1. The Absolute Value
Lesson 3.06

2. After completing this lesson, you will be able to say:
I can determine the absolute value of numbers by finding the distance from 0.
I can interpret the absolute value of a number in a real-world situation.
I can compare absolute value statements about order.

3. Key Terms
Absolute value:
A number’s distance from zero on a number line; represented as |a| to mean the absolute value of a, where “a” represents any number.
Magnitude:
Size of a number, regardless of whether it is positive or negative.

4. Absolute Value
Even though −4 and 4 are different numbers, they have the same absolute value, which is 4.
The −4 and 4 are plotted on this number line. The distance from −4 to 0 and 0 to 4 are both 4 spaces. Distance will always be positive.

5. Absolute Value Absolute value facts
Because absolute value is a measure of distance, the absolute value of any number (except 0) will always be positive.
Because 0 is zero units away from itself, the absolute value of 0 is 0.
The symbol for absolute value is two vertical bars, one on each side of a number: |a| means the absolute value of a.
Finding the absolute value of a number gives you the magnitude for the number.

6. Absolute Value - Caution
A common error is to change the sign of the number inside the absolute value symbols to its opposite.
|6| ≠ −6
Absolute value is a number's distance from 0 and will always be a positive answer.
|6| = 6

7. Absolute Value Examples

8. Absolute Value Example
There is a fish located at a position of −9 feet relative to sea level. Write a statement that shows the distance the fish is from sea level.
The fish is 9 feet below sea level. Since sea level is considered 0, in order to find the distance, you need to calculate the absolute value.
|−9| = 9
This means that the distance the fish is located from the water surface is 9 feet.

9. Try It
A rabbit is located at a position of −3.2 feet relative to the ground. Write a statement that shows the rabbit's distance from ground level.

10. This means the rabbit is at a distance of 3.2 feet from ground level.
Check your work
A rabbit is located at a position of −3.2 feet relative to the ground. Write a statement that shows the rabbit's distance from ground level.
|−3.2| = 3.2
This means the rabbit is at a distance of 3.2 feet from ground level.

11. Comparing Magnitudes
Use > or < to compare |8| ___ |−10| and determine what this means for the two numbers in relation to 0.
The number 8 is 8 units from zero. |8| = 8.
The number −10 is 10 units from zero. |−10| = 10.
Because 8 < 10, this means that |8| < |−10|.
When comparing the two numbers 8 and −10, this reveals that the number 8 is closer to 0 on a number line.

12. Comparing Magnitudes - Example
A brother and a sister are looking at two points on a number line. The brother said that the absolute values of the numbers represented by the two points are the same. The sister said that the absolute values of the numbers represented by the two points are different. Who is correct?
The absolute value of |−0.7| = 0.7 and |0.7| = 0.7, so both of the absolute values are the same. The points are the same distance away from 0 so the brother is correct.

13. Try it Choose the correct answer
Bank Account 1 is −4.35 and Bank Account 2 is−3.39 at the end of the day. Bank Account 1 has more debt than Bank Account 2.
On Monday, John spent OnTuesday, he spent On Tuesday, John spent more money, because |−$48.92| > |−45.38|.
A turtle is climbing on land at 4 feet above sea level while a fish is swimming 10 feet below sea level. The fish is further from sea level because |−10| > |4| . This shows that 4 is closer to 0 on a number line.

14. Try it Choose the correct answer
Bank Account 1 is −4.35 and Bank Account 2 is−3.39 at the end of the day. Bank Account 1 has (more or less) debt than Bank Account 2.
On Monday, John spent OnTuesday,hespent On Tuesday, John spent more money, because |−$48.92| (< or >) |−45.38|.
A turtle is climbing on land at 4 feet above sea level while a fish is swimming 10 feet below sea level. The fish is (further from or closer too) sea level because |−10| (< or >) |4| . This shows that (-10 or 4) is closer to 0 on a number line.

15. Now that you completed this lesson, you should be able to say:
I can determine the absolute value of numbers by finding the distance from 0.
I can interpret the absolute value of a number in a real-world situation.
I can compare absolute value statements about order.