The absolute value of the real number `a`, denoted by `|a|`, is the distance between `a` and `0` on the number line. For example, `|2|=2` and `|-2|=2`. In general, if `a >= 0`, then `|a|=a`; however, if `a < 0`, then `|a|=-a` because `-a` is positive when `a < 0`. This leads us to the following definition.
| Definition of Absolute Value |
| The absolute value of the real number `a` is defined by |
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`,
More complicated expressions can also be simplified by using these theorems. For example, given `-1 < x < 1`,
| Write `|(2x)/(|x|+|x-2|)|`, given `0 < x < 2`, without absolute value symbols. |