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Absolute values make the expression within nonnegative. Earlier, we saw absolute values with sums.

Let's look at a few examples of how to perform multiplication with absolute values.

Example 1

$$|-8| \times |2|$$ $$8 \times 2$$ $$16$$

The product of these two values is $16$.

Example 2

$$|(-3 \times 3) + 1|$$

We have a set of parentheses inside the absolute value signs. According to the Order of Operations (PEMDAS), we need to handle the stuff inside the parentheses first.

$$|(-9) + 1|$$

Now, we're all set to perform the operation inside the absolute value sign to evaluate the expression.

$$|-8| = 8$$

Example 3: Variables

$$4|-3x| = 24$$

First, we need to isolate the $x$ term on one side of the equation. (Hopefully this sounds familiar $-$ it's the same approach we take almost every time we solve for $x$!)

$$|-3x| = 6$$

Recall that the quantity inside the absolute value lines produces a nonnegative result, no matter what. Because of this, though, we don't know if our $-3x$ inside the absolute value lines is equal to $6$, or if our $-3x$ actually equals $-6$.

So, we need to account for both outcomes:

$$\begin{align} -3x &= 6\\ -3x &= -6 \end{align}$$

Now, just solve for $x$ in both equations.

$$\begin{align} x = -2\\ x = 2 \end{align}$$

Notice that plugging in either value for $x$ ($-2$ or $2$) into the original equation gives us the same answer of $6$.

Remember, plugging in both values is a good way to double check your work.

(Video) Solving a multi step absolute value equation

by Brian McLogan

This video by Brian McLogan works through a complex absolute value problem.

$\large{\text{Summary}}$

First of all, check out a review of finding the solution to an absolute value equation with sums.

Now, lets solve this equation.

$$8x=2|2x-2|$$

First, we have to get the absolute value alone. To do that, we divide by $2$.

$$4x=|2x-2|$$

Now, we can split this into two equations.

Variables

Simple cases

Absolute value equations with products are similar to absolute value equations with sums. The process is identical in the most basic examples. For example, lets solve for $x$ in the equation:

$$7 \times |x| = 21$$

First, we need to isolate the absolute value expression on one side of the equation. To do that, we can divide both sides by $7$:

$$\begin{align} |x| &= \frac{21}{7}\\ \\ |x| &= 3 \end{align}$$

Now that we've isolated $|x|$ on one side, we can look at the two possible cases when we remove the absolute value brackets:

$$\begin{align} x &= 3\\ x &= -3 \end{align}$$

So, $x$ equals $3$ or $-3$.

More complex cases

In many absolute value equations, $x$ will be part of a larger absolute value expression. In these cases, you will have to first isolate the absolute value expression on one side of the equation, then unpack it into your two cases, and then isolate the variable you're trying to solve for. For example, let's look at the equation:

$$5 + |2x| = 13$$

Good. Now that we've got $|2x| = 8$, we can look for our two cases (one where $2x$ is positive ultimately positive, and one where it is ultimately negative):

$$\begin{align} 2x &= 8\\ 2x &= -8 \end{align}$$

Now, all we have to do is solve for $x$.

Right, the two answers are $4$ and $-4$.