Let's solve the equation:

$3 - 6|5x - 4| = -21$

Step 1: Isolate the absolute value

First, subtract 3 from both sides:

$-6|5x - 4| = -21 - 3$

$-6|5x - 4| = -24$

Step 2: Solve for $|5x - 4|$

Now, divide both sides by -6 to isolate the absolute value:

$|5x - 4| = \frac{-24}{-6}$

$|5x - 4| = 4$

Step 3: Break it into two cases

Since we are dealing with an absolute value, we need to consider two cases:

1. $5x - 4 = 4$
2. $5x - 4 = -4$

Case 1: $5x - 4 = 4$

Solve for $x$:

$5x = 4 + 4$ $5x = 8$ $x = \frac{8}{5}$

Case 2: $5x - 4 = -4$

Solve for $x$:

$5x = -4 + 4$ $5x = 0$ $x = 0$

Final Answer:

The solutions are:

$x = \frac{8}{5} \quad \text{or} \quad x = 0$

Let me know if you'd like further details or have any questions!

Here are 5 related questions you might explore:

1. How do you solve equations with absolute values?
2. What are the rules for solving linear equations with fractions?
3. How do you graph solutions involving absolute values?
4. Can absolute values ever equal negative numbers?
5. How can you check solutions to absolute value equations?

Tip: Always remember to check if the absolute value equation has any valid solutions before proceeding!