To solve the inequality \( 2x^2 - 6x - x + 3 > 0 \), let's go through the steps systematically:

1. **Combine like terms**: 
   \[
   2x^2 - 7x + 3 > 0
   \]

2. **Find the roots of the quadratic equation** \( 2x^2 - 7x + 3 = 0 \) to identify critical points. Use the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 2 \), \( b = -7 \), and \( c = 3 \):
   \[
   x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \times 2 \times 3}}{2 \times 2}
   \]
   \[
   x = \frac{7 \pm \sqrt{49 - 24}}{4}
   \]
   \[
   x = \frac{7 \pm \sqrt{25}}{4}
   \]
   \[
   x = \frac{7 \pm 5}{4}
   \]

   The roots are:
   \[
   x = \frac{7 + 5}{4} = 3
   \]
   \[
   x = \frac{7 - 5}{4} = \frac{1}{2}
   \]

3. **Determine intervals**: The roots divide the number line into three intervals: 
   \[
   (-\infty, \frac{1}{2}), \quad \left(\frac{1}{2}, 3\right), \quad (3, \infty)
   \]

4. **Test the intervals**: Select a test point from each interval to determine where the inequality \( 2x^2 - 7x + 3 > 0 \) holds.
   - For \( x = 0 \) in the interval \( (-\infty, \frac{1}{2}) \):
     \[
     2(0)^2 - 7(0) + 3 = 3 > 0
     \]
   - For \( x = 2 \) in the interval \( \left(\frac{1}{2}, 3\right) \):
     \[
     2(2)^2 - 7(2) + 3 = 8 - 14 + 3 = -3 < 0
     \]
   - For \( x = 4 \) in the interval \( (3, \infty) \):
     \[
     2(4)^2 - 7(4) + 3 = 32 - 28 + 3 = 7 > 0
     \]

5. **Determine the solution set**: The quadratic is positive in the intervals \( (-\infty, \frac{1}{2}) \) and \( (3, \infty) \).

Therefore, the solution to the inequality is:
\[
x \in (-\infty, \frac{1}{2}) \cup (3, \infty)
\]

Would you like a more detailed explanation or have any other questions?

Here are 5 related questions:
1. How do you solve quadratic inequalities graphically?
2. What is the significance of the quadratic formula?
3. How do you determine when a quadratic inequality has no solution?
4. How do you solve inequalities involving absolute values?
5. How do you interpret the roots of a quadratic equation?

**Tip**: Always check your intervals by picking test points to see where the inequality holds true.

Learn how to solve the quadratic inequality (2x^2) - 7x + 3 > 0 using the quadratic formula. This guide covers key algebraic concepts, such as finding the roots, testing intervals, and determining the solution set. Suitable for students in Grades 9-11.

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