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The equation given is:

\[
|1 - 4x| - 1 = 5
\]

### Steps to solve:

1. **Isolate the absolute value expression**:
   \[
   |1 - 4x| = 5 + 1
   \]
   \[
   |1 - 4x| = 6
   \]

2. **Remove the absolute value** by considering two cases:
   - **Case 1**: \(1 - 4x = 6\)
   - **Case 2**: \(1 - 4x = -6\)

### Solving Case 1:
\[
1 - 4x = 6
\]
Subtract 1 from both sides:
\[
-4x = 5
\]
Divide by -4:
\[
x = -\frac{5}{4}
\]

### Solving Case 2:
\[
1 - 4x = -6
\]
Subtract 1 from both sides:
\[
-4x = -7
\]
Divide by -4:
\[
x = \frac{7}{4}
\]

### Final Solution:
The solutions to the equation are:
\[
x = -\frac{5}{4}, \quad x = \frac{7}{4}
\]

Would you like more details or clarifications on this solution?

Here are some related questions:
1. What are the properties of absolute value equations?
2. How do we check solutions in absolute value equations?
3. What if there was no solution to the absolute value equation?
4. Can absolute value equations have more than two solutions?
5. How do we solve absolute value inequalities?

**Tip**: Always check your solutions by substituting them back into the original equation to ensure they satisfy the condition.

Learn how to solve the absolute value equation |1 - 4x| - 1 = 5 with step-by-step solutions. This problem involves key algebraic concepts such as isolating absolute values and solving linear equations. Suitable for students in Grades 8-10.

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