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

17Calculus Precalculus - Absolute Value

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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.

Dr Chris Tisdell - What is the absolute value function? (part 1) [1min-57secs]

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.

Dr Chris Tisdell - What is the absolute value function? (part 2) [about 9min]

video by Dr Chris Tisdell

Solving Absolute Value Equations

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

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.

\(\abs{x-2}=4\)

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

2087 video

video by Dr Chris Tisdell

Final Answer

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

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

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

2088 video

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

2093 video

video by PatrickJMT

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Find the values of x that solve the equation \(|7-2x| = 10\).

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

2094 video

video by PatrickJMT

Final Answer

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

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

Problem Statement

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

Solution

2560 video

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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

2561 video

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

Problem Statement

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

Solution

2562 video

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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

2886 video

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

2887 video

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

2888 video

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

2889 video

Final Answer

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

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Unless otherwise instructed, find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 2|3x - 4| + 5 = 27 \).

Solution

2890 video

Final Answer

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

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Unless otherwise instructed, find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|4x - 1| - 5 = 16 \).

Solution

2891 video

Final Answer

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

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Unless otherwise instructed, find the values of \(x\) that solve \( |x-4| + 8 = 3 \).

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x-4| + 8 = 3 \).

Final Answer

no solution

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( |x-4| + 8 = 3 \).

Solution

2892 video

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

2893 video

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

2894 video

Final Answer

\( x = 4 \)

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Solving Absolute Value Inequalities

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

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

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

2089 video

video by Dr Chris Tisdell

Final Answer

\(-2 < x < 6\)

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

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

2090 video

video by Dr Chris Tisdell

Final Answer

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

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

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

2091 video

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

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

2092 video

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

Problem Statement

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

Solution

2558 video

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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

2559 video

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

2895 video

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

2896 video

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).

Solution

2897 video

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

2898 video

Final Answer

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

close solution

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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

2899 video

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

2900 video

Final Answer

no solution

close 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

2901 video

Final Answer

all real numbers

close solution

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Unless otherwise instructed, find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \).

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \).

Final Answer

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

Problem Statement

Unless otherwise instructed, find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \).

Solution

2902 video

Final Answer

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

close solution

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