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Absolute value

Absolute value (sometimes called modulus or magnitude) is how far a number, term, polynomial, or expression is from zero, regardless of whether it is positive or negative. For example: $$4$$ and $$-4$$ are the same distance from $$0$$, so they both have an absolute value of $$4$$.


Absolute value is represented by two bars, one of each side of the number, term, polynomial, or expression. For example, the absolute value of $$-4$$ would be written as $$|-4|$$

Properties of absolute values

• Non-negativity: $$\left|x\right|\ge 0$$
  Absolute value is always non-negative, meaning it always yields zero or a positive.


• $$\left|x\right|=\sqrt{x^2}$$: Squaring a number makes it positive (or zero if the number is zero), and by taking the square root of a squared number we get a positive solution (or zero if the number is zero). This only works when $$x$$ is a real number.


• Multiplicativity: $$|x·y|=\left|x\right|·\left|y\right|$$
  The absolute value of a product of two numbers equals the product of the absolute value of each number.


• Subadditivity: $$|x+y|\le \left|x\right|+\left|y\right|$$
  The absolute value of the sum of two real numbers is less than or equal to the sum of the absolute values of the two numbers.


• $$\left|x\right|=y\to x=\pm y$$ or $$\left|x\right|=\pm x$$: If the absolute value of $$x$$ equals $$y$$ then $$x$$ equals plus or minus $$y$$. This rule is used for solving most absolute value questions.

Absolute value equations

Absolute value equations are equations in which the variable is within an absolute value operator.
For example: $$|x-4|=10$$
Because the value of $$x-4$$ can be $$10$$ or $$-10$$, both of which have an absolute value of $$10$$, we need to consider both cases: $$x-4=10$$ and $$x-4=-10$$. This can also be written as $$x-4=\pm 10$$.

So, $$|x-4|=10$$ has two solutions:
$$x-4=10$$ → $$x=14$$
$$x-4=-10$$ → $$x=-6$$

Because absolute values are always non-negative, it is possible to have equations with no solutions.
For example: $$|x-5|=-9$$

Absolute Value Equations and Inequalities are solved and explained step by step by the Tiger Algebra Absolute Value module.