We are given the equation:

$- |1 - 5x| + 3 = -6$

Let's solve for $x$ step by step.

Step 1: Isolate the absolute value term

Subtract 3 from both sides:

$- |1 - 5x| = -6 - 3$

Simplify the right-hand side:

$- |1 - 5x| = -9$

Step 2: Remove the negative sign

Multiply both sides of the equation by $-1$ to eliminate the negative sign:

$|1 - 5x| = 9$

Step 3: Solve the absolute value equation

The equation $|1 - 5x| = 9$ means that $1 - 5x$ can either be 9 or $-9$. So, we have two cases to consider:

Case 1: $1 - 5x = 9$

Solve for $x$:

$1 - 5x = 9$

Subtract 1 from both sides:

$-5x = 8$

Now, divide by $-5$:

$x = -\frac{8}{5}$

Case 2: $1 - 5x = -9$

Solve for $x$:

$1 - 5x = -9$

Subtract 1 from both sides:

$-5x = -10$

Now, divide by $-5$:

$x = 2$

Final Answer:

The solutions are:

$x = -\frac{8}{5} \quad \text{or} \quad x = 2$

Do you have any questions or would you like further details?

Here are 5 related questions to expand your understanding:

1. How do you solve absolute value inequalities?
2. What happens if the absolute value term is equal to zero?
3. How would you graph the solutions of this equation?
4. Can absolute value equations have no solution?
5. How do you solve absolute value equations with variables on both sides?

Tip: Always remember that an absolute value equation can have two possible cases—positive and negative—when isolating the absolute value term.