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

17Calculus Precalculus - Absolute Value

Algebra

Polynomials

Functions

Basics of Functions

Specific Functions

Rational Functions

Graphing

Matrices

Systems

Trigonometry

Complex Numbers

Applications

Learning Tools

Articles

Absolute values in functions are not hard to handle. You just need to get used to the notation and learn how to break up a function into different parts.

What Does 'Absolute Value' Mean?

The absolute value signs are two vertical lines on both sides of an expression, for example \(\abs{x+3}\). When an expression is contained in a set of absolute value signs the idea is that the sign of the expression is stripped off and replaced with a positive sign. There are two possible cases.
Case 1 If the expression is already positive or zero, the absolute value signs do nothing and can be dropped.
Case 2 If the expression is less than zero, then the expression is replaced by the negative of itself and the absolute value signs can then be dropped.

Example 1: \(\abs{3} = 3\)
Since \(3\) is already positive, the absolute value signs do nothing and therefore can be dropped with no change in the expression, in this case \(3\).
Example 2: \(\abs{-3} = 3\)
In this case, the expression \(-3\) is stripped of the negative sign and replaced with the positive version of the expression, in this case the new value is \(3\).

These examples are pretty easy to see what needs to be done with the sign of the expression but most times, we do not know if the expression is positive or negative.
Example 3: \(\abs{x}\)
Since x is a variable and it's value has not been limited as positive, negative or even real, we can do nothing about the absolute value signs. The absolute value signs are a shorthand way of saying that when \(x\) is positive, the expression is just \(x\) but when \(x\) is negative, the expression is \(-x\).

This video clip explains this idea in more detail very well and shows a great way to write the absolute value function as a piecewise function.

Dr Chris Tisdell - What is the absolute value function? (part 1) [1min-57secs]

video by Dr Chris Tisdell

Notice he said in the video that the absolute value tells us the distance between things. This is a great way of looking at it.

Graphs of Absolute Value Functions

Let's watch a bit more of the same video above discussing how to graph absolute value functions and how to visualize the distance idea on the graph.

Dr Chris Tisdell - What is the absolute value function? (part 2) [about 9min]

video by Dr Chris Tisdell

Solving Absolute Value Equations

MIP4U - Absolute Value Equations [9min-30secs]

video by MIP4U

After that video, you should be ready for some practice problems.

Practice Solving Equalities

Basic

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

\(\abs{x-2}=4\)

Problem Statement

Find the values of \(x\) that solve \(|x-2|=4\)

Final Answer

\(x=-2, ~ x=6\)

Problem Statement

Find the values of \(x\) that solve \(|x-2|=4\)

Solution

2087 video

video by Dr Chris Tisdell

Final Answer

\(x=-2, ~ x=6\)

close solution

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 \(|3x-2|=4\).

Final Answer

\(x=-2/3, ~ x=2\)

Problem Statement

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

Solution

2088 video

video by Dr Chris Tisdell

Final Answer

\(x=-2/3, ~ x=2\)

close solution

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

\(|3x+2| = 8\)

Problem Statement

Find the values of \(x\) that solve the equation \(|3x+2| = 8\).

Solution

2093 video

video by PatrickJMT

close solution

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

\(|7-2x| = 10\)

Problem Statement

Find the values of \(x\) that solve the equation \(|7-2x| = 10\).

Final Answer

\(x=-3/2, x=17/2\)

Problem Statement

Find the values of \(x\) that solve the equation \(|7-2x| = 10\).

Solution

2094 video

video by PatrickJMT

Final Answer

\(x=-3/2, x=17/2\)

close solution

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

\( |3x-2| + 4 = 4 \)

Problem Statement

Find the values of \(x\) that solve the equation \( |3x-2| + 4 = 4 \).

Solution

2560 video

close solution

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

\( |x+1| + 6 = 2 \)

Problem Statement

Find the values of \(x\) that solve the equation \( |x+1| + 6 = 2 \).

Solution

2561 video

close solution

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

\( 3|2x-1| = 21 \)

Problem Statement

Find the values of \(x\) that solve the equation \( 3|2x-1| = 21 \).

Solution

2562 video

close solution

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

\( |3x| = 18 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x| = 18 \).

Final Answer

\( x = 6 \) and \( x = -6 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x| = 18 \).

Solution

2886 video

Final Answer

\( x = 6 \) and \( x = -6 \)

close solution

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

\( |5x + 7| = 42 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |5x + 7| = 42 \).

Final Answer

\( x = 7 \) and \( x = -49/5 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |5x + 7| = 42 \).

Solution

2887 video

Final Answer

\( x = 7 \) and \( x = -49/5 \)

close solution

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

\( |2x - 3 | = 9 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x - 3 | = 9 \).

Final Answer

\( x = 6 \) and \( x = -3 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x - 3 | = 9 \).

Solution

2888 video

Final Answer

\( x = 6 \) and \( x = -3 \)

close solution

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

\(\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}\)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \(\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}\).

Final Answer

\( x = 6 \) and \( x = -9 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \(\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}\).

Solution

2889 video

Final Answer

\( x = 6 \) and \( x = -9 \)

close solution

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

\( 2|3x - 4| + 5 = 27 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Final Answer

\( x = 5 \) and \( x = -7/3 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Solution

2890 video

Final Answer

\( x = 5 \) and \( x = -7/3 \)

close solution

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

\( 3|4x - 1| - 5 = 16 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Final Answer

\( x = 2 \) and \( x = -3/2 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Solution

2891 video

Final Answer

\( x = 2 \) and \( x = -3/2 \)

close solution

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

\( |x-4| + 8 = 3 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x-4| + 8 = 3 \)

Final Answer

no solution

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x-4| + 8 = 3 \)

Solution

2892 video

Final Answer

no solution

close solution

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

\( |7x + 2| = 4x + 11 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |7x + 2| = 4x + 11 \).

Final Answer

\( x = 3 \) and \( x = -13/11 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |7x + 2| = 4x + 11 \).

Solution

2893 video

Final Answer

\( x = 3 \) and \( x = -13/11 \)

close solution

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

\( |x + 3| = |x - 11| \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x + 3| = |x - 11| \).

Final Answer

\( x = 4 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x + 3| = |x - 11| \).

Solution

2894 video

Final Answer

\( x = 4 \)

close solution

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

Intermediate

\( |2x-1| = 3|4-8x| - 12 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x-1| = 3|4-8x| - 12 \) giving your answer in exact form.

Solution

3884 video

video by Michael Penn

close solution

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

\( |3x+6| + |x-1| = 10 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x+6| + |x-1| = 10 \) giving your answer in exact form.

Solution

3885 video

video by Michael Penn

close solution

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

Solving Absolute Value Inequalities

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 these practice problems.

Practice Solving Inequalities

Unless otherwise instructed, find all values of \(x\) that solve each equation or 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

2089 video

video by Dr Chris Tisdell

Final Answer

\(-2 < x < 6\)

close solution

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

2090 video

video by Dr Chris Tisdell

Final Answer

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

close solution

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

2091 video

video by Dr Chris Tisdell

Final Answer

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

close solution

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

2092 video

video by Dr Chris Tisdell

Final Answer

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

close solution

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

2558 video

close 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

2559 video

close solution

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

\( |x| < 4 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x| < 4 \).

Final Answer

\( -4 < x < 4 \)

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x| < 4 \).

Solution

2895 video

Final Answer

\( -4 < x < 4 \)

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x - 3| \geq 8 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x - 3| \geq 8 \).

Solution

2896 video

Final Answer

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

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Solution

2897 video

Final Answer

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

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x + 5| > 11 \)

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |2x + 5| > 11 \)

Solution

2898 video

Final Answer

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

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x - 4| \leq 17 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x - 4| \leq 17 \).

Solution

2899 video

Final Answer

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

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x + 5| \leq -3 \)

Final Answer

no solution

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |3x + 5| \leq -3 \)

Solution

2900 video

Final Answer

no solution

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |4x - 3| > -4 \)

Final Answer

all real numbers

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |4x - 3| > -4 \)

Solution

2901 video

Final Answer

all real numbers

close solution

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

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)

Solution

2902 video

Final Answer

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

close solution

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

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

Problem Statement

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

Solution

3862 video

video by Michael Penn

close solution

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

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

Problem Statement

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

Solution

3863 video

video by Michael Penn

close solution

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

Finally, if you want a full lecture on absolute value equations, inequalities and other inequalities, here is a video.

freeCodeCamp.org - [39mins-18secs]

video by freeCodeCamp.org

Really UNDERSTAND Precalculus

Topics You Need To Understand For This Page

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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.

calculus motivation - music and learning

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

Math Word Problems Demystified

Save 20% on Under Armour Plus Free Shipping Over $49!

Shop Amazon - Used Textbooks - Save up to 90%

As an Amazon Associate I earn from qualifying purchases.

Practice Instructions

Unless otherwise instructed, find all values of \(x\) that solve each equation or 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 © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics
17Calculus
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.