We are given the inequality:

$5|x + 6| - 2 \geq 28$

Step 1: Isolate the absolute value term

First, add 2 to both sides to eliminate the constant on the left:

$5|x + 6| \geq 28 + 2$

This simplifies to:

$5|x + 6| \geq 30$

Step 2: Divide by 5

Now, divide both sides by 5:

$|x + 6| \geq \frac{30}{5}$

$|x + 6| \geq 6$

Step 3: Solve the absolute value inequality

The inequality $|x + 6| \geq 6$ means that the expression inside the absolute value, $x + 6$, is either greater than or equal to 6 or less than or equal to -6. So, we solve two separate inequalities:

1. $x + 6 \geq 6$
2. $x + 6 \leq -6$

Case 1: $x + 6 \geq 6$

Subtract 6 from both sides:

$x \geq 0$

Case 2: $x + 6 \leq -6$

Subtract 6 from both sides:

$x \leq -12$

Step 4: Combine the solutions

The solution to the inequality is:

$x \geq 0 \quad \text{or} \quad x \leq -12$

This can be written in interval notation as:

$(-\infty, -12] \cup [0, \infty)$

Would you like further details or clarifications?

Here are some related questions:

1. What are the steps for solving absolute value inequalities?
2. How does the solution change if the inequality is strict (e.g., $<$ or $>$)?
3. Can we solve this graphically? What would the graph look like?
4. How do absolute value inequalities relate to distance on a number line?
5. What are some real-world examples where absolute value inequalities are used?

Tip: When solving absolute value inequalities, always split them into two cases, one for the positive and one for the negative scenario.