}{3}$ and cannot equal $-\frac{5}{3}$ , yet CAN equal 2. Therefore $-\frac{5}{3}$ should have a parentheses around it, and 2 should have a bracket: is in

$\left ( -\frac{5}{3}, 2 \right ]$

Report an Error

Example Question #3 : Solving Inequalities

Solve -2x+3<7 .

Possible Answers:

$(-2, \infty )$

$(2, \infty )$

$[-2, \infty )$

$(- \infty , -2)$

$[-2, \infty ]$

Correct answer:

$(-2, \infty )$

Explanation:

-2x+3<7

Subtract 3 from both sides: -2x<4

Divide both sides by : x>-2

Remember: when dividing by a negative number, reverse the inequality sign!

Now we need to decide if our numbers should have parentheses or brackets. is strictly greater than , so should have a parentheses around it. Since there is no upper limit here, is in $(-2, \infty )$ .

Note: Infinity should ALWAYS have a parentheses around it, NEVER a bracket.

Report an Error

Example Question #4 : Solving Inequalities

Solve $(x-1)^{2}(x+4)<0$ .

Possible Answers:

$[-\infty , -4]$

$(-\infty , -4]$

$(-\infty , -4)$

$[4, \infty )$

$(-\infty , 4)$

Correct answer:

$(-\infty , -4)$

Explanation:

$(x-1)^{2}$ must be positive, except when x=1 . When x=1 , $(x-1)^{2}=0$ .

Then we know that the inequality is only satisfied when x+4<0 , and $x\neq 1$ . Therefore x<-4 , which in interval notation is $(-\infty , -4)$ .

Note: Infinity must always have parentheses, not brackets. has a parentheses around it instead of a bracket because is less than , not less than or equal to .

Report an Error

Example Question #5 : Solving Inequalities

Solve x(x-1)(x+1)>0 .

Possible Answers:

x>1

-1<x<1

$-1<x<0\ or\ x>1$

$x=-1\ or\ x>1$

$x<-1\ or\ x>1$

Correct answer:

$-1<x<0\ or\ x>1$

Explanation:

The roots we need to look at are $x=0, x=1,\ and\ x=-1.$

Try x=-2

-2(-2-1)(-2+1)=-4<0 , so

x<-1 does not satisfy the inequality.

-1<x<0 :

Try $x=-\frac{1}{2}$

$-\frac{1}{2}\cdot (-\frac{1}{2}-1)(-\frac{1}{2}+1)=\frac{3}{8}>0$

so -1<x<0 does satisfy the inequality.

0<x<1 :

Try $x=\frac{1}{2}$

$\frac{1}{2}\left (\frac{1}{2}-1 \right )\left ( \frac{1}{2}+1 \right )=-\frac{3}{8}<0$

so 0<x<1 does not satisfy the inequality.

x>1 :

Try x=2 .

2(2-1)(2+1)=6>0

so x>1 satisfies the inequality.

Therefore the answer is -1<x<0 and x>1 .

Report an Error

Example Question #3 : Solving Inequalities

Find the domain of $y=\sqrt{x^{2}-4}$ .

Possible Answers:

all positive real numbers

$x>2\ or\ x<-2$

$x\geq 2, x\leq -2$

all non-negative real numbers

all real numbers

Correct answer:

$x\geq 2, x\leq -2$

Explanation:

We want to see what values of x satisfy the equation. $x^{2}-4$ is under a radical, so it must be positive.

$x^{2}-4\geq 0$

$x^{2}\geq 4$

$x\geq 2, x\leq -2$

Report an Error

Example Question #4 : Solving Inequalities

Solve the inequality:

$-3x+7\geq13$

Possible Answers:

$x\geq-2$

$x\geq2$

$x\leq2$

$x\leq-2$

Correct answer:

$x\leq-2$

Explanation:

$-3x+7\geq13$

$-3x\geq6$

When multiplying or dividing by a negative number on both sides of an inequality, the direction of the inequality changes.

$x\leq-2$

Report an Error

Example Question #1 : Solving Inequalities

Find the solution set for :

4 < -x + 7 < 15

Possible Answers:

$x \in (-3,8)$

$x \in (-\infty , -3) \cup (8, \infty )$

$x \in (-8,3)$

$x \in (3,8)$

$x \in (-\infty , 3) \cup (8, \infty )$

Correct answer:

$x \in (-8,3)$

Explanation:

4 < -x + 7 < 15

Subtract 7:

4-7 < -x + 7 -7 < 15-7

-3 < -x < 8

Divide by -1. Don't forget to switch the direction of the inequality signs since we're dividing by a negative number:

$-3 \cdot (-1) > -x \cdot (-1) > 8\cdot (-1)$

Simplify:

3 > x > -8 or in interval form, (-8,3) .

Report an Error

Example Question #2 : Solving Inequalities

Which of the following is equivalent to 2x-5 > 5x+4 ?

Possible Answers:

x > 3

x < 2

$x > -\frac{1}{9}$

x < -9

x < -3

Correct answer:

x < -3

Explanation:

To solve this problem we need to isolate our variable .

We do this by subtracting from both sides and subtracting from both sides as follows:

2x-5>5x+4

-5>3x+4

-9>3x

Now by dividing by 3 we get our solution.

-3>x or x<-3

Report an Error

Example Question #5 : Solving Inequalities

How many integers satisfy the following inequality:

4< 4m-5<13

Possible Answers:

Two

Four

One

Five

Three

Correct answer:

Two

Explanation:

4< 4m-5<13

9<4m<18

$\frac{9}{4}<m<\frac{9}{2}$

2.25<m<4.5

There are two integers between 2.25 and 4.5, which are 3 and 4.

Report an Error

All GMAT Math Resources

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

Popular Subjects

Statistics Tutors in Phoenix, Reading Tutors in San Francisco-Bay Area, ACT Tutors in Los Angeles, LSAT Tutors in San Francisco-Bay Area, French Tutors in Philadelphia, LSAT Tutors in Chicago, GRE Tutors in Boston, GRE Tutors in Chicago, Spanish Tutors in Denver, MCAT Tutors in San Diego

Popular Courses & Classes

ISEE Courses & Classes in Boston, ACT Courses & Classes in Seattle, GRE Courses & Classes in Chicago, GMAT Courses & Classes in Seattle, SAT Courses & Classes in Atlanta, GMAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Houston, SSAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Miami, MCAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

SSAT Test Prep in Atlanta, SAT Test Prep in Atlanta, GMAT Test Prep in Los Angeles, LSAT Test Prep in Atlanta, ACT Test Prep in Seattle, SAT Test Prep in Boston, GMAT Test Prep in Seattle, ISEE Test Prep in Philadelphia, LSAT Test Prep in Denver, GRE Test Prep in San Diego

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 : Solving inequalities

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #3 : Solving Inequalities

Example Question #4 : Solving Inequalities

Example Question #5 : Solving Inequalities

Example Question #3 : Solving Inequalities

Example Question #4 : Solving Inequalities

Example Question #1 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #5 : Solving Inequalities

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details