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, \left| -2 \right| 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: \left| -4 \right|

 

“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.

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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.

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

 

Example 2

Find the value: \left| -3.8 \right|

 

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.

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Example 3

Find the value: \left| 24 \right|

 

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: -\left| -10 \right|

 

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”

-\left| -10 \right| = -(10) \leftarrow 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.

-(10) = -10

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

 

Example 5

Simplify: -2\left| -5 - 4 \right|

 

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

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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.

-5 -4 = -5 + (-4) = -9

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

-2 \left| -9 \right|

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.

-2 \left| -9 \right| = -2(9)

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

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