Absolute Value Calculator
Absolute value |x| is the distance of a number from zero on the number line, always non-negative. |3| = 3, |−3| = 3, |0| = 0. In complex numbers, absolute value is the distance from the origin: |a+bi| = √(a²+b²).
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Tip: To solve |f(x)| = k: split into f(x) = k and f(x) = −k. For |f(x)| < k: solve −k < f(x) < k. For |f(x)| > k: solve f(x) < −k or f(x) > k.
- 1|x| = x if x ≥ 0; |x| = −x if x < 0
- 2|x − y| = distance between x and y on the number line
- 3Triangle inequality: |a + b| ≤ |a| + |b|
- 4Properties: |ab| = |a||b|, |a/b| = |a|/|b|, |a|² = a²
|−7.5|=7.5Distance from zero, always positive
|3 − 8|=5Distance between 3 and 8 on number line
| Equation | Solution | Why |
|---|---|---|
| |x| = 5 | x = 5 or x = −5 | Two numbers at distance 5 from 0 |
| |x − 3| = 4 | x = 7 or x = −1 | Two numbers at distance 4 from 3 |
| |x| < 3 | −3 < x < 3 | All numbers within 3 of zero |
| |x| > 3 | x < −3 or x > 3 | All numbers more than 3 from zero |
| |x| = −1 | No solution | Absolute value is never negative |
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Fun Fact
Absolute value is central to error analysis, machine learning loss functions (MAE — Mean Absolute Error), and the L1 norm. In optimization, the L1 norm creates "sparse" solutions — useful in data compression and feature selection.
References
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