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.
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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.
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.
video by Dr Chris Tisdell 

Solving Absolute Value Equations
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.
\(\abs{x2}=4\)
Problem Statement 

Find the values of \(x\) that solve \(x2=4\)
Final Answer 

\(x=2, ~ x=6\)
Problem Statement 

Find the values of \(x\) that solve \(x2=4\)
Solution 

video by Dr Chris Tisdell 

Final Answer 

\(x=2, ~ x=6\)
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\(3x2=4\)
Problem Statement 

Find the values of \(x\) that solve \(3x2=4\).
Final Answer 

\(x=2/3, ~ x=2\)
Problem Statement 

Find the values of \(x\) that solve \(3x2=4\).
Solution 

video by Dr Chris Tisdell 

Final Answer 

\(x=2/3, ~ x=2\)
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\(3x+2 = 8\)
Problem Statement 

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

video by PatrickJMT 

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\(72x = 10\)
Problem Statement 

Find the values of \(x\) that solve the equation \(72x = 10\).
Final Answer 

\(x=3/2, x=17/2\)
Problem Statement 

Find the values of \(x\) that solve the equation \(72x = 10\).
Solution 

video by PatrickJMT 

Final Answer 

\(x=3/2, x=17/2\)
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\( 3x2 + 4 = 4 \)
Problem Statement 

Find the values of \(x\) that solve the equation \( 3x2 + 4 = 4 \).
Solution 

video by Freshmen Math Doctor 

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\( x+1 + 6 = 2 \)
Problem Statement 

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

video by Freshmen Math Doctor 

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\( 32x1 = 21 \)
Problem Statement 

Find the values of \(x\) that solve the equation \( 32x1 = 21 \).
Solution 

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\( 3x = 18 \)
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 

Final Answer 

\( x = 6 \) and \( x = 6 \)
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\( 5x + 7 = 42 \)
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 

Final Answer 

\( x = 7 \) and \( x = 49/5 \)
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\( 2x  3  = 9 \)
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 

Final Answer 

\( x = 6 \) and \( x = 3 \)
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\(\displaystyle{ \left \frac{4x+6}{5} \right = 6}\)
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 

Final Answer 

\( x = 6 \) and \( x = 9 \)
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\( 23x  4 + 5 = 27 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 23x  4 + 5 = 27 \).
Final Answer 

\( x = 5 \) and \( x = 7/3 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 23x  4 + 5 = 27 \).
Solution 

Final Answer 

\( x = 5 \) and \( x = 7/3 \)
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\( 34x  1  5 = 16 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 34x  1  5 = 16 \).
Final Answer 

\( x = 2 \) and \( x = 3/2 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 34x  1  5 = 16 \).
Solution 

Final Answer 

\( x = 2 \) and \( x = 3/2 \)
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\( x4 + 8 = 3 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( x4 + 8 = 3 \)
Final Answer 

no solution
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( x4 + 8 = 3 \)
Solution 

Final Answer 

no solution
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\( 7x + 2 = 4x + 11 \)
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 

Final Answer 

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

Final Answer 

\( x = 4 \)
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Intermediate
\( 2x1 = 348x  12 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 2x1 = 348x  12 \) giving your answer in exact form.
Solution 

video by Michael Penn 

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\( 3x+6 + x1 = 10 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 3x+6 + x1 = 10 \) giving your answer in exact form.
Solution 

video by Michael Penn 

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Solving Absolute Value Inequalities
This guy does a great job of explaining how to solve inequalities containing absolute values.
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.
\(x2 < 4\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(x2 < 4\).
Final Answer 

\(2 < x < 6\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(x2 < 4\).
Solution 

video by Dr Chris Tisdell 

Final Answer 

\(2 < x < 6\)
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\(3x2 < 4\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(3x2 < 4\).
Final Answer 

\(2/3 < x < 2\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(3x2 < 4\).
Solution 

video by Dr Chris Tisdell 

Final Answer 

\(2/3 < x < 2\)
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\(13x \leq 2\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(13x \leq 2\).
Final Answer 

\(1/3 \leq x \leq 1\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(13x \leq 2\).
Solution 

video by Dr Chris Tisdell 

Final Answer 

\(1/3 \leq x \leq 1\)
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\(54x > 2\)
Problem Statement 

Find the values of \(x\) that solve the inequality \(54x > 2\)
Final Answer 

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

Find the values of \(x\) that solve the inequality \(54x > 2\)
Solution 

video by Dr Chris Tisdell 

Final Answer 

\(x < 3/4\) or \(x > 7/4\)
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\( 3x+7 > 27 \)
Problem Statement 

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

video by Freshmen Math Doctor 

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\( x+3 \leq 4 \)
Problem Statement 

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

video by Freshmen Math Doctor 

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\( x < 4 \)
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 

Final Answer 

\( 4 < x < 4 \)
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\( 2x  3 \geq 8 \)
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 

Final Answer 

\( x \geq 5.5 \) or \( x \leq 2.5 \)
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\( 5  34x+1 \geq 9 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 5  34x+1 \geq 9 \).
Final Answer 

\( 17/12 \leq x \leq 11/12 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 5  34x+1 \geq 9 \).
Solution 

Final Answer 

\( 17/12 \leq x \leq 11/12 \)
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\( 2x + 5 > 11 \)
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 

Final Answer 

\( x > 3 \) or \( x < 8 \)
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\( 3x  4 \leq 17 \)
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 

Final Answer 

\( 13/3 \leq x \leq 7 \)
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\( 3x + 5 \leq 3 \)
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 

Final Answer 

no solution
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\( 4x  3 > 4 \)
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 

Final Answer 

all real numbers
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\( 37x  4 +5 > 17 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 37x  4 +5 > 17 \)
Final Answer 

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

Unless otherwise instructed, find the values of \(x\) that solve \( 37x  4 +5 > 17 \)
Solution 

Final Answer 

\(x > 8/7 \) or \( x < 0 \)
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\( 2 \leq 3x+5 \leq 10 \)
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 

video by Michael Penn 

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\( 3x1+4 \lt 8 \)
Problem Statement 

Unless otherwise instructed, find the values of \(x\) that solve \( 3x1+4 \lt 8 \) giving your answer in interval notation in exact form.
Solution 

video by Michael Penn 

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

Really UNDERSTAND Precalculus
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  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle 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
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
Trig Integrals
\(\int{\sin(x)~dx} = \cos(x)+C\) 
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\) 
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) 
\(\int{\csc(x)~dx} = \) \( \ln\abs{\csc(x)+\cot(x)}+C\) 
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Practice Instructions
Unless otherwise instructed, find all values of \(x\) that solve each equation or inequality.