What Does Absolute Value Mean in Math

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An absolute value equation is any equation that contains an absolute value expression. The absolute value of a variable $x$ is denoted as $|x|$ , and it is always positive, except for zero, which is neither positive nor negative. An absolute value equation is solved using the same rules as any other algebraic equation; however, this type of equation has two potential results, derived from a positive equation and a negative equation.

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Understand what an absolute value represents. The absolute value of a number represents how far away from 0 the number is on a number line.^[2] Absolute value is denoted by bars surrounding the term or terms ( $|x|$ ). The absolute value of a number is always positive.^[3]

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Isolate the absolute value term(s) in your equation. The absolute value should be on one side of the equation. Any numbers that are not included inside the absolute value symbols should be moved to the other side of the equation.^[4] Note that an absolute value can never equal a negative number, so if, after isolating the absolute value, your absolute value equals a negative number, the equation has no solution.^[5]

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Set up the equation for the positive value. An equation involving absolute value will have two possible solutions. To set up the positive equation, simply remove the absolute value bars, and solve the equation as normal.^[6]
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Solve the positive equation. To do this, use algebra to solve for the variable. This will give you the first possible solution to the equation.
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Set up the equation for the negative value. To set up the negative equation, rewrite the equation without the absolute value bars, and take the negative value of the number on the other side of the equation.^[7]
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Solve the negative equation. Use algebra to solve for the variable as you would for any other equation. The result will be your second possible solution to the equation.

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Check the result of your positive equation. You must always plug possible solutions back into the original equation to verify that they are real solutions.^[8] To check your positive equation, plug the value for $x$ derived from the positive equation back into the original absolute value equation. If both sides of the equation are equal, the solution is true.
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Check the result of your negative equation. Just because one solution is true, it does not mean that both are true. You must also plug the solution from the negative equation back into the original equation to verify it is a real solution.
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Note your valid solutions. A solution is valid if, after plugging it back into the original equation, it yields a true equation. It is possible to have two valid solutions, but you may have one solution, or no solution.

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Remember that absolute value bars are distinct from parentheses and function differently.

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To solve absolute value equations, first isolate the absolute value terms by moving anything outside of the vertical bars to the other side of the equation. Next, solve for the positive value of the equation by isolating the variable. Since the absolute variable can represent 2 numbers, then solve for the negative value by putting a negative sign outside the vertical bars. Then, move the negative by dividing both sides by -1 and solve for the variable. If you want to learn how to check your answers for an absolute value equation, keep reading the article!

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Source: https://www.wikihow.com/Solve-Absolute-Value-Equations

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What Does Absolute Value Mean in Math