We are given the equation:

\[
-5|5x - 1| - 6 = -41
\]

### Step 1: Isolate the absolute value term
We first add 6 to both sides:

\[
-5|5x - 1| = -41 + 6
\]
\[
-5|5x - 1| = -35
\]

### Step 2: Divide both sides by -5
To isolate the absolute value expression, divide both sides by -5:

\[
|5x - 1| = \frac{-35}{-5}
\]
\[
|5x - 1| = 7
\]

### Step 3: Solve the absolute value equation
The expression \( |5x - 1| = 7 \) has two cases:

1. \( 5x - 1 = 7 \)
2. \( 5x - 1 = -7 \)

#### Case 1: \( 5x - 1 = 7 \)
Add 1 to both sides:
\[
5x = 8
\]
Divide by 5:
\[
x = \frac{8}{5} = 1.6
\]

#### Case 2: \( 5x - 1 = -7 \)
Add 1 to both sides:
\[
5x = -6
\]
Divide by 5:
\[
x = \frac{-6}{5} = -1.2
\]

### Final Solution:
The solutions are:
\[
x = 1.6 \quad \text{or} \quad x = -1.2
\]

Would you like any further details or explanations?

Here are five related questions for practice:
1. Solve \( |3x + 2| = 10 \).
2. Solve \( 2|4x - 3| + 1 = 9 \).
3. What is the general process for solving absolute value equations?
4. How does isolating the absolute value term help in solving such equations?
5. What happens when an absolute value equation has no solution?

**Tip:** When solving absolute value equations, always remember to split the equation into two cases—one for the positive and one for the negative scenario!

Learn how to solve the absolute value equation -5|5x − 1| − 6 = −41 with a step-by-step solution. This problem focuses on isolating the absolute value term and solving linear equations. Suitable for students in Grades 8-10.

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