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

17Calculus Precalculus - Solving Absolute Value Inequalities

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Solving Absolute Value Inequalities

Solving inequalities requires special techniques that you didn't need when you were solving equations.

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This guy does a great job of explaining how to solve inequalities containing absolute values.

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

video by MIP4U

Now try the practice problems.

College Algebra DeMYSTiFieD, 2nd Edition

Practice Solving Inequalities

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

\(|x-2| < 4\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|x-2| < 4\).

Final Answer

\(-2 < x < 6\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|x-2| < 4\).

Solution

Dr Chris Tisdell - 2089 video solution

video by Dr Chris Tisdell

Final Answer

\(-2 < x < 6\)

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

Problem Statement

Find the values of \(x\) that solve the inequality \(|3x-2| < 4\).

Final Answer

\(-2/3 < x < 2\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|3x-2| < 4\).

Solution

Dr Chris Tisdell - 2090 video solution

video by Dr Chris Tisdell

Final Answer

\(-2/3 < x < 2\)

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\(|1-3x| \leq 2\)

Problem Statement

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

Final Answer

\(-1/3 \leq x \leq 1\)

Problem Statement

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

Solution

Dr Chris Tisdell - 2091 video solution

video by Dr Chris Tisdell

Final Answer

\(-1/3 \leq x \leq 1\)

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\(|5-4x| > 2\)

Problem Statement

Find the values of \(x\) that solve the inequality \(|5-4x| > 2\)

Final Answer

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

Problem Statement

Find the values of \(x\) that solve the inequality \(|5-4x| > 2\)

Solution

Dr Chris Tisdell - 2092 video solution

video by Dr Chris Tisdell

Final Answer

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

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\( 3|x+7| > 27 \)

Problem Statement

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

Solution

Freshmen Math Doctor - 2558 video solution

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\( |x+3| \leq 4 \)

Problem Statement

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

Solution

Freshmen Math Doctor - 2559 video solution

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\( |x| < 4 \)

Problem Statement

Find the values of \(x\) that solve \( |x| < 4 \).

Final Answer

\( -4 < x < 4 \)

Problem Statement

Find the values of \(x\) that solve \( |x| < 4 \).

Solution

2895 video solution

Final Answer

\( -4 < x < 4 \)

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\( |2x - 3| \geq 8 \)

Problem Statement

Find the values of \(x\) that solve \( |2x - 3| \geq 8 \).

Final Answer

\( x \geq 5.5 \) or \( x \leq -2.5 \)

Problem Statement

Find the values of \(x\) that solve \( |2x - 3| \geq 8 \).

Solution

2896 video solution

Final Answer

\( x \geq 5.5 \) or \( x \leq -2.5 \)

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\( 5 - 3|4x+1| \geq -9 \)

Problem Statement

Find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Final Answer

\( -17/12 \leq x \leq 11/12 \)

Problem Statement

Find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Solution

2897 video solution

Final Answer

\( -17/12 \leq x \leq 11/12 \)

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\( |2x + 5| > 11 \)

Problem Statement

Find the values of \(x\) that solve \( |2x + 5| > 11 \)

Final Answer

\( x > 3 \) or \( x < -8 \)

Problem Statement

Find the values of \(x\) that solve \( |2x + 5| > 11 \)

Solution

2898 video solution

Final Answer

\( x > 3 \) or \( x < -8 \)

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\( |3x - 4| \leq 17 \)

Problem Statement

Find the values of \(x\) that solve \( |3x - 4| \leq 17 \).

Final Answer

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

Problem Statement

Find the values of \(x\) that solve \( |3x - 4| \leq 17 \).

Solution

2899 video solution

Final Answer

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

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\( |3x + 5| \leq -3 \)

Problem Statement

Find the values of \(x\) that solve \( |3x + 5| \leq -3 \)

Final Answer

no solution

Problem Statement

Find the values of \(x\) that solve \( |3x + 5| \leq -3 \)

Solution

2900 video solution

Final Answer

no solution

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\( |4x - 3| > -4 \)

Problem Statement

Find the values of \(x\) that solve \( |4x - 3| > -4 \)

Final Answer

all real numbers

Problem Statement

Find the values of \(x\) that solve \( |4x - 3| > -4 \)

Solution

2901 video solution

Final Answer

all real numbers

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\( 3|7x - 4| +5 > 17 \)

Problem Statement

Find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)

Final Answer

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

Problem Statement

Find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)

Solution

2902 video solution

Final Answer

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

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\( 2 \leq |3x+5| \leq 10 \)

Problem Statement

Find the values of \(x\) that solve \( 2 \leq |3x+5| \leq 10 \) giving your answer in interval notation in exact form.

Solution

Michael Penn - 3862 video solution

video by Michael Penn

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\( ||3x-1|+4| \lt 8 \)

Problem Statement

Find the values of \(x\) that solve \( ||3x-1|+4| \lt 8 \) giving your answer in interval notation in exact form.

Solution

Michael Penn - 3863 video solution

video by Michael Penn

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Practice Instructions

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

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