Solving an Absolute Value Equation

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

[latex]\begin{array}{lll}2x - 6=8\hfill & \text{ or }\hfill & 2x - 6=-8\hfill \\ 2x=14\hfill & \hfill & 2x=-2\hfill \\ x=7\hfill & \hfill & x=-1\hfill \end{array}[/latex]

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.

A General Note: Absolute Value Equations

The absolute value of x is written as [latex]|x|[/latex]. It has the following properties:

[latex]\begin{array}{l}\text{If } x\ge 0,\text{ then }|x|=x.\hfill \\ \text{If }x<0,\text{ then }|x|=-x.\hfill \end{array}[/latex]

For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex], will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]. If [latex]B<0[/latex], the equation [latex]|A|=B[/latex] has no solution. An absolute value equation in the form [latex]|ax+b|=c[/latex] has the following properties:

[latex]\begin{array}{l}\text{If }c<0,|ax+b|=c\text{ has no solution}.\hfill \\ \text{If }c=0,|ax+b|=c\text{ has one solution}.\hfill \\ \text{If }c>0,|ax+b|=c\text{ has two solutions}.\hfill \end{array}[/latex]

How To: Given an absolute value equation, solve it.

  1. Isolate the absolute value expression on one side of the equal sign.
  2. If [latex]c>0[/latex], write and solve two equations: [latex]ax+b=c[/latex] and [latex]ax+b=-c[/latex].

Example 8: Solving Absolute Value Equations

Solve the following absolute value equations:

a. [latex]|6x+4|=8[/latex]
b. [latex]|3x+4|=-9[/latex]
c. [latex]|3x - 5|-4=6[/latex]
d. [latex]|-5x+10|=0[/latex]

Solution

a. [latex]|6x+4|=8[/latex]

Write two equations and solve each:

[latex]\begin{array}{ll}6x+4\hfill&=8\hfill& 6x+4\hfill&=-8\hfill \\ 6x\hfill&=4\hfill& 6x\hfill&=-12\hfill \\ x\hfill&=\frac{2}{3}\hfill& x\hfill&=-2\hfill \end{array}[/latex]

The two solutions are [latex]x=\frac{2}{3}[/latex], [latex]x=-2[/latex].

b. [latex]|3x+4|=-9[/latex]

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

c. [latex]|3x - 5|-4=6[/latex]

Isolate the absolute value expression and then write two equations.

[latex]\begin{array}{lll}\hfill & |3x - 5|-4=6\hfill & \hfill \\ \hfill & |3x - 5|=10\hfill & \hfill \\ \hfill & \hfill & \hfill \\ 3x - 5=10\hfill & \hfill & 3x - 5=-10\hfill \\ 3x=15\hfill & \hfill & 3x=-5\hfill \\ x=5\hfill & \hfill & x=-\frac{5}{3}\hfill \end{array}[/latex]

There are two solutions: [latex]x=5[/latex], [latex]x=-\frac{5}{3}[/latex].

d. [latex]|-5x+10|=0[/latex]

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

[latex]\begin{array}{l}-5x+10\hfill&=0\hfill \\ -5x\hfill&=-10\hfill \\ x\hfill&=2\hfill \end{array}[/latex]

There is one solution: [latex]x=2[/latex].

Try It 7

Solve the absolute value equation: [latex]|1 - 4x|+8=13[/latex].

Solution