GMAT Math : Equations

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 #11 : Equations

Solve for :

4 (x +8) + 13 = 6 (x + 9) - x

Possible Answers:

x = -9

$x = 4 \frac{1}{2}$

The equation has no solution.

$x = -4 \frac{1}{2}$

x=9

Correct answer:

x = -9

Explanation:

4 (x +8) + 13 = 6 (x + 9) - x

$4 \cdot x +4 \cdot 8 + 13 = 6 \cdot x + 6 \cdot 9 - x$

4 x +32 + 13 = 6x + 54 - x

4 x +45 = 5x + 54

4 x +45 -4x -54 = 5x + 54 -4x -54

-9 = x

Report an Error

Example Question #1408 : Problem Solving Questions

Give all real solutions of the following equation:

$x^{4} - 10x^{2} + 9 = 0$

Possible Answers:

$\left \{ -3, -1, 1, 3\right \}$

$\left \{ 1, 3\right \}$

$\left \{1, 9\right \}$

$\left \{ -9, -1, 1, 9\right \}$

The equation has no solution.

Correct answer:

$\left \{ -3, -1, 1, 3\right \}$

Explanation:

By substituting $u = x^{2}$ and, subsequently, $u^{2} =\left ( x^{2} \right )^{2} = x^{4}$ , this equation be rewritten as a quadratic equation, and solved as such:

$x^{4} - 10x^{2} + 9 = 0$

$u^{2} - 10u + 9 = 0$

We are looking to factor the quadratic expression as (u+?)(u+?) , replacing the two question marks with integers with a product of 9 and a sum of ; these integers are -9,-1 .

(u-9)(u-1) = 0

Substitute back:

$(x^{2}-9)(x^{2}-1) = 0$

These factors can themselves be factored as the difference of squares:

(x+3)(x-3)(x+1)(x-1) = 0

Set each factor to zero and solve:

$x-3 = 0 \Rightarrow x = 3$

$x-1 = 0 \Rightarrow x = 1$

$x+1 = 0 \Rightarrow x = - 1$

$x+3 = 0 \Rightarrow x = - 3$

The solution set is $\left \{ -3, -1, 1, 3\right \}$ .

Report an Error

Example Question #1409 : Problem Solving Questions

Solve for :

$x - 7 \sqrt{x} + 12 = 0$

Possible Answers:

The equation has no solution.

$x = \sqrt{3} \textrm{ or } x = 2$

$x = 3 \textrm{ or } x = 4$

$x = 9 \textrm{ or } x = 16$

$x = 9 \textrm{ or } x = -9$

Correct answer:

$x = 9 \textrm{ or } x = 16$

Explanation:

Substitute $u = \sqrt{x}$ , and, subsequently, $u^{2} = x$ , to rewrite this equation as quadratic, then solve by factoring.

$x - 7 \sqrt{x} + 12 = 0$

$u^{2} - 7 u + 12 = 0$

We can rewrite the quadratic expression as (x + ?) (x + ?) , where the question marks are replaced with integers whose product is 12 and whose sum is ; these integers are $-3\ and -4$ .

(u-3) (u-4)= 0

Set each factor to zero and solve for ; then substitute back and solve for :

u-3 = 0

u=3

$\sqrt{x} =3$

x = 9

u-4 = 0

u=4

$\sqrt{x} =4$

x = 16

The solution set, which can be confirmed by substitution, is $\left \{ 9,16\right \}$ .

Report an Error

Example Question #1410 : Problem Solving Questions

Find all real solutions to the following equation:

$\frac{1}{x^{2}} - \frac{8}{x} +12 = 0$

Possible Answers:

The equation has no solution.

$x = 2 \textrm{ or } x = 6$

$x = -6 \textrm{ or } x = -2$

$x = \frac{1}{6} \textrm{ or } x = \frac{1}{2}$

$x = - \frac{1}{2} \textrm{ or } x = - \frac{1}{6}$

Correct answer:

$x = \frac{1}{6} \textrm{ or } x = \frac{1}{2}$

Explanation:

This can be best solved by substituting $u = \frac{1}{x}$ , and, subsequently, $u ^{2}=\left ( \frac{1}{x} \right ) ^{2} =\frac{1}{x ^{2}}$ , then solving the resulting quadratic equation.

$\frac{1}{x^{2}} - \frac{8}{x} + 12 = 0$

$\frac{1}{x^{2}} - 8 \cdot \frac{1}{x} + 12 = 0$

$u^{2} - 8u + 12 = 0$

Factor the expression on the left by finding two integers whose product is 12 and whose sum is :

(u-6)(u-2) = 0

Set each linear binomial factor to 0, solve separately for , and substitute back:

u-6 = 0

u=6

$\frac{1}{x} = 6$

$x = \frac{1}{6}$

u-2 = 0

u=2

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

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

Report an Error

Example Question #12 : Equations

The period of a pendulum - that is, the time it takes for the pendulum to swing once and back - varies directly as the square root of its length.

The pendulum of a giant clock is 18 meters long and has period 8.5 seconds. If the pendulum is lengthened to 21 meters, what will its period be, to the nearest tenth of a second?

Possible Answers:

$11.2 \textrm{ sec}$

$7.9 \textrm{ sec}$

$11.6 \textrm{ sec}$

$10.4 \textrm{ sec}$

$9.2 \textrm{ sec}$

Correct answer:

$9.2 \textrm{ sec}$

Explanation:

The variation equation for this situation is

$\frac{P_{1}}{\sqrt{L_{1}}} = \frac{P_{2}}{\sqrt{L}_{2}}$

Set $P_{1} = 8.5 , L_{1} = 18, {L}_{2}= 21$ , and solve for $P_{2}$ ;

$\frac{8.5}{\sqrt{18}} = \frac{P_{2}}{\sqrt{21}}$

$\frac{8.5}{\sqrt{18}} \cdot{\sqrt{21}} = \frac{P_{2}}{\sqrt{21}} \cdot{\sqrt{21}}$

$P_{2} = \frac{8.5}{\sqrt{18}} \cdot{\sqrt{21}} \approx 9.2 \textrm{ sec}$

Report an Error

Example Question #13 : Equations

$x = 8^{N}$

Which of these expressions is equal to $\log_{32} x$ ?

Possible Answers:

$N\sqrt[3]{N^{2}}$

$\sqrt[5]{N^{3}}$

$\frac{3}{5} N$

$\ln \frac{5}{3}N$

$\frac{5}{3} N$

Correct answer:

$\frac{3}{5} N$

Explanation:

$2^{5} = 32$

$2 = \sqrt[5]{32}$

$8 = 2^{3} = \left ( \sqrt[5]{32} \right ) ^{3} = 32^{ \frac{3}{5} }$

$x = 8^{N}= \left (32^{ \frac{3}{5} } \right \)^ {N } =32^{ \frac{3}{5} N }$

$\log_{32} x =\log_{32}32^{ \frac{3}{5} N } = \frac{3}{5} N$

Report an Error

Example Question #14 : Equations

Solve for :

| 4x - 18 | - 18 = 57

Possible Answers:

x=23.25

$x= -14.25 \textrm{ or }x=23.25$

$x= -14.25 \textrm{ or }x=14.25$

x= -14.25

$x= -23.25 \textrm{ or }x=23.25$

Correct answer:

$x= -14.25 \textrm{ or }x=23.25$

Explanation:

First, isolate the absolute value expression on one side:

| 4x - 18 | - 18 = 57

| 4x - 18 | -18+ 18 = 57+18

| 4x - 18 | = 75

Rewrite as a compound sentence:

$4x - 18 = - 75 \textrm{ or } 4x - 18 = 75$

Solve each separately:

4x - 18 = - 75

4x - 18+18 = -75+18

4x= - 57

$4x \div 4= - 57 \div 4$

x= - 14.25

4x - 18 = 75

4x - 18+18 = 75+18

4x= 93

$4x \div 4=93 \div 4$

x = 23.25

The solution set is $\left \{ -14.25, 23.25 \right \}$

Report an Error

Example Question #15 : Equations

Solve for :

| 4x - 18 | + 18 = 57

Possible Answers:

$x=-5.25 \textrm{ or }x=14.25$

$x=-5.25 \textrm{ or }x=5.25$

x=14.25

x=-14.25

$x=-14.25\textrm{ or }x=14.25$

Correct answer:

$x=-5.25 \textrm{ or }x=14.25$

Explanation:

First, isolate the absolute value expression on one side:

| 4x - 18 | + 18 = 57

| 4x - 18 | + 18 -18 = 57-18

| 4x - 18 | = 39

Rewrite as a compound sentence:

$4x - 18 = - 39 \textrm{ or } 4x - 18 = 39$

Solve each separately:

4x - 18 = - 39

4x - 18+18 = - 39+18

4x= - 21

$4x \div 4= - 21 \div 4$

x= - 5.25

4x - 18 = 39

4x - 18+18 = 39+18

4x= 57

$4x \div 4=57 \div 4$

x = 14.25

The solution set is $\left \{ -5.25, 14.25\right \}$ .

Report an Error

Example Question #332 : Algebra

Give the solution set of the equation:

$\left | x ^{2} - 2x \right | = 3x -6$

Possible Answers:

$x \in \left \{ 2, 3\right \}$

$x \in \left \{ -3, 2, 3\right \}$

$x \in \left \{ -3, 3\right \}$

The equation has no solution.

$x \in \left \{ -3, 2\right \}$

Correct answer:

$x \in \left \{ 2, 3\right \}$

Explanation:

Write this as a compound equation and solve each separately.

$\left | x ^{2} - 2x \right | = 3x -6$

$x ^{2} - 2x = 3x -6 \textrm{ or }x ^{2} - 2x = -\left ( 3x -6 \right )$

$x ^{2} - 2x = 3x -6$

$x ^{2} - 2x - 3x + 6 = 3x -6 - 3x + 6$

$x ^{2} - 5x + 6 =0$

(x-3) (x-2) =0

$x - 3 = 0 \Rightarrow x = 3$

$x - 2 = 0 \Rightarrow x =2$

$x ^{2} - 2x = -\left ( 3x -6 \right )$

$x ^{2} - 2x = -3x +6$

$x ^{2} - 2x+ 3x - 6= -3x +6 + 3x - 6$

$x ^{2} +x - 6= 0$

(x + 3) (x - 2)= 0

$x + 3 = 0 \Rightarrow x = -3$

$x - 2 = 0 \Rightarrow x =2$

This gives us three possible solutions - $\left \{ -3, 2, 3\right \}$ . We check all three.

x = -3

$\left | x ^{2} - 2x \right | = 3x -6$

$\left | \left ( -3\right ) ^{2} - 2 (-3) \right | = 3 (-3) -6$

$\left | \left9 - (-6) \right | = -9-6$

$\left |15| = -15$

15 = -15

This is a false statement so we can eliminate as a false "solution".

x = 2

$\left | x ^{2} - 2x \right | = 3x -6$

$\left |2^{2} - 2 \cdot 2 \right | = 2 \cdot 3 -6$

$\left | \left 4-4 \right | = 6-6$

|0| = 0

0=0

2 is a solution.

x = 3

$\left | x ^{2} - 2x \right | = 3x -6$

$\left | 3 ^{2} - 2 \cdot 3 \right | = 3 \cdot 3 -6$

$\left | \left 9 - 6\right | = 9-6$

$\left |3 | = 3$

3= 3

3 is a solution.

The solution set is $x \in \left \{ 2, 3\right \}$ .

Report an Error

Example Question #16 : Equations

Solve for :

$4 ^{5x - 7} = 8 ^{2x -1}$

Possible Answers:

x = 2.75

x = -1.0625

The equation has no solution

x = 4.25

x = 0.6875

Correct answer:

x = 2.75

Explanation:

Since $2 ^{2} = 4$ and $2^{3} = 8$ , replace, and use the exponent rules:

$4 ^{5x - 7} = 8 ^{2x -1}$

$\left ( 2 ^{2} \right ) ^{5x - 7} = \left ( 2^{3} \right ) ^{2x -1}$

$2 ^{2 \left ( 5x - 7 \right )}=2 ^{3 \left ( 2x - 1 \right )}$

Set the exponents equal to each other and solve for :

$2 \left ( 5x - 7 \right )=3 \left ( 2x - 1 \right )$

10x -14=6x - 3

10x-6x -14+14=6x -6x - 3+14

4x = 11

$4x \div 4 = 11 \div 4$

x = 2.75

Report an Error

All GMAT Math Resources

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

Popular Subjects

GMAT Tutors in Houston, GMAT Tutors in New York City, SAT Tutors in Philadelphia, English Tutors in Denver, Spanish Tutors in Denver, ISEE Tutors in Los Angeles, Calculus Tutors in Seattle, GRE Tutors in New York City, Statistics Tutors in Houston, ACT Tutors in Dallas Fort Worth

Popular Courses & Classes

SAT Courses & Classes in Chicago, LSAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Phoenix, SSAT Courses & Classes in Miami, MCAT Courses & Classes in San Diego, MCAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Houston, LSAT Courses & Classes in Los Angeles, LSAT Courses & Classes in San Diego, LSAT Courses & Classes in Seattle

Popular Test Prep

MCAT Test Prep in Los Angeles, GMAT Test Prep in Miami, LSAT Test Prep in San Diego, ISEE Test Prep in New York City, SAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Philadelphia, SSAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Washington DC, GMAT Test Prep in Los Angeles

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 : Equations

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Equations

Example Question #1408 : Problem Solving Questions

Example Question #1409 : Problem Solving Questions

Example Question #1410 : Problem Solving Questions

Example Question #12 : Equations

Example Question #13 : Equations

Example Question #14 : Equations

Example Question #15 : Equations

Example Question #332 : Algebra

Example Question #16 : Equations

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details