Solving Absolute Value Equations

Solving absolute value equations involves understanding that the absolute value of an expression, |x|, represents its distance from zero on the number line. This means that x could be either positive or negative. Therefore, when solving an absolute value equation, you typically need to consider two cases.

The following diagram shows how to solve an absolute value equation. Scroll down the page for more examples and solutions.

 

Absolute Value Worksheets
Practice your skills with the following worksheets:
Absolute Value Worksheets
Absolute Value Equation Worksheets

Steps on how to solve absolute value equations

1. Isolate the Absolute Value Expression:
   The first step is to isolate the absolute value expression on one side of the equation. This means getting the term with the absolute value symbol by itself.
   Example:
   Solve |2x − 1| + 3 = 8
   Subtract 3 from both sides:
   |2x − 1| = 8 − 3
   |2x − 1| = 5
   

2. Set Up Two Cases:
   Once the absolute value expression is isolated, you need to consider the two possibilities for the expression inside the absolute value: it could be equal to the positive value on the other side of the equation, or it could be equal to the negative of that value.

For |A| = b (where b ≥ 0), this leads to two equations:
Case 1: A = b
Case 2: A = −b

Example (continued):
For |2x − 1| = 5, we set up two cases:
Case 1: 2x − 1 = 5
Case 2: 2x − 1 = −5

3. Solve Each Case:
   Solve each of the two equations you set up in the previous step for the variable.
   Example (continued):
   Case 1:
   2x − 1 = 5
   2x = 6
   x = 3
   Case 2:
   2x − 1 = −5
   2x = −4
   x = −2
   

4. Check Your Solutions:
   It’s crucial to check your solutions by substituting them back into the original absolute value equation to ensure they are valid. Absolute value equations can sometimes yield extraneous solutions (solutions that satisfy the transformed equations but not the original one).

Absolute Value Equations
In this video, we learn that when solving an absolute value equation, the first step is to isolate the absolute value. The next step is to split the equation up into two separate equations (one positive and the other negative). Finally, the last step is to solve each of the remaining equations.

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Solving Absolute Value Equations
In math, absolute value equations are best solved by isolating the absolute value portion from the rest of the equation. Find out why the lowest number on the opposite side of an absolute value equation should be zero with help from a math teacher in this free video series on math help and lessons.

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How to Solve Multi-Step Equations with Absolute Values?

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Solving Absolute Value Equations
In math, the absolute value of a number is its distance from zero. Solve absolute value equations with tips from a math teacher in this free video on solving math problems.

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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