Absolute Value: Distance from Zero
The absolute value of a number is its distance from zero on the number line. Negative numbers become positive; positive numbers and zero stay unchanged. |x| is always non-negative.
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|a - b| gives the distance between a and b on the number line. Triangle inequality: |a + b| โค |a| + |b| holds for all real numbers. |x| < 5 means -5 < x < 5โa bounded interval.
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Why: Absolute value answers: How far is this from zero? It strips away sign and gives magnitudeโessential for error analysis, distances, and inequalities.
How: If the number is non-negative, |x| = x. If negative, |x| = -x. For |ax+b|, evaluate the expression first, then take absolute value of the result.
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๐ Step-by-Step Breakdown
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
|xy| = |x||y| โ absolute value of product equals product of absolute values.
Triangle inequality is used in analysis, physics, and optimization.
๐ Key Takeaways
- โข |x| = distance of x from zero on the number line
- โข |x| = x if x โฅ 0; |x| = -x if x < 0
- โข |x| โฅ 0 always; |x| = 0 only when x = 0
- โข |-x| = |x| (symmetric about zero)
๐ก Did You Know?
๐ How It Works
The absolute value of a number is its distance from zero. Negative numbers become positive; positive numbers and zero stay the same. For expressions like |ax + b|, first evaluate ax + b, then take the absolute value.
๐ Worked Example: |-7|
Step 1: -7 is negative
Step 2: |-7| = 7 (distance from 0)
๐ฏ Expert Tips
- When solving |x| = a (a โฅ 0), solutions are x = a or x = -a.
- |x| < a means -a < x < a; |x| > a means x < -a or x > a.
- For |ax + b|, find when ax + b = 0 to split into cases.
- Use absolute value for tolerance: |error| < 0.01.
โ FAQ
What is absolute value?
The distance of a number from zero. Always non-negative. |โ5| = 5, |5| = 5.
Why is |x| always โฅ 0?
Distance cannot be negative. Zero is the minimum (at x = 0).
How to solve |x| = 7?
x = 7 or x = -7. Both are 7 units from zero.
What does |ax + b| mean?
Evaluate ax + b first, then take absolute value of that result.
When is |x| = -x?
When x โค 0. For negative x, -x is positive, so |x| = -x.
โ ๏ธ Disclaimer: This calculator provides educational estimates. For rigorous mathematical work, verify with authoritative sources.
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