To solve the equation involving absolute value:

$|6 - 4x| - 1 = 8$

Step 1: Isolate the absolute value expression.

Add 1 to both sides of the equation:

$|6 - 4x| = 9$

Step 2: Break it into two cases.

Since $|6 - 4x| = 9$, we consider two possibilities:

1. $6 - 4x = 9$
2. $6 - 4x = -9$

Case 1: $6 - 4x = 9$

Solve for $x$:

$6 - 4x = 9 \quad \Rightarrow \quad -4x = 9 - 6 \quad \Rightarrow \quad -4x = 3$ $x = \frac{3}{-4} = -\frac{3}{4}$

Case 2: $6 - 4x = -9$

Solve for $x$:

$6 - 4x = -9 \quad \Rightarrow \quad -4x = -9 - 6 \quad \Rightarrow \quad -4x = -15$ $x = \frac{-15}{-4} = \frac{15}{4}$

Final Answer:

The solutions are:

$x = -\frac{3}{4}, \frac{15}{4}$

Would you like more details or have any questions?

---

Here are 5 related questions you might want to explore:

1. How do you graph absolute value functions?
2. What happens when there is no solution to an absolute value equation?
3. How do you solve absolute value inequalities?
4. What are the properties of absolute value functions?
5. How can absolute value equations represent real-world scenarios?

Tip: Always check both cases when solving absolute value equations, as both positive and negative values need to be considered.