https://file.aiquickdraw.com/mathbot/file/1725487089245240uy1q1.jpg

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.

Learn how to solve the quadratic inequality 5x^2 - 45 ≥ 0 with a detailed, step-by-step solution. This problem covers key algebraic concepts including factoring and properties of inequalities, suitable for students in Grades 9-10.

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