Understanding the Absolute Value Function | Definition, Properties, and Applications

Absolute value function

The absolute value function is a mathematical function that returns the distance of a number from zero on the number line

The absolute value function is a mathematical function that returns the distance of a number from zero on the number line. It is denoted by vertical bars or two straight lines surrounding the number. For example, the absolute value of -5 is written as |-5| = 5, which means that -5 is 5 units away from zero.

The absolute value function can be defined as follows:

For any real number x, the absolute value of x, denoted as |x|, is given by:
– If x is greater than or equal to zero, then |x| = x.
– If x is less than zero, then |x| = -x.

In simpler terms, the absolute value function removes the negative sign from a negative number, making it positive.

The absolute value function has several important properties:
1. Non-negativity: The absolute value of any number is always non-negative. |x| ≥ 0 for any real number x.
2. Symmetry: The absolute value function is symmetric about the y-axis. |x| = |-x|.
3. Triangle inequality: For any two real numbers x and y, |x + y| ≤ |x| + |y|. This means that the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

The absolute value function has many applications in mathematics and real-life situations, such as finding distances, solving absolute value equations, and representing magnitudes.

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