## Absolute Value Equations

In this section, you will explore methods for solving absolute value equations and how to analyze solutions to determine their feasibility. This section will help you become familiar with the algebra of absolute value equations in preparation for functions.

### Solving an Absolute Value Equation

Next, we will learn how to solve an absolute value equation. To solve an equation such as , we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is 8 or . This leads to two different equations we can solve independently.

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 is written as . It has the following properties:

For real numbers and , an equation of the form , with , will have solutions when or . If , the equation has no solution.

An **absolute value equation** in the form has the following properties:

#### HOW TO

Given an absolute value equation, solve it.

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

2. If , write and solve two equations: and .

#### EXAMPLE 8

Solving Absolute Value Equations

Solve the following absolute value equations:

#### Solution

(a)

Write two equations and solve each:

b)

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

(c)

Isolate the absolute value expression and then write two equations.

There are two solutions: , and .

(d)

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

#### TRY IT #7

Solve the absolute value equation: .

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

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