Solve the Quadratic Inequality: x² + 9 > 0

Let's explore this problem step-by-step:

The inequality given is $x^2 + 9 > 0$.

1. To understand this inequality, we start by considering the expression $x^2$. We know that for any real number $x$, $x^2 \geq 0$. This means $x^2$ is always non-negative.

2. Since $x^2 \geq 0$ for every real number, adding 9 to $x^2$ will necessarily make the expression greater than zero, because a non-negative number plus a positive number gives a positive result: $x^2 + 9 \geq 9 > 0$.

3. Therefore, the inequality $x^2 + 9 > 0$ holds true for all real numbers $x$. There is no value of $x$ that makes the left side equal to or less than zero.

4. Thus, the solution to the inequality is that it holds for all values of $x$.

Consequently, the correct choice from the options provided is:

• All values of $x$

Therefore, the solution is that the inequality $x^2 + 9 > 0$ is true for all values of $x$.