The definition of distance between any two points on a real number line makes use of absolute value.

Distance Between Points on a Real Number Line

For any real numbers `a` and `b`, the distance between the graph of `a` and the graph of `b` is denoted by `d(a,b)`, where `d(a,b)=|a-b|`

The following theorems can be derived by using the definition of absolute value.

Absolute Value Theorems
For all real numbers `a` and `b`,
Nonnegative	`|a| >= 0`
Product	`|ab| = |a| |b|`
Quotient	`|a/b| = |a| / |b|`, ` \ \ \ \ \ b!=0`
Triangle inequality	`|a+b| <= |a|+|b|`
Difference	`|a-b| = |b-a|`

The definition of absolute value and the absolute value theorems can be used to write some expressions without absolute value symbols. For instance, because `1-pi < 0`,

`|1-pi|=-(1-pi)=pi-1`

More complicated expressions can also be simplified by using these theorems. For example, given `-1 < x < 1`,

`|x+3|-|x-2|=(x+3)-[-(x-2)]`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (x+3)+(x-2)`
` = 2x+1`