## 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.

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

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:

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

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:

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

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

1. Isolate the absolute value expression on one side of the equal sign.
2. 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.

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

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.

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

There is one solution: $x=2$.

### Try It 7

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

Solution