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Concept

# Absolute Value

The absolute value of a number, $a,$ expressed as $|a|,$ is the non-negative value of $a.$ Therefore, the absolute value of a negative number changes its sign whereas the absolute value of a non-negative number is equal to the number itself. $|\text{-} 3|=3 \quad \quad |3|=3$ A compact way to write the absolute value of a real number is by using a curly bracket. $|a| = \begin{cases} a & \quad \text{if } a\geq0\\ \text{-} a & \quad \text{if } a<0 \end{cases}$ The absolute value of a number $a$ can be thought of as the distance between $a$ and $0$ on the number line. Since the absolute value of $\text{-}3$ and $3$ are equal, these numbers are also the same distance from $0.$

Concept

## Properties and Identities of Absolute Value

There are several properties and identities that can be useful when simplifying expressions or solving equations including absolute values. For real values of $a$ and $b,$ these relationships are true.

• $|a|\geq0$
• $|\text{-} a|=|a|$
• $|a\cdot b|=|a|\cdot |b|$
• $\left|\dfrac{a}{b}\right|=\dfrac{|a|}{|b|}$
• $|a+b|\leq |a|+|b|$
• $|a|=\sqrt{a^2}$