## 17Calculus Precalculus - Absolute Value

### Functions

Functions

Polynomials

Rational Functions

Matrices

Systems

Trigonometry

Complex Numbers

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

### 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

Instructions - Unless otherwise instructed, find all values of $$x$$ that solve each equation.

$$\abs{x-2}=4$$

Problem Statement

Find the values of $$x$$ that solve $$|x-2|=4$$.

$$x=-2, ~ x=6$$

Problem Statement

Find the values of $$x$$ that solve $$|x-2|=4$$.

Solution

### 2087 video

video by Dr Chris Tisdell

$$x=-2, ~ x=6$$

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

Find the values of x that solve $$|3x-2|=4$$.

Problem Statement

Find the values of x that solve $$|3x-2|=4$$.

$$x=-2/3, ~ x=2$$

Problem Statement

Find the values of x that solve $$|3x-2|=4$$.

Solution

### 2088 video

video by Dr Chris Tisdell

$$x=-2/3, ~ x=2$$

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

Find the values of x that solve the equation $$|3x+2| = 8$$.

Problem Statement

Find the values of x that solve the equation $$|3x+2| = 8$$.

Solution

### 2093 video

video by PatrickJMT

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

Find the values of x that solve the equation $$|7-2x| = 10$$.

Problem Statement

Find the values of x that solve the equation $$|7-2x| = 10$$.

$$x=-3/2, x=17/2$$

Problem Statement

Find the values of x that solve the equation $$|7-2x| = 10$$.

Solution

### 2094 video

video by PatrickJMT

$$x=-3/2, x=17/2$$

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

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

Problem Statement

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

Solution

### 2560 video

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

Find the values of x that solve the equation $$|x+1| + 6 = 2$$.

Problem Statement

Find the values of x that solve the equation $$|x+1| + 6 = 2$$.

Solution

### 2561 video

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

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

Problem Statement

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

Solution

### 2562 video

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

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

Problem Statement

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

$$x = 6$$ and $$x = -6$$

Problem Statement

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

Solution

### 2886 video

$$x = 6$$ and $$x = -6$$

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

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

Problem Statement

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

$$x = 7$$ and $$x = -49/5$$

Problem Statement

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

Solution

### 2887 video

$$x = 7$$ and $$x = -49/5$$

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

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

Problem Statement

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

$$x = 6$$ and $$x = -3$$

Problem Statement

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

Solution

### 2888 video

$$x = 6$$ and $$x = -3$$

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

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

Problem Statement

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

$$x = 6$$ 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

$$x = 6$$ and $$x = -9$$

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

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

Problem Statement

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

$$x = 5$$ and $$x = -7/3$$

Problem Statement

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

Solution

### 2890 video

$$x = 5$$ and $$x = -7/3$$

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

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

Problem Statement

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

$$x = 2$$ and $$x = -3/2$$

Problem Statement

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

Solution

### 2891 video

$$x = 2$$ and $$x = -3/2$$

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

Unless otherwise instructed, find the values of $$x$$ that solve $$|x-4| + 8 = 3$$.

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x-4| + 8 = 3$$.

no solution

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x-4| + 8 = 3$$.

Solution

### 2892 video

no solution

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

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

Problem Statement

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

$$x = 3$$ and $$x = -13/11$$

Problem Statement

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

Solution

### 2893 video

$$x = 3$$ and $$x = -13/11$$

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

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

Problem Statement

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

$$x = 4$$

Problem Statement

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

Solution

### 2894 video

$$x = 4$$

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

Solving Absolute Value Inequalities

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

video by MIP4U

Now try these practice problems.

Practice

Instructions - Unless otherwise instructed, find the values of $$x$$ that solve each inequality.

Find the values of x that solve the inequality $$|x-2| < 4$$.

Problem Statement

Find the values of x that solve the inequality $$|x-2| < 4$$.

$$-2 < x < 6$$

Problem Statement

Find the values of x that solve the inequality $$|x-2| < 4$$.

Solution

### 2089 video

video by Dr Chris Tisdell

$$-2 < x < 6$$

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

Find the values of x that solve the inequality $$|3x-2| < 4$$.

Problem Statement

Find the values of x that solve the inequality $$|3x-2| < 4$$.

$$-2/3 < x < 2$$

Problem Statement

Find the values of x that solve the inequality $$|3x-2| < 4$$.

Solution

### 2090 video

video by Dr Chris Tisdell

$$-2/3 < x < 2$$

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

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

Problem Statement

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

$$-1/3 \leq x \leq 1$$

Problem Statement

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

Solution

### 2091 video

video by Dr Chris Tisdell

$$-1/3 \leq x \leq 1$$

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

Find the values of x that solve the inequality $$|5-4x| > 2$$.

Problem Statement

Find the values of x that solve the inequality $$|5-4x| > 2$$.

$$x < 3/4$$ 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

$$x < 3/4$$ or $$x > 7/4$$

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

Find the values of x that solve the inequality $$3|x+7| > 27$$.

Problem Statement

Find the values of x that solve the inequality $$3|x+7| > 27$$.

Solution

### 2558 video

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

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

Problem Statement

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

Solution

### 2559 video

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

Unless otherwise instructed, find the values of $$x$$ that solve $$|x| < 4$$.

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x| < 4$$.

$$-4 < x < 4$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x| < 4$$.

Solution

### 2895 video

$$-4 < x < 4$$

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

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

Problem Statement

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

$$x \geq 5.5$$ or $$x \leq -2.5$$

Problem Statement

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

Solution

### 2896 video

$$x \geq 5.5$$ or $$x \leq -2.5$$

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

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

Problem Statement

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

$$-17/12 \leq x \leq 11/12$$

Problem Statement

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

Solution

### 2897 video

$$-17/12 \leq x \leq 11/12$$

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

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x + 5| > 11$$.

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x + 5| > 11$$.

$$x > 3$$ or $$x < -8$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x + 5| > 11$$.

Solution

### 2898 video

$$x > 3$$ or $$x < -8$$

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

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

Problem Statement

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

$$13/3 \leq x \leq 7$$

Problem Statement

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

Solution

### 2899 video

$$13/3 \leq x \leq 7$$

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

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

Problem Statement

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

no solution

Problem Statement

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

Solution

### 2900 video

no solution

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

Unless otherwise instructed, find the values of $$x$$ that solve $$|4x - 3| > -4$$.

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|4x - 3| > -4$$.

all real numbers

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|4x - 3| > -4$$.

Solution

### 2901 video

all real numbers

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

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

Problem Statement

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

$$x > 8/7$$ or $$x < 0$$

Problem Statement

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

Solution

### 2902 video

$$x > 8/7$$ or $$x < 0$$

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

### absolute value 17calculus youtube playlist

Really UNDERSTAND Precalculus

 piecewise functions

### 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

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

### 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.

 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.
 What Does 'Absolute Value' Mean? Graphs of Absolute Value Functions Solving Absolute Value Equations Practice Solving Absolute Value Inequalities Practice

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.