We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

PSAT Math : Equations / Inequalities

Study concepts, example questions & explanations for PSAT Math

Create An Account

All PSAT Math Resources

10 Diagnostic Tests 421 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 … 5 6 7 8 9 10 11 12 13 … 22 23 Next →

Example Question #1 : Linear / Rational / Variable Equations

Consider the equation

$\frac{x+9}{x}+\frac{x+3}{x-2} = \frac{4x+11}{x}$

Which of the following is true?

Possible Answers:

The equation has exactly one solution, which is positive.

The equation has exactly two solutions, which are of like sign.

The equation has exactly two solutions, which are of unlike sign.

The equation has no solution.

The equation has exactly one solution, which is negative.

Correct answer:

The equation has exactly two solutions, which are of unlike sign.

Explanation:

Multiply the equation on both sides by LCM $\frac{x(x-2)}{1}$ :

$\frac{x+9}{x}+\frac{x+3}{x-2} = \frac{4x+11}{x}$

$\frac{x(x-2)}{1} \cdot \frac{x+9}{x}+\frac{x(x-2)}{1} \cdot\frac{x+3}{x-2} = \frac{x(x-2)}{1} \cdot\frac{4x+11}{x}$

$(x-2) \left ( x+9 \right ) + x( x+3 )= (x-2) (4x+11)$

$\left ( x^{2}+7x -18\right ) + ( x^{2}+3x )= 4x^{2}+3x -22$

$4x^{2}+3x -22 = x^{2}+7x -18 + x^{2}+3x$

$4x^{2}+3x -22 = 2x^{2}+10x -18$

$4x^{2}+3x -22 - 2x^{2}-10x +18 = 2x^{2}+10x -18 - 2x^{2}-10x +18$

$2x^{2}-7x -4 = 0$

(x-4)(2x+1)= 0

x-4 = 0

x= 4

2x+1 = 0

2x = -1

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

Substitution confirms that these are the solutions.

There are two solutions of unlike sign.

Report an Error

Example Question #2 : Linear / Rational / Variable Equations

Which of the following equations has no solution?

Possible Answers:

6- |x + 8 | = 8

6- |x | = 8

6- |x + 6 | = 8

6- |x + 2 | = 8

Each of the equations in the other responses has no solution.

Correct answer:

Each of the equations in the other responses has no solution.

Explanation:

The problem is basically asking for what value of the equation

6- |x + A | = 8

has no solution.

We can simplify as folllows:

6- |x + A | - 6 = 8 - 6

- |x + A | = 2

|x + A | = - 2

Since the absolute value of a number must be nonnegative, regardless of the value of , this equation can never have a solution. Therefore, the correct response is that none of the given equations has a solution.

Report an Error

Example Question #1 : Linear / Rational / Variable Equations

Consider the equation

$\frac{4}{y+3} = \frac{y+6}{10}$

Which of the following is true?

Possible Answers:

The equation has no real solutions.

The equation has exacty one real solution, which is negative.

The equation has exacty one real solution, which is positive.

The equation has exactly two real solutions, which are of unlike sign.

The equation has exactly two real solutions, which are of like sign.

Correct answer:

The equation has exactly two real solutions, which are of unlike sign.

Explanation:

Multiply both sides by LCD $\frac{10(y+3)}{1}$ :

$\frac{4}{y+3} = \frac{y+6}{10}$

$\frac{4}{y+3}\cdot \frac{10(y+3)}{1} = \frac{y+6}{10} \cdot \frac{10(y+3)}{1}$

$4 \cdot 10 = (y+6) (y+3)$

$y^{2}+9y+18 = 40$

$y^{2}+9y+18- 40 = 40 - 40$

$y^{2}+9y-22 = 0$

(y+11)(y-2) = 0

$y+11 \Rightarrow y = -11$

$y-2 = 0 \Rightarrow y = 2$

There are two solutions of unlike sign.

Report an Error

Example Question #1 : How To Find Out When An Equation Has No Solution

All of the following equations have no solution except for which one?

Possible Answers:

4x + 3(x-10) = 2x + 5 (x - 8)

4x + 3(x-10) = 2x + 5 (x - 6)

4x + 3(x-10) = 2x + 5 (x - 12)

4x + 3(x-10) = 2x + 5 (x - 10)

4x + 3(x-10) = 2x + 5 (x - 4)

Correct answer:

4x + 3(x-10) = 2x + 5 (x - 6)

Explanation:

Since all of the equations have the same symbols save for one number, the problem is essentially as follows:

For what value of does the equation

4x + 3(x-10) = 2x + 5 (x - A)

have a solution set other than the empty set?

We can simplify as follows:

4x + 3x-30= 2x + 5 x - 5A

7x-30=7x - 5A

-30 = -5A

If -5A and are not equivalent expressions, the solution set is the empty set. If -5A and are equivalent expressions, the solution set is the set of all real numbers; this happens if and only if:

-5A = -30

$-5A \div (-5)= -30 \div (-5)$

A = 6

Therefore, the only equation among the given choices whose solution set is not the empty set is the equation

4x + 3(x-10) = 2x + 5 (x - 6)

which is the correct choice.

Report an Error

Example Question #5 : Linear / Rational / Variable Equations

Which of the following equations has no real solutions?

Possible Answers:

$-14 - \sqrt[4]{x - 7 } = 7$

Each of the equations given in the other choices has at least one real solution.

$14 - \sqrt[4]{x - 7 } = -7$

$-14 - \sqrt[3]{x - 7 } = 7$

$14 - \sqrt[3]{x - 7 } = -7$

Correct answer:

$-14 - \sqrt[4]{x - 7 } = 7$

Explanation:

We can examine each individually.

$14 - \sqrt[3]{x - 7 } = -7$

$14 - \sqrt[3]{x - 7 } + 7 + \sqrt[3]{x - 7 } = -7 + 7 + \sqrt[3]{x - 7 }$

$21 = \sqrt[3]{x - 7 }$

$21^{3} =\left ( \sqrt[3]{x - 7 } \right )^{3}$

x-7 = 9,261

x = 9,268

This equation has a solution.

$-14 - \sqrt[3]{x - 7 } = 7$

$-14 - \sqrt[3]{x - 7 } - 7 + \sqrt[3]{x - 7 } = 7 - 7 + \sqrt[3]{x - 7 }$

$-21 = \sqrt[3]{x - 7 }$

$\left (-21 \right )^{3} =\left ( \sqrt[3]{x - 7 } \right )^{3}$

x-7 =- 9,261

x =- 9,254

This equation has a solution.

$14 - \sqrt[4]{x - 7 } = -7$

$14 - \sqrt[4]{x - 7 } + 7 + \sqrt[4]{x - 7 } = -7 + 7 + \sqrt[4]{x - 7 }$

$21 = \sqrt[4]{x - 7 }$

$21^{4} =\left ( \sqrt[4]{x - 7 } \right )^{4}$

x-7 = 194,481

x = 194,488

This equation has a solution.

$-14 - \sqrt[4]{x - 7 } = 7$

$-14 - \sqrt[4]{x - 7 } - 7 + \sqrt[4]{x - 7 } = 7 - 7 + \sqrt[4]{x - 7 }$

$-21 = \sqrt[4]{x - 7 }$

This equation has no solution, since a fourth root of a number must be nonnegative.

The correct choice is $-14 - \sqrt[4]{x - 7 } = 7$ .

Report an Error

Example Question #1 : Equations / Inequalities

Solve $\left | 3-4x\right |<0$ .

Possible Answers:

$x<\frac{4}{3}$

No solutions

$x>\frac{4}{3}$

$x<\frac{3}{4}$

$x>\frac{3}{4}$

Correct answer:

No solutions

Explanation:

By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.

Report an Error

Example Question #51 : Gre Quantitative Reasoning

Rational_5

Possible Answers:

–1

–2

Correct answer:

Explanation:

Rational_2

Rational_3

Rational_4

Report an Error

Example Question #51 : Gre Quantitative Reasoning

Linesmb1

Possible Answers:

–b/(m²– 1)

–bm/(m²+ 1)

–b/(m + 1)

b/(m²+ 1)

bm/(m²+ 1)

Correct answer:

b/(m²+ 1)

Explanation:

Linesmb5

Linesmb4

Report an Error

Example Question #3 : How To Find The Solution To A Rational Equation With Lcd

In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

$\frac{1}{m^3}-\frac{1}{k^2}=\frac{1}{p}$

Possible Answers:

$m=(\frac{p+k^2}{2})^3$

$m=(\frac{p+k^2}{pk^2})^{\frac{1}{3}}$

$m=\frac{p^2k^3}{p+k^2}$

$m=(\frac{pk^2}{p+k^2})^{\frac{1}{3}}$

$m=p^{\frac{1}{2}}-k^{\frac{1}{3}}$

Correct answer:

$m=(\frac{pk^2}{p+k^2})^{\frac{1}{3}}$

Explanation:

Pkm_7-21-13

Pkm2_7-21-13

Report an Error

Example Question #1 : Algebra

If 6x = 42 and xk = 2, what is the value of k?

Possible Answers:

1/7

1/6

2/7

Correct answer:

2/7

Explanation:

Solve the first equation for x by dividing both sides of the equation by 6; the result is 7. Solve the second equation for k by dividing both sides of the equation by x, which we now know is 7. The result is 2/7.

Report an Error

← Previous 1 2 … 5 6 7 8 9 10 11 12 13 … 22 23 Next →

View PSAT Mathematics Tutors

Eklavya
Certified Tutor

Indira Gandhi National Open University, Bachelor, Physics.

View PSAT Mathematics Tutors

Becca
Certified Tutor

Brigham Young University-Provo, Bachelor of Science, Electrical Engineering.

View PSAT Mathematics Tutors

Antonia
Certified Tutor

University of Michigan-Ann Arbor, Bachelor of Engineering, Nuclear Engineering. University of Wisconsin-Madison, Current Grad...

All PSAT Math Resources

10 Diagnostic Tests 421 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Chemistry Tutors in Los Angeles, English Tutors in Chicago, SAT Tutors in Miami, Reading Tutors in San Diego, Spanish Tutors in Houston, SSAT Tutors in Denver, Computer Science Tutors in Dallas Fort Worth, Computer Science Tutors in Seattle, French Tutors in Seattle, Algebra Tutors in Dallas Fort Worth

Popular Courses & Classes

GMAT Courses & Classes in Seattle, GMAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Philadelphia, MCAT Courses & Classes in Chicago, MCAT Courses & Classes in Philadelphia, MCAT Courses & Classes in Boston, MCAT Courses & Classes in Washington DC, Spanish Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Atlanta, SAT Courses & Classes in Chicago

Popular Test Prep

LSAT Test Prep in San Diego, GMAT Test Prep in Houston, MCAT Test Prep in New York City, ISEE Test Prep in New York City, MCAT Test Prep in Atlanta, LSAT Test Prep in Seattle, ISEE Test Prep in Boston, SSAT Test Prep in San Francisco-Bay Area, SAT Test Prep in Philadelphia, SSAT Test Prep in Houston

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)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

PSAT Math : Equations / Inequalities

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Linear / Rational / Variable Equations

Example Question #2 : Linear / Rational / Variable Equations

Example Question #1 : Linear / Rational / Variable Equations

Example Question #1 : How To Find Out When An Equation Has No Solution

Example Question #5 : Linear / Rational / Variable Equations

Example Question #1 : Equations / Inequalities

Example Question #51 : Gre Quantitative Reasoning

Example Question #51 : Gre Quantitative Reasoning

Example Question #3 : How To Find The Solution To A Rational Equation With Lcd

Example Question #1 : Algebra

All PSAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: