Solving Absolute Value Equations

Learning Objective

Introduction

Absolute valueThe value of a number without regard to its sign. 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 absolute value is the opposite of the expression inside the absolute value bars: `|-3|=-(-3) = 3`. In terms of a variable, if the variable `x` is negative, then `|x|=-(x)=-x`.

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.

`|x|=3`

The number line goes from negative 4 to positive 4 with 7 marks in between. Each mark increases by one. Solid points are shown at negative 3 and positive 3.

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)` 

 

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.

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`

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)

The number line goes from negative 4 to positive 4 with 7 marks in between. Each mark increases by one. Solid points are shown at negative 2 and positive 2.

 

B)

The number line goes from negative 4 to positive 4 with 7 marks in between. Each mark increases by one. Solid points are shown at negative 4 and positive 4.

 

C)

The number line goes from negative 4 to positive 4 with 7 marks in between. Each mark increases by one. There is a solid point shown at positive 4.

 

D)

The number line goes from negative 4 to positive 4 with 7 marks in between. Each mark increases by one. There is a solid point shown at negative 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.