## Absolute Value Computation

## Introduction

The concept of absolute value of x, written as |x|, can be described as the distance from a number to zero. Since distance is always positive, the result of an absolute value must always be positive (except for the case of

*|0| = 0.)*

Although we normally think of absolute values as the distance from zero to a single number, they can also be expressed as part of a larger expression. Absolute values that are found within expressions generally follow the same rules as parenthesis. However, there are some differences between problems with parenthesis and those with absolute values.

## Lesson

To simplify expressions containing an absolute value, follow these three steps:

1) Simplify all numbers inside the absolute value (bars) using the order of operations

2) Evaluate the absolute value once it has been simplified down to a single number inside the bars

3) Simplify the remaining problem using the order of operations

Following the three steps above, see how the following problems are done.

Simplify -2 + |4 – 8| - 3

Simplify 32 - |8 – (2 + 4)| + 11

Simplify (6 – 12) + |6 – 12|

When simplifying problems, one major difference between parenthesis and absolute value occurs when you have a double negative.

Problems where most numbers are positive are fairly straightforward. As you encounter more and more negatives, problems involving absolute value should be approached more carefully. As always, showing every step of work gives you the best chance of getting the correct answer.

Simplify 15 - |-3| - |-12 + 6| - (-4)

Simplify (13 – 21) - |24 – 42| - 8

The key to simplifying expression that contain absolute values is to use the order of operations to simplify everything *inside* the absolute value bars first, then evaluate the absolute value(s) themselves. Finally, evaluate everything that is left using the order of operations.

## Try It

Simplify each expression:

1) 34 - |-34|

2) 21 – (-21)

3) 13 - |2 + (-9)| + |(-18) – (-11)|

4) 33 - |15 – 11 - |12 – 8||

5) |24 + ((-17) + 11| - |11 – 14|

Scroll Down for Answers…

Answers:

1) 34 - |-34| = 34 – 34 = 0.

2) 21 – (-21) = 21 + 21 = 42.

3) 13 - |2 + (-9)| + |(-18) – (-11)| = 13 - |-7| + |-18 + 11| = 13 – 7 + |-7| = 6 + 7 = 13.

4) 33 - |15 – 11 - |12 – 8|| = 33 - |15 – 11 - |4|| = 33 - |0| = 33.

5) |24 + ((-17) + 11)| - |11 – 14| = |24 + (-6)| -|-3| = |18| - 3 = 18 – 3 = 15.

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