Math101 · Independent Ontario learning library
Math101learn.math101.caAbsolute 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.
Worked example
Representations and interpretation
On a number line, $|x|$ ignores direction and records only distance from zero. The graph $y=|x|$ forms a V with vertex $(0,0)$: the left branch reflects negative inputs upward, while the right branch follows $y=x$.
Reasoning about variations
Absolute value does not simply “remove a minus sign.” For $|2-7|$, the inside becomes negative and its opposite is taken; for $|-3|^2$, absolute value gives $3$ before squaring. In $-|{-3}|$, the outside negative remains, producing $-3$.
Common mistakes
How to check your work
- Confirm that a standalone absolute value result is nonnegative.
- Plot the input and count its unit distance from zero.
- Use $|x|^2=x^2$ as a numerical check when a square is convenient.
Practice
- Evaluate $|-17|$.
- Find $|6-14|$.
- Evaluate $-|{-9}|+2$.
Answers and brief solutions
Show answers
- $17$ $-17$ lies $17$ units from zero.
- $8$ The inside is $-8$, whose absolute value is $8$.
- $-7$ $|{-9}|=9$, so the outside negative gives $-9+2=-7$.
Synthesis and transfer
A machine part specified within $0.04$ mm of a target length can be modelled by an absolute-value inequality, with the endpoints representing the largest permitted deviations.
If $L$ is the measured length and $T$ the target, the requirement is $|L-T|\le0.04$. Unpacking distance gives the interval $T-0.04\le L\le T+0.04$, including both endpoints when the tolerance is allowed exactly. The signed error $L-T$ still matters diagnostically because it distinguishes an oversized part from an undersized one, even though acceptance depends only on its magnitude. A calibration shift changes the centre $T$, while improved precision narrows the radius of the interval. This connection between absolute value and symmetric intervals becomes especially useful when specifications, measurement uncertainty, or numerical approximation are described as “within” a fixed amount.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Evaluate $|-17|$.
- $-17$ lies $17$ units from zero.
End of lesson
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