## Absolute Value

## Introduction

When you first learned to count, the numbers went from 1 to 10. Later you learned up to 20, 100, 1000, and higher. The numbers that were larger than what you were learning seemed SO big… until you eventually learned how to count up to those bigger numbers. You measured the size of a number by the time it took to count up to the number.

Later, you learned to represent numbers on a number line and were able to estimate how large one number was compared to another. You could now measure the ** magnitude** of a number and compare its size to the size of other numbers using a diagram.

Then you learned about negative numbers and how on the number line they looked like a mirror image of their corresponding positive number. For example, 5 simply means 5 to the right of zero while -5 means 5 to the left of zero. The distance from the number to zero is the same, but the direction is different for 5 compared to -5. The distance from a number to zero is called the absolute value of the number, so in this case the ** absolute value** of 5 and -5 must be the same. Each has an absolute value of 5.

It is easy to see that the distance is 5 units away from zero for each of the above points on the number line.

## Lesson

The distance between two locations must always be a positive number. Therefore, the absolute value of a number is always positive. The absolute value of a positive number is the same as the original number. The absolute value of a negative number can be found by simply removing the negative signs. Example 1 shows some examples of absolute value.

**Example 1****:** Find the absolute value of each number:

|5| = 5 |-14| = 14 |325| = 325

|-5| = 5 |0| = 0 |-105| = 105

The absolute value of a number can be shown by putting bars around the number. When adding or subtracting numbers that include absolute values, treat the bars as you would treat parenthesis and apply them before doing anything else in the problem. Example 2 shows several different ways to simplify absolute value expressions.

**Example 2****:** Find the value of each expression:

Each of the problems in example 2 can be done using the order of operations as well as the properties of absolute value.

## Try It

Simplify each expression:

1) |-15|

2) |21 – 37|

3) |-6| - |-12|

4) 24 - |10 – 17|

5) |31 – 46| - |12 + 16| + |-10 + 5|

6) |9 – (-9)| + |-6| - |-14|

Scroll Down for Answers…

Answers:

1) |-15| = 15

2) |21 – 37| = |-16| = 16

3) |-6| - |-12| = 6 – 12 = -6

4) 24 - |10 – 17| = 24 - |-7| = 24 – 7 = 17

5) |31 – 46| - |12 + 16| + |-10 + 5| = |-15| - |28| + |-5| = 15 – 28 + 5 = -8

6) |9 – (-9)| + |-6| - |-14| = |9 + 9| + 6 – 14 = 18 + 6 – 14 = 10

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