Solving Absolute Value Equations
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On this page, we cover how to solve equations (with an equal sign) that contain absolute values. There is one key idea that you need to keep in mind as you solve these equations. To get started, let's watch this video.
video by MIP4U 

It is important to notice one important detail that he did in this video. The key idea is that, if you have only one absolute value expression in the equation, move all terms that do not contain absolute value to one side of the equal sign, leaving only the absolute value term on the other side. Only after doing that, can you set up the two equations without absolute values and solve.
Now you should be ready for some practice problems. After that, you will be ready to solve absolute value inequalities on the next page.
Practice Solving Equations
Basic
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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\(3x2=4\)
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 

Find the values of \(x\) that solve \( 3x = 18 \).
Final Answer 

\( x = 6 \) and \( x = 6 \)
Problem Statement
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 

Find the values of \(x\) that solve \( 5x + 7 = 42 \).
Final Answer 

\( x = 7 \) and \( x = 49/5 \)
Problem Statement
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 

Find the values of \(x\) that solve \( 2x  3  = 9 \).
Final Answer 

\( x = 6 \) and \( x = 3 \)
Problem Statement
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 

Find the values of \(x\) that solve \(\displaystyle{ \left \frac{4x+6}{5} \right = 6}\).
Final Answer 

\( x = 6 \) and \( x = 9 \)
Problem Statement
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 

Find the values of \(x\) that solve \( 23x  4 + 5 = 27 \).
Final Answer 

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

Find the values of \(x\) that solve \( 34x  1  5 = 16 \).
Final Answer 

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

Find the values of \(x\) that solve \( x4 + 8 = 3 \)
Final Answer 

no solution
Problem Statement
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 

Find the values of \(x\) that solve \( 7x + 2 = 4x + 11 \).
Final Answer 

\( x = 3 \) and \( x = 13/11 \)
Problem Statement
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 

Find the values of \(x\) that solve \( x + 3 = x  11 \).
Hint 

This seems more difficult than it really is. Try all four possible cases with positive and negative signs, then solve each one. After that, look back at your work and notice that really only two cases are required.
Problem Statement 

Find the values of \(x\) that solve \( x + 3 = x  11 \).
Final Answer 

\( x = 4 \)
Problem Statement
Find the values of \(x\) that solve \( x + 3 = x  11 \).
Hint
This seems more difficult than it really is. Try all four possible cases with positive and negative signs, then solve each one. After that, look back at your work and notice that really only two cases are required.
Solution
The four cases are
1. Both sides positive
2. Both sides negative
3. Left side positive and right side negative
4. Right side positive and left side negative.
Cases 1 and 2 are the same. Cases 3 and 4 are the same.
Final Answer
\( x = 4 \)
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Intermediate
\( 2x1 = 348x  12 \)
Problem Statement
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
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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Really UNDERSTAND Precalculus
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