Call Now to Set Up Tutoring:

(888) 888-0446

Algebra 1 : Absolute Value Inequalities

Study concepts, example questions & explanations for Algebra 1

Create An Account

All Algebra 1 Resources

10 Diagnostic Tests 557 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 Next →

Example Question #1 : Absolute Value Inequalities

Solve for :

| 8y + 2 | < -38

Possible Answers:

$\left ( -5, -4 \frac{1}{2} \right )$

The inequality has no solution.

$\left ( -5, 4 \frac{1}{2} \right )$

$\left ( 4 \frac{1}{2}, 5 \right )$

$\left ( - 4 \frac{1}{2}, 5 \right )$

Correct answer:

The inequality has no solution.

Explanation:

The absolute value of a number must always be nonnegative, so | 8y + 2 | can never be less than . This means the inequality has no solution.

Report an Error

Example Question #1 : Absolute Value Inequalities

Solve the inequality $\left | 4x-1\right |+7\leq14$ .

Possible Answers:

$-2\leq x\leq 1.5$

$-1.5\leq x\leq2$

$x\leq1$

$-6\leq x\leq 0.5$

$x\geq-3$

Correct answer:

$-1.5\leq x\leq2$

Explanation:

First, we can simplify this inequality by subtracting 7 from both sides. This gives us

$\left | 4x-1\right |\leq7$

Next, however, we need to make two separate inequalities due to the presence of an absolute value expression. What this inequality actually means is that

$4x-1\leq7$

and

$4x-1\geq -7$

(Be careful with the inequality signs here! The second sign must be switched to allow for the effect of absolute value on negative numbers. In other words, the inequality must be greater than because, after the absolute value is applied, it will be less than 7.) When we solve the two inequalities, we get two solutions:

$x\leq2$

and

$x\geq-1.5$

For the original statement to be true, both of these inequalities must be fulfilled. We're left with a final answer of

$-1.5\leq x\leq2$

Report an Error

Example Question #2 : Absolute Value Inequalities

Solve the inequality:

$\small \left |15x - 39\right | < -42$

Possible Answers:

$\small \left ( -\infty, \infty \right )$

$\small \left ( -\infty , -\frac{1}{5}\right ) \cup \left (\frac{27}{5} , \infty \right )$

$\small \left(-\frac{1}{5}, \frac{61}{5} \right )$

$\small \varnothing$ (no solution)

Correct answer:

$\small \varnothing$ (no solution)

Explanation:

The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, $\small \left |15x -39 \right| < -42$ can never happen. There is no solution.

Report an Error

Example Question #1 : Solving Inequalities

Solve this inequality.

$\left | \frac{4x+3}{x+6}\right |\eq>1$

Possible Answers:

$x\eq>-\frac{9}{5}, x \eq<1$

$x\eq>-\frac{9}{5}, x\eq>1$

$x\eq<\frac{9}{5}, x\eq>-1$

$x \eq< -\frac{9}{5}, \hspace x \eq>1$

$x<-\frac{9}{5}, x\eq<1$

Correct answer:

$x \eq< -\frac{9}{5}, \hspace x \eq>1$

Explanation:

Split the inequality into two possible cases as follows, based on the absolute values.

First case: $\frac{4x+3}{x+6}\eq>1$

Second case: $\frac{4x+3}{x+6} \eq< -1$

Let's find the inequality of the first case.

$\frac{4x+3}{x+6} \eq>1$

Multiply both sides by x + 6.

$4x+3 \eq>x+6$

Subtract x from both sides, then subtract 3 from both sides.

$3x \eq>3$

Divide both sides by 3.

$x\eq>1$

Let's find the inequality of the second case.

$\frac{4x+3}{x+6} \eq< -1$

Multiply both sides by x + 6.

$4x+3 \eq< -1(x+6)$

Simplify.

4x+3< -x-6

Add x to both sides, then subtract 3 from both sides.

$5x\eq< -9$

Divide both sides by 5.

$x \eq<-\frac{9}{5}$

So the range of x-values is $x<-\frac{9}{5}$ and $x \eq>1$ .

Report an Error

Example Question #1 : Solving Absolute Value Equations

Solve for :

$\left |2x+3 \right |>7$

Possible Answers:

x>2, x<-5

x<2, x>-5

x>2, x<-10

x>3.5, x<-5

x>2

Correct answer:

x>2, x<-5

Explanation:

Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.

Report an Error

Example Question #1 : Absolute Value Inequalities

Give the solution set for the following equation:

| 4x - 7 | + 5 = 86

Possible Answers:

$\begin{Bmatrix} 18.5, 22 \end{Bmatrix}$

$\begin{Bmatrix} 22 \end{Bmatrix}$

$\begin{Bmatrix} 22, 24.5 \end{Bmatrix}$

$\begin{Bmatrix} -18.5, 22 \end{Bmatrix}$

$\begin{Bmatrix} -24.5, 22 \end{Bmatrix}$

Correct answer:

$\begin{Bmatrix} -18.5, 22 \end{Bmatrix}$

Explanation:

First, subtract 5 from both sides to get the absolute value expression alone.

| 4x - 7 | + 5 = 86

| 4x - 7 | = 81

Split this into two linear equations:

4x - 7 = 81

4x = 88

x = 22

4x - 7 = -81

4x = -74

$x = -74 \div 4 = -18.5$

The solution set is $\begin{Bmatrix} -18.5, 22 \end{Bmatrix}$

Report an Error

Example Question #3 : Absolute Value Inequalities

Solve for in the inequality below.

$|x-2| \leq 4$

Possible Answers:

$-2 \leq x\leq 6$

$x\leq 4$

No solutions

$x\leq -2\; \; or\; \; x\geq 6$

All real numbers

Correct answer:

$-2 \leq x\leq 6$

Explanation:

The absolute value gives two problems to solve. Remember to switch the "less than" to "greater than" when comparing the negative term.

$x-2\leq 4$ or $x-2\geq -4$

Solve each inequality separately by adding to all sides.

$x\leq 6$ or $x\geq -2$

This can be simplified to the format $-2 \leq x\leq 6$ .

Report an Error

Example Question #10 : Solving Absolute Value Equations

$\textup{If }\left | 2x+4\right |\leq10\textup{ , what are all possible values of }x\textup{?}$

Possible Answers:

$-7\leq x\leq3$

x=-7,x=3

$-3\leq x\leq7$

$-3\leq x\leq3$

$x\leq-7\textup{ and }x\geq3$

Correct answer:

$-7\leq x\leq3$

Explanation:

$\textup{Set up two inequalities. Reverse the inequality sign for the negative one.}$

$2x+4\leq10\;\;\;\;\;\;\;\;\;\;\;\;\;2x+4\geq-10$

$2x\leq6\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \; 2x\geq-14$

$x\leq3\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\geq-7$

Report an Error

Example Question #3 : Absolute Value Inequalities

Solve the inequality.

$\small \left | x+2 \right |>5$

Possible Answers:

$\small x<-7\ or\ x>3$

$\small -7<x<3$

$\small -3<x<7$

$\small x<-3\ or\ x>7$

Correct answer:

$\small x<-7\ or\ x>3$

Explanation:

$\small \left | x+2 \right |>5$

Remove the absolute value by setting the term equal to either $\small 5$ or $\small -5$ . Remember to flip the inequality for the negative term!

$\small x+2>5\ or\ x+2<-5$

Solve each scenario independently by subtracting $\small 2$ from both sides.

$\small x+2-2>5-2\ or\ x+2-2<-5-2$

$\small x>3\ or\ x<-7$

Report an Error

Example Question #1 : Absolute Value Inequalities

Solve for :

| 4y + 7 | > -5

Possible Answers:

$(-\infty , -3) \cup \left ( \frac{1}{2}, \infty \right)$

$\left(-\infty , -\frac{1}{2}\right) \cup \left ( \frac{1}{2}, \infty \right)$

$(-\infty , \infty )$

$\left(-\infty , -\frac{1}{2}\right) \cup \left (3, \infty \right)$

$(-\infty , -3) \cup (3, \infty )$

Correct answer:

$(-\infty , \infty )$

Explanation:

The absolute value of any number is nonnegative, so | 4y + 7 | must always be greater than . Therefore, any value of makes this a true statement.

Report an Error

← Previous 1 2 Next →

View Algebra Tutors

Maja
Certified Tutor

Estrella Mountain Community College, Aeronautical & Astronautical Engineer, Computer Software Engineering.

View Algebra Tutors

Glauco
Certified Tutor

University of Mount Union, Bachelor of Science, Electrical Engineering.

View Algebra Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

All Algebra 1 Resources

10 Diagnostic Tests 557 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Calculus Tutors in Dallas Fort Worth, Math Tutors in Philadelphia, Spanish Tutors in Philadelphia, SAT Tutors in Washington DC, Statistics Tutors in Miami, ACT Tutors in Philadelphia, Biology Tutors in Chicago, Computer Science Tutors in Phoenix, Reading Tutors in Atlanta, SAT Tutors in Denver

Popular Courses & Classes

LSAT Courses & Classes in Phoenix, LSAT Courses & Classes in Atlanta, GRE Courses & Classes in Phoenix, ISEE Courses & Classes in Washington DC, GRE Courses & Classes in San Diego, GMAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Boston, ACT Courses & Classes in Miami, ISEE Courses & Classes in Phoenix, SSAT Courses & Classes in Seattle

Popular Test Prep

SSAT Test Prep in Miami, ISEE Test Prep in Phoenix, MCAT Test Prep in Phoenix, ACT Test Prep in Washington DC, GRE Test Prep in New York City, GRE Test Prep in Seattle, GMAT Test Prep in Los Angeles, ACT Test Prep in Dallas Fort Worth, SSAT Test Prep in San Francisco-Bay Area, ACT Test Prep in New York City

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

Algebra 1 : Absolute Value Inequalities

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : Absolute Value Inequalities

Example Question #1 : Absolute Value Inequalities

Example Question #2 : Absolute Value Inequalities

Example Question #1 : Solving Inequalities

Example Question #1 : Solving Absolute Value Equations

Example Question #1 : Absolute Value Inequalities

Example Question #3 : Absolute Value Inequalities

Example Question #10 : Solving Absolute Value Equations

Example Question #3 : Absolute Value Inequalities

Example Question #1 : Absolute Value Inequalities

All Algebra 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: