https://file.aiquickdraw.com/mathbot/file/17294724760320dxas6j4.jpg

The inequality presented in the image is:

\[
2x^2 < 3x + 9
\]

### Let's solve this step by step:

1. **Rearrange the inequality**: Bring all terms to one side to set the inequality to 0.
   \[
   2x^2 - 3x - 9 < 0
   \]

2. **Solve the quadratic equation**: To understand the boundary points, solve the equation \( 2x^2 - 3x - 9 = 0 \).

   Using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \(a = 2\), \(b = -3\), and \(c = -9\).

   \[
   x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-9)}}{2(2)}
   \]
   \[
   x = \frac{3 \pm \sqrt{9 + 72}}{4}
   \]
   \[
   x = \frac{3 \pm \sqrt{81}}{4}
   \]
   \[
   x = \frac{3 \pm 9}{4}
   \]
   Therefore, the two solutions are:
   \[
   x = \frac{3 + 9}{4} = 3 \quad \text{and} \quad x = \frac{3 - 9}{4} = -\frac{3}{2}
   \]

3. **Test intervals**: Now that we know the boundary points are \(x = -\frac{3}{2}\) and \(x = 3\), we test values from the intervals \( (-\infty, -\frac{3}{2}) \), \( (-\frac{3}{2}, 3) \), and \( (3, \infty) \) to see where the inequality holds.

   - For \(x = -2\) (in \( (-\infty, -\frac{3}{2}) \)):
     \[
     2(-2)^2 - 3(-2) - 9 = 8 + 6 - 9 = 5 > 0
     \]
     So this interval does **not** satisfy the inequality.
   
   - For \(x = 0\) (in \( (-\frac{3}{2}, 3) \)):
     \[
     2(0)^2 - 3(0) - 9 = -9 < 0
     \]
     So this interval **does** satisfy the inequality.
   
   - For \(x = 4\) (in \( (3, \infty) \)):
     \[
     2(4)^2 - 3(4) - 9 = 32 - 12 - 9 = 11 > 0
     \]
     So this interval does **not** satisfy the inequality.

4. **Conclusion**: The inequality \(2x^2 - 3x - 9 < 0\) holds in the interval \((- \frac{3}{2}, 3)\).

Thus, the solution set in interval notation is:

\[
\boxed{(- \frac{3}{2}, 3)}
\]

Would you like more details on this process or have further questions?

---

Here are 5 related questions for further exploration:

1. What is the general form of a quadratic inequality, and how do you solve it?
2. Can you explain how to determine the intervals for testing solutions in quadratic inequalities?
3. How does the quadratic formula work, and why is it useful for solving equations?
4. How do you represent solutions in interval notation, and what do the brackets mean?
5. How would the solution change if the inequality were \(2x^2 > 3x + 9\) instead?

**Tip**: Always remember to check the boundary points when solving inequalities—they tell you where the expression equals zero!

Solve the inequality 2x^2 < 3x + 9 and provide the solution set in interval notation.

Learn how to solve the quadratic inequality 2x^2 < 3x + 9 using the quadratic formula and interval testing. This step-by-step solution covers key algebraic concepts and is suitable for students in Grades 9-11.

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