Let's solve the absolute inequality step by step.

We are given:

$|2x + 2| \geq 10$

Step 1: Break the absolute value inequality

The general rule for absolute value inequalities is:

$|A| \geq B \quad \text{implies} \quad A \geq B \quad \text{or} \quad A \leq -B$

So, for $|2x + 2| \geq 10$, we have two cases:

1. $2x + 2 \geq 10$
2. $2x + 2 \leq -10$

Step 2: Solve each inequality

Case 1: $2x + 2 \geq 10$

$2x \geq 10 - 2$ $2x \geq 8$ $x \geq 4$

Case 2: $2x + 2 \leq -10$

$2x \leq -10 - 2$ $2x \leq -12$ $x \leq -6$

Step 3: Combine the solutions

From the two cases, we have: $x \geq 4 \quad \text{or} \quad x \leq -6$

Thus, the solution to the inequality is: $x \leq -6 \quad \text{or} \quad x \geq 4$

Step 4: Verify your options

Looking at the provided options, the correct one is:

$x \leq -6 \quad \text{or} \quad x \geq 4$

This matches the inequality solution.

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Do you want more details or have any questions? Here are some related questions to explore:

1. How do you solve absolute value inequalities in general?
2. What happens if the inequality was $|2x+2| \leq 10$?
3. How would the solution change if the inequality sign was strict (i.e., $|2x+2| > 10$)?
4. Can you graph the solution $x \leq -6 \text{ or } x \geq 4$?
5. What is the difference between absolute value equations and inequalities?

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