Mastering the Absolute Value Function: Essential Properties and Applications in Calculus, Algebra, and Trigonometry.

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The absolute value of a number is the distance of that number from zero on the number line. It is represented by two vertical bars enclosing the number between them, like |x|. The absolute value of a positive number and its negative counterpart is the same, for example, the absolute value of 5 and -5 is both 5. The absolute value of zero is also 0.

The absolute value function is used in many branches of mathematics, particularly in calculus, algebra, and trigonometry. It is a common tool to solve equations and inequalities, simplify expressions, and find the distance between two points on a plane.

Here are some properties of absolute value that are commonly used in solving equations and performing mathematical operations:

1.|-x| = |x|
2.|x*y| = |x|*|y|
3.|x+y| ≤ |x|+|y|
4.|x-y| ≥ |x|-|y|
5.|x| < a if and only if -a < x < a, where a is a positive number. In summary, the absolute value is a mathematical function used to determine the distance of a number from zero on a number line. It has many important applications in various fields of mathematics and is widely used in solving equations and inequalities.

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