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Absolute Values and their Properties

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Absolute Value Equations

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Absolute Values and their Properties

Algebra-1

Learning Made Easy

You will be instructed on the meaning and definition of absolute value and its properties, which are applied to problems of varying difficulty.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Define the absolute value of a number.
  • Evaluate the absolute value of a number.
  • Use the properties of absolute value to evaluate expressions involving absolute value.
  • Use the properties of absolute values to compare quantities.

Everything You'll Have Covered

The absolute value of a real number is a measure of the number's magnitude. Given a real number a, the absolute value of a is denoted by and equal to the number's distance from zero on the real number line. For example, the absolute value of -3 is equal to 3, which is demonstrated on the number line below.

The absolute value is also used to determine the distance, or amount of space, between two numbers. If a and b are any two real numbers, then returns the distance between a and b on the real number line. As a measure of distance, the absolute value is fundamental to our conception of the real numbers as a geometric space.

The absolute value, by virtue of its geometric definition, satisfies a number of important properties. The most important of these guarantees that it is sensible to describe the absolute value as a magnitude or measure of distance. In what follows, let b and a be any two real numbers.

The geometric definition of the absolute value poses computational and algebraic challenges. The following algebraic formula is used to circumvent such challenges.

According to this formula, the absolute value of a number can be evaluated by identifying the number's sign and then applying the appropriate case. For example, a negative number such as -3 is evaluated by using the second case.

With the algebraic formula in hand, one can straightforwardly verify the following algebraic facts by separately considering the negative and positive cases (again, let b and a be any two real numbers):

These laws, combined with the algebraic formula above, can help to calculate or simplify the absolute value of algebraic expressions.

These computational feats are useful, but more importantly, properties 1-4 are evidence of a deep relation between the geometric and algebraic features of the real numbers.

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts Learners should understand the concepts of positive and negative numbers and know how to order negative and positive numbers on a number line.
Course Algebra-1
Type of Tutorial Concept Development
Key Vocabulary absolute value, distance, magnitude