Absolute value

Absolute value is easy to spot because it uses vertical bars: |these vertical lines|. The rules are straightforward. The key idea is this:

After you finish simplifying what’s inside the absolute value bars, the result becomes non-negative.

Absolute value represents distance from zero on a number line, and distance is never negative.

Let’s look at the absolute value of a few numbers:

• $|2|$, the absolute value of $2$, simplifies to $2$.
• $|-2|$, the absolute value of $-2$, simplifies to $2$.

Sometimes, though, you’ll need to do more algebra.

How about $3-|2-6|$? Use PEMDAS, and treat absolute value bars like parentheses.

1. Simplify inside the absolute value bars: $|2-6|=|-4|$
2. Apply absolute value: $|-4|=4$
3. Finish the expression: $3-4=-1$

Notice that the overall answer can be negative (here, $-1$). What must be non-negative is the value that comes out of the absolute value bars (here, $4$).

Layered absolute value questions

This is an example of an absolute value expression inside another absolute value expression. Like nested parentheses, you work from the inside out: evaluate the inner absolute value bars first.

Try this one on your own using the same rule:

$$|3-|5-3(12)||$$

As you can see, you treat absolute value bars like parentheses for order of operations (inside to outside). The extra step is that after each absolute value is evaluated, its result is non-negative.

Absolute value solutions

Some problems ask you to find solutions to an equation. That means you’re looking for value(s) of $x$ that make the equation true.

When $x$ is inside absolute value bars, you usually get two solutions, because the expression inside the bars could be positive or negative and still produce the same absolute value.

Find the solutions to the following equation: $4+|x-12|=12$

First, isolate the absolute value expression by subtracting $4$ from both sides:

$|x-12|=8$

Now use this key principle: if $|A|=8$, then $A$ can be $8$ or $-8$. So you write two equations:

$$(x-12)=8\text{ and }-(x-12)=8$$

Try solving both equations. You should get two solutions.

No-solution cases with absolute value

There’s one important situation to watch for when solving absolute value equations: sometimes there is no solution.

Remember the fundamental rule:

Absolute value can never be negative.

So if you ever end up with an equation like:

$$\mid x-5\mid =-3$$

you should immediately recognize that something is impossible:

• The left side, an absolute value expression, is always non-negative.
• The right side is negative.

Since a non-negative number can never equal a negative number, this equation has no solution.

Absolute value graphs

One important trait to recognize is the shape of an absolute value graph. Absolute value graphs look like a “V,” either right-side up or upside down: