- DU Libraries
- Library Guides
- Math Skills Overview Guide
- Absolute Value

Menu

- Home
- Basic Operations Toggle Dropdown
- Order of Operations
- Math Properties Toggle Dropdown
- Factors & Multiples Toggle Dropdown
- Fractions Toggle Dropdown
- Decimals
- Percents Toggle Dropdown
- Ratios & Proportions Toggle Dropdown
- Exponents Toggle Dropdown
- Scientific Notation
- Averages
- Equation Basics
- Polynomials Toggle Dropdown
- Linear Equations Toggle Dropdown
- Absolute Value
- Rational Expressions
- Roots & Radicals
- Quadratic Equation Toggle Dropdown
- Functions Toggle Dropdown
- Algebraic Ratios & Proportions
- Equations & Inequalities
- Logarithms Toggle Dropdown
- Imaginary Numbers
- Sequences & Series
- Introduction to Matrices
- Geometry Toggle Dropdown
- Trigonometry Toggle Dropdown
- Math Documents
- Get Help From Jessica

**The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.**

For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line. Furthermore, the absolute value of the difference of two real numbers is the distance between them.

**The absolute value has the following four fundamental properties:**

Non-negativity $$|a| ≥ 0$$ Positive-definiteness $$|a| = 0 ⇔ a = 0$$ Multiplicativeness $$|ab| = |a||b|$$ Subadditivity $$|a + b| ≤ |a| + |b|$$

**Other important properties of the absolute value include:**

Idempotence (the absolute value of the absolute value is the absolute value) $$||a|| = |a|$$ Symmetry $$|-a| = |a|$$ Identity of indiscernibles (equivalent to positive-definiteness) $$|a - b| = 0 ⇔ a = b$$ Triangle inequality (equivalent to subadditivity) $$|a - b| ≤ |a - c| + |c - b|$$ Preservation of division (equivalent to multiplicativeness) $$|a / b| = |a| / |b| \space\space if \space\space b ≠ 0$$ (equivalent to subadditivity) $$|a - b| ≥ ||a| - |b||$$

**Two other useful properties concerning inequalities are:** $$|a| ≤ b ⇔ -b ≤ a ≤ b$$ $$|a| ≥ b ⇔ a ≤ -b \space or \space b ≤ a$$

**These relations may be used to solve inequalities involving absolute values. For example:** $$|x - 3| ≤ 9 ⇔ -9 < x - 3 < 9$$ $$⇔ -6 < x < 12$$

The absolute value of 5 is 5, it is the distance from 0, 5 units.

The absolute value of -5 is 5 it is the distance from 0, 5 units.

$$|x| = 2 $$

$$|x| > 2$$

$$For \space |x| < 2, -2 < x < 2$$ $$For \space |x| = 4, -4 = x = 4$$

|7| = 7 means the absolute value of 7 is 7.

|-7| = 7 means the absolute value of -7 is 7.

|-2 - x| means the absolute value of -2 minus x.

-|x| means the negative of the absolute value of x.

Watch a Khan Academy Video »

*Length: 5:39 *

Watch a Khan Academy Video »

*Length: 2:22 *

Watch a Khan Academy Video »

*Length: 10:41 *

- Last Updated: Dec 11, 2020 9:55 AM
- URL: https://davenport.libguides.com/math-skills-overview
- Print Page