The given equation is:

$|1 - 4x| - 1 = 5$

Let's solve it step by step:

Step 1: Isolate the absolute value

Add 1 to both sides to isolate the absolute value term:

$|1 - 4x| = 6$

Step 2: Solve for the two cases of the absolute value

Since the absolute value of a number can be either positive or negative, we split it into two cases:

1. $1 - 4x = 6$
2. $1 - 4x = -6$

Case 1: $1 - 4x = 6$

$1 - 4x = 6$ Subtract 1 from both sides: $-4x = 5$ Now divide by -4: $x = -\frac{5}{4}$

Case 2: $1 - 4x = -6$

$1 - 4x = -6$ Subtract 1 from both sides: $-4x = -7$ Now divide by -4: $x = \frac{7}{4}$

Final Solution

The solutions for $x$ are: $x = -\frac{5}{4} \quad \text{or} \quad x = \frac{7}{4}$

Would you like more details or further explanation on any of the steps?

Here are some related questions you might find helpful:

1. What is the general approach to solving absolute value equations?
2. How do you handle more complex absolute value equations with multiple terms inside?
3. What happens if there's a quadratic term inside the absolute value?
4. How can you verify the solutions of absolute value equations?
5. How do you graph absolute value functions?

Tip: Always remember to check both cases when solving absolute value equations, as they can yield different results!