Solving an Absolute Value Equation

Next, we will learn how to solve an absolute value equation. To solve an equation such as $|2x - 6|=8$ , 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{matrix} 2 x - 6 = 8 & or & 2 x - 6 = - 8 2 x = 14 & 2 x = - 2 x = 7 & x = - 1 \end{matrix}

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

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 $|x|$ . It has the following properties:

\begin{matrix} If x \geq 0, then | x | = x . If x < 0, then | x | = - x . \end{matrix}

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

For real numbers $A$ and $B$ , an equation of the form $|A|=B$ , with $B\ge 0$ , 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 $|ax+b|=c$ has the following properties:

\begin{matrix} If c < 0, | a x + b | = c has no solution . If c = 0, | a x + b | = c has one solution . If c > 0, | a x + b | = c has two solutions . \end{matrix}

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

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

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

Example 8: Solving Absolute Value Equations

Solve the following absolute value equations:

a. $|6x+4|=8$
b. $|3x+4|=-9$
c. $|3x - 5|-4=6$
d. $|-5x+10|=0$

Solution

a. $|6x+4|=8$

Write two equations and solve each:

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

The two solutions are $x=\frac{2}{3}$ , $x=-2$ .

b. $|3x+4|=-9$

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

c. $|3x - 5|-4=6$

Isolate the absolute value expression and then write two equations.

\begin{matrix} | 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 = - \frac{5}{3} \end{matrix}

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

There are two solutions: $x=5$ , $x=-\frac{5}{3}$ .

d. $|-5x+10|=0$

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

\begin{matrix} - 5 x + 10 & = 0 - 5 x & = - 10 x & = 2 \end{matrix}

$\begin{array}{l}-5x+10\hfill&=0\hfill \\ -5x\hfill&=-10\hfill \\ x\hfill&=2\hfill \end{array}$

There is one solution: $x=2$ .

Try It 7

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

Solution

Other Types of Equations