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Absolute Value Equation

Concept

Absolute Value Equation

An absolute value equation is an equation that contains the absolute value of a variable expression. 2x8=5 \left| 2x-8 \right|=5 Equations of the type x=a,|x|=a, where a0,a\geq0, can be solved by thinking of the absolute value of a number aa as the distance between aa and 00 on the number line. For example, x=4|x|=4 are all values of xx which are 44 units away from 0.0.

As there are two points on the number line that fulfill this requirement, there are two solutions to the equation x=4,|x|=4, namely x=4x=4 and x=-4.x=\text{-} 4.

Concept

Number of Solutions

The absolute value equation x=4|x|=4 has two solutions. Equations of the same type, x=a,|x|=a, will have 0,0, 1,1, or 22 solutions, depending on a.a. More complex absolute value equations may have more than 22 solutions.

Equation Number of solutions
x=-4|x|=\text{-} 4 00
x=0|x|=0 11
x=4|x|=4 22
x24=2\left|x^2-4 \right|=2 44
Concept

Solving Absolute Value Equations

When solving absolute value equations algebraically, it is necessary to take into consideration that the absolute value of both a negative number and a positive yields a positive. One possible strategy is to, when removing the absolute value, split the equation into two cases: one case where the expression inside the absolute value is negative and one where it is positive.

Absolute value equations can also be solved graphically or numerically.