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.

Learn how to solve the absolute value inequality 5|x + 6| - 2 ≥ 28 with this step-by-step solution. The problem involves isolating the absolute value term, dividing, and solving two cases. This concept is crucial in algebra and is suitable for students in Grades 9-11.

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