Let's solve the inequality \( 5|x + 6| - 2 \geq 28 \).

### Step 1: Isolate the absolute value expression
First, add 2 to both sides of the inequality:
\[
5|x + 6| \geq 30
\]

Now, divide both sides by 5:
\[
|x + 6| \geq 6
\]

### Step 2: Break the absolute value inequality into two cases
The expression \( |x + 6| \geq 6 \) means that the value inside the absolute value can either be greater than or equal to 6, or less than or equal to -6. So we have two cases to solve:

#### 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 3: Combine the solutions
The solution to the inequality is:
\[
x \geq 0 \quad \text{or} \quad x \leq -12
\]
In interval notation, the solution is:
\[
(-\infty, -12] \cup [0, \infty)
\]

This is the final solution.

Do you have any questions or need further details?

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Here are 5 related questions:
1. How would the solution change if the inequality was \( 5|x + 6| - 2 \leq 28 \)?
2. Can you explain how to solve absolute value inequalities in general?
3. What happens if the inequality sign changes direction in absolute value inequalities?
4. How can we graph the solution set for \( 5|x + 6| - 2 \geq 28 \)?
5. How does the solution change if we have \( 5|x - 6| - 2 \geq 28 \)?

**Tip:** When solving absolute value inequalities, always break the problem into two cases to account for both the positive and negative scenarios of the expression inside the absolute value.

Learn how to solve the absolute value inequality 5|x + 6| - 2 ≥ 28 step by step. This problem explores absolute value inequalities and the theorem for splitting the inequality into two cases. Suitable for students in Grades 9-12.

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