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.

Try It

Find the solution(s) to each absolute value equation:

1)  |f| = 5

2)  |g| = -2

3)  |2h – 1| = 11

4)  |3i| - 2 = 13

5)  5|2j – 4| = 30

6)  |-2k + 10| = 10

Scroll Down for Answers…

Answers:

1)  f = 5 or f = -5

2)  No solution.

3)  h = 6 or h = -5

4)  i = 5 or i = -5

5)  j = 5 or j = -1

6)  k = 0 or k = 10