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**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.

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