https://file.aiquickdraw.com/mathbot/file/17272843312038rdew0kq.jpg

The equation displayed in the image is:

\[
|3x - 1| = 8x
\]

Let's solve this step by step:

### Step 1: Split the absolute value equation
We have two cases to consider when solving an equation with absolute values.

#### Case 1: \( 3x - 1 = 8x \)
In this case, the expression inside the absolute value is equal to \(8x\).
- Subtract \(3x\) from both sides:
  \[
  -1 = 5x
  \]
- Solve for \(x\):
  \[
  x = -\frac{1}{5}
  \]

#### Case 2: \( 3x - 1 = -8x \)
In this case, the expression inside the absolute value is equal to the negative of \(8x\).
- Add \(8x\) to both sides:
  \[
  11x - 1 = 0
  \]
- Add 1 to both sides:
  \[
  11x = 1
  \]
- Solve for \(x\):
  \[
  x = \frac{1}{11}
  \]

### Step 2: Check for extraneous solutions
We must check both solutions to ensure they satisfy the original equation.

#### Check \( x = -\frac{1}{5} \):
\[
|3(-\frac{1}{5}) - 1| = |-\frac{3}{5} - 1| = |-1.6| = 1.6
\]
\[
8(-\frac{1}{5}) = -1.6
\]
The equation does not hold, so \(x = -\frac{1}{5}\) is **not a solution**.

#### Check \( x = \frac{1}{11} \):
\[
|3(\frac{1}{11}) - 1| = |\frac{3}{11} - 1| = |-\frac{8}{11}| = \frac{8}{11}
\]
\[
8(\frac{1}{11}) = \frac{8}{11}
\]
The equation holds, so \(x = \frac{1}{11}\) is a valid solution.

### Final Answer:
The solution is \( x = \frac{1}{11} \), so the correct option is **A**.

Let me know if you need any further explanation or details! Here are some related questions you might find useful:

1. How do you solve more complex absolute value equations?
2. What are extraneous solutions, and why do they occur?
3. How can you graphically verify the solution to an absolute value equation?
4. How does changing the coefficients of the absolute value expression affect the solutions?
5. What other types of problems can be solved using similar methods?

**Tip:** Always check for extraneous solutions when dealing with absolute value equations by plugging them back into the original equation!

Learn how to solve the absolute value equation |3x - 1| = 8x step by step. This problem involves key algebraic concepts such as splitting into two cases and checking for extraneous solutions. Suitable for students in Grades 9-10.

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