The inequality given in the image is:

$5x^2 - 45 \geq 0$

Steps to solve the inequality:

1. Simplify the inequality: Factor out the common factor from the terms:

   $5(x^2 - 9) \geq 0$

2. Factor further: Recognize that $x^2 - 9$ is a difference of squares:

   $5(x - 3)(x + 3) \geq 0$

3. Analyze the inequality: We need to find the values of $x$ where the product $(x - 3)(x + 3)$ is non-negative (i.e., greater than or equal to 0).

   The critical points (where the expression is zero) occur at $x = 3$ and $x = -3$.

4. Test intervals: The number line is divided into three intervals by the critical points: $(-\infty, -3)$, $(-3, 3)$, and $(3, \infty)$.

   • For $x \in (-\infty, -3)$: Pick $x = -4$, then $(x - 3)(x + 3) = (-4 - 3)(-4 + 3) = (-7)(-1) = 7 > 0$.
   • For $x \in (-3, 3)$: Pick $x = 0$, then $(x - 3)(x + 3) = (0 - 3)(0 + 3) = (-3)(3) = -9 < 0$.
   • For $x \in (3, \infty)$: Pick $x = 4$, then $(x - 3)(x + 3) = (4 - 3)(4 + 3) = (1)(7) = 7 > 0$.

   Therefore, the expression $(x - 3)(x + 3)$ is non-negative in the intervals $(-\infty, -3]$ and $[3, \infty)$.

5. Conclusion: The solution to the inequality $5x^2 - 45 \geq 0$ is:

   $x \in (-\infty, -3] \cup [3, \infty)$

Would you like more details on any step or do you have any further questions?

Related Questions:

1. How do you solve quadratic inequalities in general?
2. What is the significance of the difference of squares in factoring?
3. How can the sign of a product of two binomials be determined?
4. Can you graph the inequality $5x^2 - 45 \geq 0$ on a number line?
5. How does solving quadratic inequalities differ from solving quadratic equations?

Tip:

When solving inequalities, always test intervals between critical points to determine where the inequality holds true.