![]() ![]() For example: $$|x - 3| ≤ 9 ⇔ -9 < x - 3 < 9$$ $$⇔ -6 < x < 12$$įrom: en.wikipedia. To solve absolute value equations, find x values that make the expression inside the absolute value positive or negative the constant. These relations may be used to solve inequalities involving absolute values. Two other useful properties concerning inequalities are: $$|a| ≤ b ⇔ -b ≤ a ≤ b$$ $$|a| ≥ b ⇔ a ≤ -b \space or \space b ≤ a$$ 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) $$|a - b| ≥ ||a| - |b||$$ Other important properties of the absolute value include: The absolute value has the following four fundamental properties: ![]() Furthermore, the absolute value of the difference of two real numbers is the distance between them. ![]() The absolute value of a number may be thought of as its distance from zero along real number line. The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.įor example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. 3.1 Graphing 3.2 Lines 3.3 Circles 3.4 The Definition of a Function 3.5 Graphing Functions 3.6 Combining Functions 3.7 Inverse Functions 4.
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