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.
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
Instructions  Unless otherwise instructed, find all values of \(x\) that solve each equation.
\(\abs{x2}=4\)
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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Find the values of x that solve \(3x2=4\).
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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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 

video by PatrickJMT 

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Find the values of x that solve the equation \(72x = 10\).
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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Find the values of x that solve the equation \( 3x2 + 4 = 4 \).
Problem Statement 

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

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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 

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

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

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

Unless otherwise instructed, find the values of \(x\) that solve \( 3x = 18 \).
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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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 \( 5x + 7 = 42 \).
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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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 \( 2x  3  = 9 \).
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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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 \(\displaystyle{ \left \frac{4x+6}{5} \right = 6}\).
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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Unless otherwise instructed, find the values of \(x\) that solve \( 23x  4 + 5 = 27 \).
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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Unless otherwise instructed, find the values of \(x\) that solve \( 34x  1  5 = 16 \).
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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Unless otherwise instructed, find the values of \(x\) that solve \( x4 + 8 = 3 \).
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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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 \( 7x + 2 = 4x + 11 \).
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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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 \( x + 3 = x  11 \).
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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Solving Absolute Value Inequalities 

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 \(x2 < 4\).
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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Find the values of x that solve the inequality \(3x2 < 4\).
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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Find the values of x that solve the inequality \(13x \leq 2\).
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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Find the values of x that solve the inequality \(54x > 2\).
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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Find the values of x that solve the inequality \( 3x+7 > 27 \).
Problem Statement 

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

close solution

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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 

close solution

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

Unless otherwise instructed, find the values of \(x\) that solve \( x < 4 \).
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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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 \( 2x  3 \geq 8 \).
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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Unless otherwise instructed, find the values of \(x\) that solve \( 5  34x+1 \geq 9 \).
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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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 \( 2x + 5 > 11 \).
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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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 \( 3x  4 \leq 17 \).
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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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 \( 3x + 5 \leq 3 \).
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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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 \( 4x  3 > 4 \).
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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Unless otherwise instructed, find the values of \(x\) that solve \( 37x  4 +5 > 17 \).
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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