https://file.aiquickdraw.com/mathbot/file/1727859267369sax8p7rv.jpg

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!

Learn to solve the absolute value equation |1 - 4x| - 1 = 5 with a step-by-step solution. This guide covers absolute value concepts, isolation techniques, and dealing with two possible cases. Suitable for students in Grades 9-10.

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