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Expressions inside of an absolute value become nonnegative. For instance a negative number becomes positive.

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

Example 1

$$|-4| + |2| - 3$$

Okay, so the first thing we need to do is handle our absolute values. There are no actual math operations for us to perform inside the absolute values in this example, so we can just re-write our expression by making every absolute value term nonnegative.

Notice how the absolute value sign turned our negative $4$ into a positive $4$. Also, the $2$ inside the other set of absolute value bars was already positive, so we don't need to do anything to it. All that's left is to evaluate the expression: $$4 + 2 - 3 = 3$$

Example 2

$$\frac{|2 - 5| + 7}{|-10|}$$

Here, we actually have an operation to perform inside the absolute value in the numerator (the top part of the fraction). Think of the absolute value signs as a type of parentheses. This means that you need to handle them FIRST in the order of operations (PEMDAS), before you do anything else.

$$\frac{|2 - 5| + 7}{|-10|} =\frac{|-3| + 7}{|-10|} =\frac{3 + 7}{10} =\frac{10}{10} =1$$

Example 3: Variables

$$|x + 4| = 9$$

Now we've introduced variables to the mix. We're going to be solving for $x$ like we usually do, but there's one extra step we need to take because of the absolute value.

Recall that the absolute value makes the quantity inside of it nonnegative. What we don't know, though, is whether or not the quantity would be nonnegative without the absolute value signs.

In our example, $x+4$ might equal $9$. But $x+4$ might also equal $-9$. The absolute value sign makes it impossible for us to tell, since it basically "erases" negatives.

Not to worry, though. All we need to do with these types of problems is account for both possible outcomes.

Take a look:

$$\begin{align} x + 4 &= 9\\ x + 4 &= -9 \end{align}$$

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

$$\begin{align} x+4&=9\\ x&=5\\ \\ x+4&=-9\\ x&=-13 \end{align}$$

Try plugging in both values into the original equation ($|x+4|=9$). Notice how both values of $x$ give us the same answer.

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

There are other approaches to solving absolute value equations that you can learn about here!

(Video) Absolute Value Equations

by yaymath

This video by yaymath goes over absolute value equations.

$\large{\text{Summary}}$

Watch the first couple problems to see the basics of how addition and subtraction works when absolute value is at play. Remember that the basic steps are:

1. Isolate the absolute value part of the equation on one side of the equals sign. Check to make sure that the rearranged equation is true based on what you know about absolute value.

2. Figure out the two possible values for the absolute value. The options should be identical except that one is negative. However, if the absolute value is $0$, then there will just be one option (equal to $0$).

3. Solve both equations. Those are your two answers! You can check your work by plugging them back into the original equation.

Now, try a problem that might throw a wrench in it. Try to solve the following equation.

$$|x + 3| + 7 = 2$$

What did you notice about that equation? Well, for one thing, when you isolate the absolute value section on one side, you get:

$$|x + 3| + 7 = 2\\\ \\ |x + 3| = -5$$

What's wrong with this picture? Absolute value equations must always be positive, since they're just measuring distance from $0$. If there's a negative number on the other side of the equals sign, then something must be wrong! In this case, it means that there are no solutions to this equation.

There's a similar pattern to absolute value equations with products!