Solution: Steps for Solving Absolute Value Equations

Absolute value equations are a fundamental concept in algebra that often appear in various mathematical problems. These equations include an absolute value expression, denoted by two vertical lines surrounding an expression, and require a different approach for solving compared to other types of equations. This article aims to provide a step-by-step guide on how to solve absolute value equations effectively.

Step 1: Identify the absolute value equation

The first step in solving an absolute value equation is recognizing that it is, indeed, an absolute value equation. Look for the presence of the absolute value symbol, “| |”, in the equation. For example, consider the equation |x – 3| = 7. Here, the absolute value symbol signifies that it is an absolute value equation.

Step 2: Isolate the absolute value expression

To make the equation easier to solve, the next step involves isolating the absolute value expression. This requires removing any additional terms or variables added or subtracted from the absolute value expression. In the equation above, start by isolating |x – 3| by subtracting 3 from both sides: |x – 3| – 3 = 7 – 3, which simplifies to |x – 3| = 4.

Step 3: Set up two equations

Since the absolute value symbol represents two possibilities, one positive and one negative, it is necessary to set up two equations to account for both scenarios. Remove the absolute value symbol and set the expression inside it equal to the positive value and then repeat the process for the negative value. For the equation |x – 3| = 4, set up:

x – 3 = 4 (Positive case)
x – 3 = -4 (Negative case)

Step 4: Solve each equation separately

Now, solve each equation independently. Begin by isolating the variable on each side of the equation. For our positive case, add 3 to both sides to solve for x, resulting in x = 7. For the negative case, add 3 to both sides as well, leading to x = -1.

Step 5: Check for extraneous solutions

To ensure the validity of the solutions, it is crucial to check for extraneous solutions – solutions that satisfy the initial equation but not the simplified equation after the absolute value expression has been removed. Substitute each solution back into the original equation and verify its validity. In our example, substituting 7 into the original equation |x – 3| = 7 yields |7 – 3| = 7 which simplifies to 4 = 7. This is false, indicating that x = 7 is an extraneous solution. However, substituting -1 into the equation yields |-1 – 3| = 7, which simplifies to 4 = 7, also false. Therefore, x = -1 is also an extraneous solution.

Step 6: State the solution

After the previous steps are completed, the final step is to state the solution to the absolute value equation. In this case, there is no valid solution as both possible solutions were extraneous. Thus, the solution to the original equation |x – 3| = 7 is an empty set, also known as “no solution.”

In conclusion, solving absolute value equations requires a systematic approach. By following the step-by-step guide provided in this article, one can effectively solve absolute value equations and arrive at valid solutions while identifying any extraneous solutions.