https://file.aiquickdraw.com/mathbot/file/173021861132182maqwqj.jpg

The inequality shown in the image is:

\[
2x^2 - 7x + 15 > 0
\]

To solve this inequality, let's first analyze the quadratic expression.

### Step 1: Check the Discriminant
For a quadratic inequality \( ax^2 + bx + c > 0 \) (or \( < 0 \)), it can be useful to check the discriminant \( D = b^2 - 4ac \). If \( D < 0 \), the quadratic does not intersect the x-axis and maintains a single sign based on the leading coefficient \( a \).

Here, \( a = 2 \), \( b = -7 \), and \( c = 15 \).

Calculating the discriminant:
\[
D = (-7)^2 - 4 \cdot 2 \cdot 15 = 49 - 120 = -71
\]

Since \( D < 0 \), the quadratic has no real roots. Because \( a = 2 > 0 \), the parabola opens upwards and thus \( 2x^2 - 7x + 15 \) is always positive for all real values of \( x \).

### Conclusion:
The inequality \( 2x^2 - 7x + 15 > 0 \) holds for all \( x \in \mathbb{R} \).

### Solution:
\[
\text{Solution: } x \in (-\infty, \infty)
\]

Would you like further clarification on any part of this solution?

Here are some related questions to explore this further:
1. What does the discriminant tell us about the nature of a quadratic equation?
2. How can you determine the direction of a parabola just by looking at the coefficient of \( x^2 \)?
3. What are the general steps to solve quadratic inequalities?
4. Why does a negative discriminant indicate no real roots?
5. How does the sign of the leading coefficient affect the inequality’s solution?

**Tip:** For inequalities involving quadratic expressions, the discriminant can be a quick tool to determine if the inequality holds for all values of \( x \) or has specific solutions.

Learn how to solve the quadratic inequality 2x^2 - 7x + 15 > 0 using discriminant analysis. This solution includes checking the discriminant and analyzing the direction of the parabola, suitable for students in Grades 10-12.

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