# Properties of Absolute Value

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. The number with the smallest absolute value is zero, thus the absolute value of a number may be thought of as its distance from zero.

\begin{align} \left | x\right | = y &\Rightarrow x = -y \; \textrm{or}\; x = y\\ \\ \left | x\right | < y&\Rightarrow-y < x < y\\ \\ \left | x\right | > y&\Rightarrow x < -y \; \textrm{or}\; x > y\\ \end{align}

1. Subadditivity - The absolute value of the sum of real numbers always returns something less than or equal to the sum of the absolute values of the real numbers:

$$\left | x+y\right | \leq\left | x\right | + \left | y\right |$$

2. The absolute value of the difference of two real numbers shall not be less than the difference between the absolute values of the minuend and subtrahend:

$$\left | x-y\right | \geq\left | x\right | - \left | y\right |$$

3. Multiplicativity - The absolute value of the product of the multiplication is equal to the absolute values of the multiplicand and the multiplier:

$$\left | x \times y\right | =\left | x\right | \times \left | y\right |$$

4. Preservation of division - the absolute value of the division of two real numbers is equal to division of the absolute values of the dividend and the divisor:

$$\left | \frac{x}{y}\right | =\frac{\left | x\right |}{\left | y\right |}$$