To solve the equation:

$-5 - 3| -8x + 6 | = -29$

we'll first isolate the absolute value term.

Step 1: Isolate the Absolute Value Term

1. Start by moving $-5$ to the other side: $-3| -8x + 6 | = -29 + 5$ $-3| -8x + 6 | = -24$

2. Next, divide both sides by $-3$: $| -8x + 6 | = 8$

Step 2: Set Up Two Cases for the Absolute Value

Since $| -8x + 6 | = 8$, we can write two separate equations:

Case 1:

$-8x + 6 = 8$

1. Subtract 6 from both sides: $-8x = 2$
2. Divide by $-8$: $x = -\frac{1}{4}$

Case 2:

$-8x + 6 = -8$

1. Subtract 6 from both sides: $-8x = -14$
2. Divide by $-8$: $x = \frac{7}{4}$

Solution

The solutions are: $x = -\frac{1}{4}, \frac{7}{4}$

Let me know if you want further clarification or have additional questions!

Additional Questions

1. How can absolute value equations have two cases?
2. Why did we divide by $-3$ to isolate the absolute value term?
3. What would happen if the right side were positive instead of negative?
4. How can you verify the solutions?
5. Are there cases where absolute value equations have no solutions?

Tip: For absolute value equations, always check if isolating the absolute term gives a feasible solution before proceeding with cases.