How to Solve Absolute Value Equations

Three Methods:Understanding Absolute ValueDetermining Possible SolutionsChecking Your Results

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. For example, an absolute value equation might look like this: |x − 1| + 4 = 0.

Part 1
Understanding Absolute Value

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    Know the mathematical definition of an absolute value. Absolute value has a specific mathematic definition. The variable p represents any number.
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    Know the geometric definition of an absolute value. Absolute value also has a geometric definition, in which |p| is expressed as the distance of p from 0 on a number line. This distance will always be positive.
    • In the example above, you can see that the distance of -3 from 0 is 3, so |−3| = 3.

Part 2
Determining Possible Solutions

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    Divide the equation into one positive equation and one negative equation. The first step to solving an absolute value equation is rewriting it so that you have one positive and one negative equation. For the positive equation, simply remove the absolute value bars and replace them with parentheses; for the negative equation, do the same, but put a negative sign in front of the parenthetical expression. Take, for example, |2x−3|+1 = 8.
    • In this example, you would first create a positive equation by removing the absolute value bars and replacing them with parentheses: (2x−3)+1 = 8.
    • Second, you would create a negative expression by repeating the process and adding a negative sign: −(2x−3)+1 = 8.
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    Solve the positive equation. Turn your attention to the positive equation you’ve just created. Solve the equation. Your answer will be one of the possible solutions for your absolute value equation.
    • In the example above, you would simply solve for x:
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    Solve the negative equation. Now turn your attention to the negative equation you’ve created. Solve this equation as well. Your answer will be the second possible solution for your absolute value equation.
    • In the example above, you would simply solve for x again:

Part 3
Checking Your Results

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    Check the result of your positive equation. In order to confirm that your answer is a real solution, you must substitute the result of your positive equation for the x in your original absolute value equation. If the two sides balance, the solution is a valid one.
    • In the example above, you would replace the x with your answer, 5, and simplify. The right side and the left side are equal, so x = 5 is one valid solution to your absolute value equation.
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    Check the result of your negative equation. You’ll also have to confirm that your second answer is a real solution. Substitute the result of your negative equation for the x in your original absolute value equation. Again, if the two sides balance, the solution is a valid one.
    • In the example above, you would replace the x with your answer, -2, and simplify. The right side and the left side are equal again, so x = -2 is also a valid solution to your absolute value equation.
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    Note your solutions. Since there are two solutions to your absolute value equation, you would write: x = 5,− 2.

Tips

  • Remember that absolute value bars are distinct from parentheses and function differently. Don’t get confused by the fact that we replace absolute value bars with parentheses in order to check potential solutions to absolute value equations.

Article Info

Categories: Algebra