## absolute value function

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

The absolute value function is a mathematical function that gives the distance of a number from zero on a number line. It is denoted by || or | |.

Mathematically, for any real number x, the absolute value function is defined as:

| x | =

x if x ≥ 0,

– x if x < 0.
In simpler terms, the absolute value of a number is the value of the number without considering its sign. It always returns a non-negative value.
For example:
- The absolute value of 5 is 5: | 5 | = 5.
- The absolute value of -7 is 7: | -7 | = 7.
- The absolute value of 0 is 0: | 0 | = 0.
Often in mathematics, the absolute value function is used to find the distance between two numbers, or to represent a magnitude or size without considering direction. It is also commonly used in solving equations and inequalities.
Properties of the absolute value function include:
1. | x | ≥ 0 for any real number x. The absolute value is always non-negative.
2. | x | = x if x is non-negative (x ≥ 0).
3. | x | = -x if x is negative (x < 0).
4. | x | = | -x | for any real number x. The absolute value of a number is equal to the absolute value of its negation.
In graphical representation, the absolute value function creates a V-shaped graph. It has a vertex at the origin (0,0) and both ends extend infinitely. The graph is symmetric with respect to the y-axis.
Overall, the absolute value function is a fundamental concept in algebra and calculus used to represent the distance or magnitude of a number.

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