Math101 · Independent Ontario learning library
Math101Absolute Value
The absolute value $|x|$ is the distance from the real number $x$ to $0$ on a number line. Distance is never negative, so $|5|=5$, $|-5|=5$, and $|0|=0$.
Absolute value expresses error, deviation, tolerance, and geometric distance without choosing a direction. It is the foundation for distance formulas and later absolute-value equations and inequalities.
Intuition and core definition
The absolute value $|x|$ is the distance from the real number $x$ to $0$ on a number line. Distance is never negative, so $|5|=5$, $|-5|=5$, and $|0|=0$. Algebraically, $|x|=x$ when $x\ge0$ and $|x|=-x$ when $x<0$; the second rule makes a negative input nonnegative.
Notation, language, and conditions
Vertical bars denote absolute value, not parentheses. The piecewise definition is $|x|=\begin{cases}x,&x\ge0\\-x,&x<0.\end{cases}$ More generally, $|a-b|$ is the distance between $a$ and $b$. Statements such as $|x|=k$ require $k\ge0$; no real number can have negative distance from zero.
Why this idea matters
Distance language makes absolute value useful for tolerances, deviations, and comparisons because it measures separation without assigning a direction.
A dependable method
- Simplify the expression inside the bars first.
- Interpret the resulting number as its distance from zero.
- If the inside value is nonnegative, keep it; if it is negative, take its opposite.
- For $|a-b|$, subtract in either order and take the nonnegative magnitude.
- Check that the final absolute value is at least zero.
