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An absolute value equation is an equation that involves the absolute value of a variable expression. | 2x-8 |=5 Equations of the form |x|=a, where a is a real number greater than zero, can be solved by looking for the numbers x whose distance from 0 in the number line equals a. For example, the solutions of |x|=4 are all values of x that are 4 units away from 0.
Since there are two points on the number line that fulfill this requirement, there are two solutions to the equation |x|=4, namely x=4 and x=- 4. However, solving an absolute value equation, in general, might require a more elaborate and structured approach.
Simple absolute value equations of the form |x|=a, can have no, one, or two solutions, depending on the value of a. However, more complex absolute value equations may have more than two solutions.
| Equation | Number of Solutions | Solution(s) |
|---|---|---|
| |x|=- 4 | Zero | No solution |
| |x|=0 | One | 0 |
| |x|=4 | Two | - 4, 4 |
| |x^2-4 |=2 | Four | - sqrt(2), sqrt(2), - sqrt(6), sqrt(6) |