Absolute Value Equations

 

Learning Objective(s)

·         Find all possible solutions for absolute value equations involving variables and variable terms.

 

Introduction

 

Absolute value describes the magnitude of a number or the distance between points, but it strips out information on the sign of the number or the direction of the distance. A positive absolute value could represent either a positive or a negative original value. When we simplify or solve equations that include absolute value expressions, we must consider both possibilities.

 

Absolute value expressions can include not just numbers, but also variables. This adds yet another wrinkle to evaluating such expressions.

 

The Absolute Value of Isolated Variables

 

Let’s look at the simple equation |x| = 3. To solve an equation like this, with a variable inside absolute value bars, we must separate the two possible cases and solve for each.

 

The expression inside the absolute value bars might be positive. In that case it equals the absolute value: x = 3.

 

Or the expression inside the absolute value bars might be negative. In that case the expression’s original value is the opposite of the absolute value: -(x) = 3. To get the value of x, we can multiply each side of the equation by -1 and we get: x = -3.

 

So solving the equation for x gives us more than one correct answer. This is generally the case for equations that include the absolute value of a variable: they have more than one solution.

 

We indicate this numerically by listing all the correct answers, separated by a comma. In this example, |x| = 3, the solution is x = -3, 3.

 

To show the solutions on a number line, we put a point in both locations.

 

 

 

Which of the following is the correct solution for |x| = 8?

 

A) x = 8, -8

B) x = -8

C) x = 8

D) x = -(-8), +(8)

 

Show/Hide Answer

A) x = 8, -8

Correct. Both 8 and -8 have an absolute value of 8. Both are possible solutions.

 

B) x = -8

Incorrect. While it’s true that the absolute value of -8 is 8, it is also true that the absolute value of 8 is 8. The equation has two possible solutions. The correct answer is x = 8, -8.

 

C) x = 8

Incorrect. While it’s true that the absolute value of 8 is 8, it is also true that the absolute value of -8 is 8. The equation has two possible solutions. The correct answer is x = 8, -8.

 

D) x = -(-8), +(8)

Incorrect. While it’s true that the absolute value of +8 is 8, -(-8) is the same as 8, not a second solution. The correct answer is x = 8, -8.

 

 

The Absolute Value of Variable Terms

 

An expression inside absolute value bars can be more complicated than just a lone variable. When these expressions include other values and operations, we must be very careful, especially when solving for their opposites.

 

To solve |-2x| = 8, for example, we must consider two possibilities—that the expression inside the absolute value bars, -2x, is positive or negative.

 

So if -2x is positive, then

-2x = 8

To solve for x, we can divide each side of the equation by -2 and we get

x = -4

 

So if -2x is negative, then

-(-2x) = 8

To solve for x, we multiply -2x by -1 and get

2x = 8

Then divide each side of the equation by 2 and we get

x = 4

 

So the solution to |-2x| = 8 is

x = -4, 4

 

Notice how very careful we have to be about positive and negative signs when working with the absolute value of an algebraic expression. We can't let ourselves be distracted by seeing negative numbers inside the absolute value bars. We must still solve for the case when the entire expression inside the absolute value bars is positive and then again, when it is negative.

 

Which graph shows the correct solution for |2n| = 4?

 

A)

B)

 

 

C)

 

D)

 

Show/Hide Answer

A) Correct. n = -2, 2.

 

B) Incorrect. -4 and 4 are the possible values of the expression 2n, not the possible values of n. The correct plot shows points at -2 and 2.

 

C) Incorrect. 4 is the absolute value of the expression 2n. n = -2, 2. The correct plot shows points at -2 and 2.

 

D) Incorrect. -2 is one possible solution to |2n| = 4. But 2 is also a solution. Both possibilities must be shown. The correct plot shows points at -2 and 2.

 

 

 

Summary

 

Absolute value changes the value of the expression within them. The absolute value of any expression is either positive or zero.

 

When we are given the absolute value of an expression containing a variable, we don’t know whether the variable is positive or negative. Both cases will satisfy the equation, so both solutions must be found.