Solving Absolute Value Equations Finding The Missing Solution

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Firoto attempted to solve the absolute value equation 6 - 4|2x - 8| = -10, but only found one solution. The goal is to identify the other solution from the provided options. This article will guide you through the process of solving absolute value equations, pinpointing Firoto's mistake, and determining the correct solutions.

Understanding Absolute Value Equations

Absolute value equations are equations where the variable is inside an absolute value symbol. The absolute value of a number is its distance from zero, meaning it is always non-negative. For example, |3| = 3 and |-3| = 3. This property leads to a critical aspect of solving absolute value equations: there are often two possible solutions. This is because the expression inside the absolute value can be either positive or negative, and both cases need to be considered to find all possible solutions.

When solving absolute value equations, the general strategy is to isolate the absolute value expression first. This involves performing algebraic operations to get the absolute value term by itself on one side of the equation. Once the absolute value is isolated, we can set up two separate equations. One equation considers the case where the expression inside the absolute value is positive, and the other considers the case where the expression is negative.

For instance, if we have an equation like |x| = a, where a is a positive number, we need to solve two equations:

  1. x = a
  2. x = -a

This dual approach ensures that we account for both possibilities that satisfy the original absolute value equation. It's a fundamental principle in solving these types of equations and a concept we'll apply to find the missing solution in Firoto's work.

Firoto's Solution: Identifying the Error

Let's examine Firoto's steps to solve the equation 6 - 4|2x - 8| = -10:

  1. -4|2x - 8| = -10
  2. -4|2x - 8| = -16 (This step is incorrect)
  3. |2x - 8| = 4
  4. 2x - 8 = 4
  5. 2x = 12
  6. x = 6

The critical error lies in step 2. Firoto seems to have missed a step in isolating the absolute value term. To correctly isolate the absolute value, we need to subtract 6 from both sides of the original equation before dividing. The correct steps should be:

  1. 6 - 4|2x - 8| = -10
  2. -4|2x - 8| = -10 - 6 (Subtract 6 from both sides)
  3. -4|2x - 8| = -16
  4. |2x - 8| = 4 (Divide both sides by -4)

While Firoto arrives at the correct simplified absolute value equation |2x - 8| = 4, the initial error obscures the complete solution process. This highlights the importance of meticulous algebraic manipulation when solving equations, especially those involving absolute values. Missing a step or performing an operation out of order can lead to an incomplete or incorrect solution.

Solving the Absolute Value Equation Correctly

Now that we've identified Firoto's mistake, let's solve the equation 6 - 4|2x - 8| = -10 correctly to find both solutions.

As we established earlier, the correct steps to isolate the absolute value are:

  1. 6 - 4|2x - 8| = -10
  2. -4|2x - 8| = -10 - 6
  3. -4|2x - 8| = -16
  4. |2x - 8| = 4

Now, we need to consider both the positive and negative cases of the expression inside the absolute value:

Case 1: 2x - 8 = 4

  1. 2x - 8 = 4
  2. 2x = 12 (Add 8 to both sides)
  3. x = 6 (Divide both sides by 2)

This gives us the solution x = 6, which is the solution Firoto found.

Case 2: 2x - 8 = -4

  1. 2x - 8 = -4
  2. 2x = 4 (Add 8 to both sides)
  3. x = 2 (Divide both sides by 2)

This gives us the other solution, x = 2. It’s crucial to remember that absolute value equations often yield two solutions because the expression inside the absolute value bars can be either positive or negative while still resulting in the same absolute value.

Identifying the Missing Solution

By correctly solving the absolute value equation, we found two solutions: x = 6 and x = 2. Firoto only found x = 6. Now, let's check the given options to find the missing solution:

A. -6 B. -4 C. 2 D. 10

The correct answer is C. 2, which matches the solution we calculated.

Key Takeaways for Solving Absolute Value Equations

To successfully solve absolute value equations and avoid errors like Firoto's, remember these key steps:

  1. Isolate the absolute value expression: This is the most crucial first step. Ensure that the absolute value term is by itself on one side of the equation before proceeding. This often involves addition, subtraction, multiplication, or division.
  2. Set up two equations: Once the absolute value is isolated, create two separate equations. In the first equation, set the expression inside the absolute value equal to the value on the other side of the equation. In the second equation, set the expression inside the absolute value equal to the negative of the value on the other side.
  3. Solve each equation: Solve each of the two equations independently using standard algebraic techniques. This will give you the potential solutions to the absolute value equation.
  4. Check your solutions: It's always a good practice to substitute your solutions back into the original equation to verify that they are correct. This helps catch any algebraic errors or extraneous solutions.

By following these steps carefully, you can confidently solve absolute value equations and find all possible solutions.

Conclusion

In summary, solving absolute value equations requires careful attention to detail and a systematic approach. Firoto's mistake highlights the importance of correctly isolating the absolute value term before proceeding. By setting up and solving two separate equations, we were able to find both solutions to the equation 6 - 4|2x - 8| = -10, which are x = 6 and x = 2. The missing solution was 2 (Option C). Remember to always isolate the absolute value, consider both positive and negative cases, and check your solutions to ensure accuracy.