The problem asks for which real values of $x$ the inequality $|x + 9| \geq 5$ holds.

Step 1: Understanding the Absolute Value Inequality

The inequality $|x + 9| \geq 5$ means that the expression $x + 9$ is either greater than or equal to 5, or less than or equal to -5. Mathematically, this translates to: $x + 9 \geq 5 \quad \text{or} \quad x + 9 \leq -5$

Step 2: Solve the Two Inequalities Separately

1. First Inequality: $x + 9 \geq 5$ Subtract 9 from both sides: $x \geq 5 - 9$ Simplifying: $x \geq -4$

2. Second Inequality: $x + 9 \leq -5$ Subtract 9 from both sides: $x \leq -5 - 9$ Simplifying: $x \leq -14$

Step 3: Combine the Results

The solution to the inequality $|x + 9| \geq 5$ is: $x \leq -14 \quad \text{or} \quad x \geq -4$ This means $x$ lies in either of the intervals $(-\infty, -14]$ or $[-4, \infty)$.

Step 4: Compare with the Options Provided

The correct answer is:

• Option B: $x \in (-\infty, -14] \cup [-4, \infty)$

Do you want more details or have any questions? Here are some related questions for further practice:

1. What happens if the inequality were $|x + 9| > 5$ instead of $\geq$?
2. How would the solution change if the inequality was $|x - 7| \geq 2$?
3. What are the solutions for $|x + 9| \leq 5$?
4. How do you solve absolute value inequalities involving a variable on both sides, like $|2x + 1| \geq |x - 3|$?
5. What if the inequality were $|x + 9| \leq -5$? How would you interpret this?

Tip: When solving absolute value inequalities, always split the problem into two cases to capture both possible scenarios of the inequality.