## Absolute Value Equations

## Introduction

Remember that the absolute value of a number means “the distance from that number to zero.” The absolute value of a number can easily be represented as a distance.

The absolute value of a variable can also be represented as a distance. Since the distance on a number line can be either right or left from zero, a problem such as |x| = 4 will have two answers, one where distance is to the right of zero and a second where distance is to the left of zero.

So equation |x| = 4 actually has two answers. You will find that most absolute value equations in this lesson also have two answers.

## Lesson

The equation |x| = 9 can be translated as “the distance from x to zero is equal to nine (units).” There are two numbers that are 9 away from zero, 9 and -9. Each of these works for x in the above equation, so each is a solution to the equation.

**Working with absolute value equations**

The equation |x – 3| = 4 can also be separated into two individual answers.

The answers here are x = 7 and x = -1. This same process of solving two separate equations can be used to solve more complex equations:

**Example 1****:** Find the solution(s) to |3x – 6| = 18

Solving an absolute value equation generally means separating it into two separate equations, one equaling a positive the other the negative answer to the original equation. Example 2 can be solved similarly.

**Example 2****:** Find the solution(s) to 2|(3 – x)| = 4

In many problems, there will be one positive and one negative answer. Example 2 is not a typical problem as it has a pair of positive answers.

**How many answers to expect**

The absolute value problems above each have two answers… but is that always the case? To answer this question, recall what the term *absolute value* really means. It can be expressed as “the distance from zero to a number.” Take a look at the three different possible results for an absolute value problem.

While two solutions are the norm when solving absolute value equations, some problems have a single solution (or none at all.) The number of solutions is always decided by whether the absolute value equals a positive, zero, or negative result.

**Related Links: **

Looking for a lesson on a similar topic? Try the links below.

**Absolute Value Lessons**

**Solving Equations Lessons**

Looking for something else? Explore our menu of general math or algebra lessons.