## Solving Equations Containing Absolute Values

### Learning Outcomes

• Solve equations containing absolute values

## Solving One-Step Equations Containing Absolute Values with Addition

The absolute value of a number or expression describes its distance from $0$ on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or $0$.

For example, $−4$ and  $4$ both have an absolute value of  $4$ because they are each $4$ units from $0$ on a number line—though they are located in opposite directions from $0$ on the number line.

When solving absolute value equations and inequalities, you have to consider both the behavior of absolute value and the properties of equality and inequality.

Because both positive and negative values have a positive absolute value, solving absolute value equations means finding the solution for both the positive and the negative values.

Let’s first look at a very basic example.

$\displaystyle \left| x \right|=5$

This equation is read “the absolute value of  $x$ is equal to five.” The solution is the value(s) that are five units away from $0$ on a number line.

You might think of $5$ right away; that is one solution to the equation. Notice that $−5$ is also a solution because $−5$ is $5$ units away from  $0$ in the opposite direction. So, the solution to this equation $\displaystyle \left| x \right|=5$ is $x = −5$ or $x = 5$.

### Solving Equations of the Form $|x|=a$

For any positive number $a$, the solution of $\left|x\right|=a$ is

$x=a$  or  $x=−a$

$x$ can be a single variable or any algebraic expression.

You can solve a more complex absolute value problem in a similar fashion.

### Example

Solve $\displaystyle \left| x+5\right|=15$.

The following video provides worked examples of solving linear equations with absolute value terms.

## Solving One-Step Equations Containing Absolute Values With Multiplication

In the last section, we saw examples of solving equations with absolute values where the only operation was addition or subtraction. Now we will see how to solve equations with absolute value that include multiplication.

Remember that absolute value refers to the distance from zero. You can use the same technique of first isolating the absolute value, then setting up and solving two equations to solve an absolute value equation involving multiplication.

### Example

Solve $\displaystyle \left| 2x\right|=6$.

### Example

Solve $\displaystyle\frac{1}{3}\left|k\right|=12$.

In the following video, you will see two examples of how to solve an absolute value equation, one with integers and one with fractions.

## Solving Multi-Step Equations With Absolute Value

We can apply the same techniques we used for solving a one-step equation which contains absolute value to an equation that will take more than one step to solve.  Let’s start with an example where the first step is to write two equations: one equal to positive  $26$ and one equal to negative $26$.

### Example

Solve for $p$. $\left|2p–4\right|=26$

### Try It

In the next video, we show more examples of solving a simple absolute value equation.

Now let’s look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.

### Example

Solve for $w$. $3\left|4w–1\right|–5=10$

In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.