# GED Mathematical Reasoning: Absolute Value

- Absolute value is a fancy phrase that simply means: distance. More specifically, absolute value means the distance from a number to zero on the number line. We denote absolute value with two vertical bars.
- For example, is read “the absolute value of -2” and the value is equal to 2.
- When evaluating the absolute value of a number, the answer will always be the POSITIVE version of the number inside the absolute value bars. This makes sense because absolute value means distance and distance is never negative.
- Also – It’s important to know that absolute value bars are considered a grouping symbol (like parenthesis or brackets). So if you are simplifying a problem with more than one operation, evaluate absolute value in the same step and in the same manner you would evaluate parentheses or brackets.

**Example 1**

Find the value:

“The absolute value of -4” can be loosely translated into the question: What is the distance from -4 to zero on the number line? Visually, you can see that the number -4 is 4 units from zero on the number line. So we say that the “absolute value of -4” is equal to 4.

It’s interesting to note that the absolute value of positive 4 is also equal to 4, because it is also 4 units from zero on the number line – it just lies in the opposite direction.

By the way, the number inside the absolute value bars can be a fraction or a decimal, too.

**Example 2**

Find the value:

What is the distance from -3.8 to zero on the number line? Visually, you can see that the number -3.8 is 3.8 units from zero on the number line. So we say that the “absolute value of -3.8” is equal to 3.8.

**Example 3**

Find the value:

What is the distance from 24 to zero on the number line? The answer is 24. So “the absolute value of 24” is 24.

**Example 4**

Simplify:

In example four, we see “the absolute value of -10” with a negative preceding it. We will first evaluate “the absolute value of -10” in accordance to the order of operations. The absolute value of -10 is positive 10.

Now we will deal with the negative sign in front. Recall that a negative sign means “the opposite.” So we can translate the original problem as: “the opposite of the absolute value of -10.” Since the absolute value of -10 is 10, this becomes: “the opposite of 10”

the opposite of 10

Notice that we will drop the absolute value bars after we have found the absolute value of -10, since they are no longer needed. It’s a good idea to replace the bars with parenthesis. The opposite of 10 is -10, thus making our final answer -10.

So as you can see, even though our example included absolute value bars, our final answer was negative.

**Example 5**

Simplify:

Notice that we have subtraction within the absolute value bars. And we also have multiplication between the -2 and the absolute value bars.

Following the order of operations, let’s first simplify what’s inside the absolute value bars.

-5 minus 4 can be re-written as -5 plus -4. -5 plus -4 is equal to -9.

So now our problem looks like this: -2 times the absolute value of -9

Next, let’s evaluate the absolute value of -9, which is equal to positive 9. Again, we will drop the absolute value bars after we have found the absolute value of -9 and replace the bars with parenthesis.

For our final answer, we will multiply -2 by 9 which is equal to -18. So the final result is: -18.