To solve the inequality $|x + 1| + 8 \geq 6$, we can start by isolating the absolute value term:

$|x + 1| + 8 \geq 6$

Subtract 8 from both sides:

$|x + 1| \geq -2$

Since the absolute value of any expression is always non-negative, $|x + 1| \geq -2$ is always true for all real numbers $x$. Therefore, the solution to this inequality is:

$x \in \mathbb{R} \quad \text{(All real numbers)}$

Would you like further details on this, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. What if the inequality was $|x + 1| + 8 \geq 10$?
2. How do you solve an equation with an absolute value, like $|x + 1| = 5$?
3. What if the inequality was $|x + 1| - 8 \geq 6$?
4. Can you explain how the graph of $y = |x + 1|$ looks?
5. What happens if you have two absolute values, such as $|x - 3| + |2x + 5| \geq 7$?

Tip: When dealing with absolute value inequalities, remember that absolute values can never be negative, and this property is often crucial in determining the solution set.