The absolute value (or modulus) |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = xfor a positive x, |x| = −x for a negative x, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

The term absolute value was introduced in 1806 and the notion |x| was introduced in year 1857 by Karl Weierstrass. The absolute value is also known as the numerical value.

Following are the fundamental properties of Absolute values:

- |a|≥ 0
- |a| = 0 <==> a=0
- |ab|=|a| |b|
- |a+b| ≤ |a|+|b|

- Relationship to the sign function
- Derivative
- Antiderivative

Example for finding absolute value:

Solve | x + 2 | = 7

For finding the absolute value, the equation is split into its two possible two cases, one case for each sign:

(x + 2) = 7 or –(x + 2) = 7

x + 2 = 7 or –x – 2 = 7

x = 5 or –9 = x

Then the solution is x = –9, 5.

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