\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Solving Absolute Value Inequalities

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Solving Absolute Value Inequalities

Solving inequalities requires special techniques that you didn't need when you were solving equations.

Recommended Books on Amazon (affiliate links)

Complete 17Calculus Recommended Books List

Prime Student 6-month Trial

This guy does a great job of explaining how to solve inequalities containing absolute values.

MIP4U - Absolute Value Inequalities [10min-29secs]

video by MIP4U

Now try the practice problems.

Trigonometry Demystified 2/E

Practice Solving Inequalities

Unless otherwise instructed, find all values of \(x\) that solve each inequality.

\(|x-2| < 4\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|x-2| < 4\).

Final Answer

\(-2 < x < 6\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|x-2| < 4\).

Solution

Dr Chris Tisdell - 2089 video solution

video by Dr Chris Tisdell

Final Answer

\(-2 < x < 6\)

Log in to rate this practice problem and to see it's current rating.

\(|3x-2| < 4\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|3x-2| < 4\).

Final Answer

\(-2/3 < x < 2\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|3x-2| < 4\).

Solution

Dr Chris Tisdell - 2090 video solution

video by Dr Chris Tisdell

Final Answer

\(-2/3 < x < 2\)

Log in to rate this practice problem and to see it's current rating.

\(|1-3x| \leq 2\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|1-3x| \leq 2\).

Final Answer

\(-1/3 \leq x \leq 1\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|1-3x| \leq 2\).

Solution

Dr Chris Tisdell - 2091 video solution

video by Dr Chris Tisdell

Final Answer

\(-1/3 \leq x \leq 1\)

Log in to rate this practice problem and to see it's current rating.

\(|5-4x| > 2\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|5-4x| > 2\)

Final Answer

\(x < 3/4\) or \(x > 7/4\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|5-4x| > 2\)

Solution

Dr Chris Tisdell - 2092 video solution

video by Dr Chris Tisdell

Final Answer

\(x < 3/4\) or \(x > 7/4\)

Log in to rate this practice problem and to see it's current rating.

\( 3|x+7| > 27 \)

Problem Statement

Find the values of \(x\) that solve the inequality \( 3|x+7| > 27 \).

Solution

Freshmen Math Doctor - 2558 video solution

Log in to rate this practice problem and to see it's current rating.

\( |x+3| \leq 4 \)

Problem Statement

Find the values of \(x\) that solve the inequality \( |x+3| \leq 4 \).

Solution

Freshmen Math Doctor - 2559 video solution

Log in to rate this practice problem and to see it's current rating.

\( |x| < 4 \)

Problem Statement

Find the values of \(x\) that solve \( |x| < 4 \).

Final Answer

\( -4 < x < 4 \)

Problem Statement

Find the values of \(x\) that solve \( |x| < 4 \).

Solution

2895 video solution

Final Answer

\( -4 < x < 4 \)

Log in to rate this practice problem and to see it's current rating.

\( |2x - 3| \geq 8 \)

Problem Statement

Find the values of \(x\) that solve \( |2x - 3| \geq 8 \).

Final Answer

\( x \geq 5.5 \) or \( x \leq -2.5 \)

Problem Statement

Find the values of \(x\) that solve \( |2x - 3| \geq 8 \).

Solution

2896 video solution

Final Answer

\( x \geq 5.5 \) or \( x \leq -2.5 \)

Log in to rate this practice problem and to see it's current rating.

\( 5 - 3|4x+1| \geq -9 \)

Problem Statement

Find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Final Answer

\( -17/12 \leq x \leq 11/12 \)

Problem Statement

Find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Solution

2897 video solution

Final Answer

\( -17/12 \leq x \leq 11/12 \)

Log in to rate this practice problem and to see it's current rating.

\( |2x + 5| > 11 \)

Problem Statement

Find the values of \(x\) that solve \( |2x + 5| > 11 \)

Final Answer

\( x > 3 \) or \( x < -8 \)

Problem Statement

Find the values of \(x\) that solve \( |2x + 5| > 11 \)

Solution

2898 video solution

Final Answer

\( x > 3 \) or \( x < -8 \)

Log in to rate this practice problem and to see it's current rating.

\( |3x - 4| \leq 17 \)

Problem Statement

Find the values of \(x\) that solve \( |3x - 4| \leq 17 \).

Final Answer

\( 13/3 \leq x \leq 7 \)

Problem Statement

Find the values of \(x\) that solve \( |3x - 4| \leq 17 \).

Solution

2899 video solution

Final Answer

\( 13/3 \leq x \leq 7 \)

Log in to rate this practice problem and to see it's current rating.

\( |3x + 5| \leq -3 \)

Problem Statement

Find the values of \(x\) that solve \( |3x + 5| \leq -3 \)

Final Answer

no solution

Problem Statement

Find the values of \(x\) that solve \( |3x + 5| \leq -3 \)

Solution

2900 video solution

Final Answer

no solution

Log in to rate this practice problem and to see it's current rating.

\( |4x - 3| > -4 \)

Problem Statement

Find the values of \(x\) that solve \( |4x - 3| > -4 \)

Final Answer

all real numbers

Problem Statement

Find the values of \(x\) that solve \( |4x - 3| > -4 \)

Solution

2901 video solution

Final Answer

all real numbers

Log in to rate this practice problem and to see it's current rating.

\( 3|7x - 4| +5 > 17 \)

Problem Statement

Find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)

Final Answer

\(x > 8/7 \) or \( x < 0 \)

Problem Statement

Find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)

Solution

2902 video solution

Final Answer

\(x > 8/7 \) or \( x < 0 \)

Log in to rate this practice problem and to see it's current rating.

\( 2 \leq |3x+5| \leq 10 \)

Problem Statement

Find the values of \(x\) that solve \( 2 \leq |3x+5| \leq 10 \) giving your answer in interval notation in exact form.

Solution

Michael Penn - 3862 video solution

video by Michael Penn

Log in to rate this practice problem and to see it's current rating.

\( ||3x-1|+4| \lt 8 \)

Problem Statement

Find the values of \(x\) that solve \( ||3x-1|+4| \lt 8 \) giving your answer in interval notation in exact form.

Solution

Michael Penn - 3863 video solution

video by Michael Penn

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Precalculus

Log in to rate this page and to see it's current rating.

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

learning and study techniques

Shop Amazon - Rent eTextbooks - Save up to 80%

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Practice Search

Practice Instructions

Unless otherwise instructed, find all values of \(x\) that solve each inequality.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics