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

17Calculus Precalculus - Solving Absolute Value Equations

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Solving Absolute Value Equations

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.

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

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.

Deep Work: Rules for Focused Success in a Distracted World

Practice Solving Equations

Basic

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 \(|x-2|=4\)

Final Answer

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

Problem Statement

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

Solution

Dr Chris Tisdell - 2087 video solution

video by Dr Chris Tisdell

Final Answer

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

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

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

Dr Chris Tisdell - 2088 video 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

PatrickJMT - 2093 video solution

video by PatrickJMT

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

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

PatrickJMT - 2094 video solution

video by PatrickJMT

Final Answer

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

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

Problem Statement

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

Solution

Freshmen Math Doctor - 2560 video solution

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

Problem Statement

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

Solution

Freshmen Math Doctor - 2561 video solution

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

Problem Statement

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

Solution

2562 video 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

2886 video 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

2887 video 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

2888 video 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

2889 video solution

Final Answer

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

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

Problem Statement

Find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Final Answer

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

Problem Statement

Find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Solution

2890 video solution

Final Answer

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

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

Problem Statement

Find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Final Answer

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

Problem Statement

Find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Solution

2891 video solution

Final Answer

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

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

Problem Statement

Find the values of \(x\) that solve \( |x-4| + 8 = 3 \)

Final Answer

no solution

Problem Statement

Find the values of \(x\) that solve \( |x-4| + 8 = 3 \)

Solution

2892 video 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

2893 video 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.

2894 video solution

Final Answer

\( x = 4 \)

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Intermediate

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

Problem Statement

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

Solution

Michael Penn - 3884 video solution

video by Michael Penn

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

Problem Statement

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

Solution

Michael Penn - 3885 video solution

video by Michael Penn

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Practice Instructions

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

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