The given equation is:

$5 | -4x - 5 | - 10 = 25$

Let's solve this step by step.

Step 1: Isolate the absolute value term

First, move the constant $-10$ to the right side of the equation:

$5 | -4x - 5 | = 25 + 10$ $5 | -4x - 5 | = 35$

Step 2: Divide both sides by 5

To simplify the equation, divide both sides by 5:

$| -4x - 5 | = \frac{35}{5}$ $| -4x - 5 | = 7$

Step 3: Solve the absolute value equation

Now, we remove the absolute value and create two equations, since the absolute value of a number can be either positive or negative:

1. $-4x - 5 = 7$
2. $-4x - 5 = -7$

Step 4: Solve both equations

Equation 1:

$-4x - 5 = 7$ Add 5 to both sides: $-4x = 7 + 5$ $-4x = 12$ Divide by $-4$: $x = \frac{12}{-4} = -3$

Equation 2:

$-4x - 5 = -7$ Add 5 to both sides: $-4x = -7 + 5$ $-4x = -2$ Divide by $-4$: $x = \frac{-2}{-4} = \frac{1}{2}$

Step 5: Conclusion

The solutions are:

$x = -3 \quad \text{or} \quad x = \frac{1}{2}$

Thus, the correct answer is:

$\boxed{x = -3, \frac{1}{2}}$

Would you like further explanation or have any questions? Here are some related questions you might find interesting:

1. How can you verify the solution of an absolute value equation?
2. What happens if the absolute value term equals a negative number?
3. How do you solve absolute value inequalities?
4. Can an absolute value equation have no solutions? Why or why not?
5. What is the general form of an absolute value equation?

Tip: Always check both solutions in the original equation to ensure they are valid.