The absolute inequality given is:

$|2x + 2| \geq 10$

To solve this, let's break it down:

1. The definition of absolute value inequality states that:

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

   Applying this to our problem:

   $2x + 2 \leq -10 \quad \text{or} \quad 2x + 2 \geq 10$

2. Solving the first inequality $2x + 2 \leq -10$:

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

3. Solving the second inequality $2x + 2 \geq 10$:

   $2x \geq 8$ $x \geq 4$

Thus, the solution to the inequality is:

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

This matches option 1: $x \leq -6$ or $x \geq 4$.

Would you like further details or explanations on any step?

Here are 5 related questions:

1. How do you solve absolute value equations?
2. What is the difference between solving $|A| \geq B$ and $|A| \leq B$?
3. Can you graph the solution of this inequality?
4. What happens if the inequality sign was $>$ instead of $\geq$?
5. How do you solve compound inequalities?

Tip: Always isolate the absolute value before applying the rules to split the inequality into two cases.