Absolute Value Equations

Next, we will learn how to solve an absolute value equation. To solve an equation such as \(|2 x-6|=8\), we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is 8 or \(-8\). This leads to two different equations we can solve independently.

\(
\begin{array}{rllr}
2 x-6 & =8 & \text { or } & 2x-6&=-8 \\
2 x & =14 && 2x&=-2 \\
x & =7 && x&=-1
\end{array}
\)

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

ABSOLUTE VALUE EQUATIONS

The absolute value of \(x\) is written as \(|x|\). It has the following properties:

\(\text{If} \, x \geq 0, \text{then} |x|=x\).

\(\text{If} \, x < 0, \text{then} |x|=-x\).

For real numbers \(A\) and \(B\), an equation of the form \(|A|=B\), with \(B \geq 0\), will have solutions when \(A=B\) or \(A=-B\). If \(B < 0\), the equation \(|A|=B\) has no solution.

An absolute value equation in the form \(|a x+b|=c\) has the following properties:

\( \begin{array}{l}

\text{If c < 0,|a x+b|=c has no solution.} \\

\text{If c = 0,|a x+b|=c has one solution.} \\

\text{If c > 0,|a x+b|=c has two solutions.} \end{array}\)

HOW TO

Given an absolute value equation, solve it.

1. Isolate the absolute value expression on one side of the equal sign.

2. If \(c > 0\), write and solve two equations: \(a x+b=c\) and \(a x+b=-c\).

EXAMPLE 8

Solving Absolute Value Equations

Solve the following absolute value equations:

(a) \(|6 x+4|=8\)

(b) \(|3 x+4|=-9\)

(c) \(|3 x-5|-4=6\)

(d) \(|-5 x+10|=0\)

Solution

(a)

\(|6 x+4|=8\)

Write two equations and solve each:

\(
\begin{array}{rlrl}
6 x+4 & =8 & 6 x+4 & =-8 \\
6 x & =4 & 6 x & =-12 \\
x & =\dfrac{2}{3} & x & =-2
\end{array}
\)

The two solutions are \(\dfrac{2}{3}\) and \(-2\).

b)

\(|3 x+4|=-9\)

There is no solution as an absolute value cannot be negative.

(c)

\(|3 x-5|-4=6\)

Isolate the absolute value expression and then write two equations.

\(
\begin{array}{rlrlr}
&& {|3 x-5|-4} =6 \\
&& |3 x-5|=10 & \\
3 x-5 & =10 & & 3 x-5=-10 \\
3 x & =15 & & 3 x=-5 \\
x & =5 & & x=-\dfrac{5}{3}
\end{array}
\)

There are two solutions: \(5\), and \(-\dfrac{5}{3}\).

(d)

(d) \(|-5 x+10|=0\)

The equation is set equal to zero, so we have to write only one equation.

\(

\begin{aligned}

-5 x+10 &=0 \\

-5 x &=-10 \\

x &=2

\end{aligned}

\)

There is one solution: \(2\).

TRY IT #7

Solve the absolute value equation: \(|1-4 x|+8=13\).

Source: Rice University, https://openstax.org/books/college-algebra/pages/2-6-other-types-of-equations
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