The question presented in the image is asking whether the solution set for the inequality $x^2 + 1 \geq 0$ is the empty set, denoted as $\{ \}$.

Let's analyze the inequality:

• The inequality is $x^2 + 1 \geq 0$.
• The term $x^2$ is the square of a real number, and the square of any real number is always non-negative ($x^2 \geq 0$).
• Adding 1 to $x^2$ will always give a number greater than or equal to 1, since $x^2 + 1 \geq 1$.

Thus, the expression $x^2 + 1$ is always greater than or equal to 1 for all real numbers $x$. Therefore, the inequality $x^2 + 1 \geq 0$ holds true for all real values of $x$.

Conclusion:

The solution set is not the empty set. In fact, the solution set is all real numbers, so the correct answer to the question should be False (since the solution set is not $\{ \}$).

Do you want more details or have any questions?

Here are five related questions:

1. What are the general properties of quadratic inequalities?
2. How do we solve an inequality of the form $x^2 + c \geq 0$ where $c$ is a constant?
3. What does the empty set represent in the context of inequalities?
4. How would the inequality change if it were $x^2 + 1 < 0$?
5. Can you think of any situation where the sum of a square and a constant could be negative?

Tip: Always check the behavior of the squared term in inequalities, as it can provide a lot of information about the solution set!