Understanding the Absolute Value Function: Definition, Properties, and Applications in Math

Absolute Value Function

The absolute value function is a mathematical function that yields the magnitude or distance of a real number from zero

The absolute value function is a mathematical function that yields the magnitude or distance of a real number from zero. It is denoted by the symbol “|” around the number or expression inside.

The absolute value function is defined as:

| x | =
x, if x ≥ 0
-x, if x < 0 In simpler terms, if the input x is positive or zero, then the absolute value of x is equal to x itself. However, if x is negative, then the absolute value of x is equal to its negative counterpart, hence -x. For example: - | 4 | = 4, because 4 is positive or equal to zero. The absolute value of 4 is therefore 4. - | -3 | = -(-3) = 3, because -3 is a negative number. In this case, we take -x, which is -(-3), resulting in 3. Properties of the absolute value function include: 1. Non-negativity: The absolute value of any number is always greater than or equal to zero. Example: | x | ≥ 0, for any real number x. 2. Symmetry: The absolute value function is symmetric about the y-axis. Example: | x | = | -x |, for any real number x. 3. Triangle inequality: For any two real numbers a and b, the absolute value of their sum is less than or equal to the sum of their absolute values. Example: | a + b | ≤ | a | + | b |, for any real numbers a and b. The absolute value function is often used in various applications, such as calculating distances, solving equations involving distances or inequalities, and representing magnitudes of quantities.

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