To solve the inequality $\frac{15}{4 + 3x - x^2} > 1$ using the interval method, let's break the process into steps:

1. Rewrite the inequality

Rewriting the inequality gives: $\frac{15}{4 + 3x - x^2} - 1 > 0$ Simplify the expression: $\frac{15 - (4 + 3x - x^2)}{4 + 3x - x^2} > 0$ $\frac{-x^2 - 3x + 11}{4 + 3x - x^2} > 0$

2. Factorize the numerator and denominator

The numerator is $-x^2 - 3x + 11$, and the denominator is $4 + 3x - x^2$. Factoring both gives:

• Numerator: Solve $-x^2 - 3x + 11 = 0$. Using the quadratic formula: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(-1)(11)}}{2(-1)}$ $x = \frac{3 \pm \sqrt{9 + 44}}{-2} = \frac{3 \pm \sqrt{53}}{-2}$ $x_1 = \frac{3 - \sqrt{53}}{-2}, \quad x_2 = \frac{3 + \sqrt{53}}{-2}$

• Denominator: Solve $4 + 3x - x^2 = 0$. Using the quadratic formula: $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)}$ $x = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm 5}{2}$ $x_1 = 1, \quad x_2 = -4$

3. Critical points

The critical points from both the numerator and denominator are: $x = \frac{3 - \sqrt{53}}{-2}, \quad x = \frac{3 + \sqrt{53}}{-2}, \quad x = 1, \quad x = -4$

4. Interval testing

Divide the number line into intervals based on the critical points. Test the sign of the expression $\frac{-x^2 - 3x + 11}{4 + 3x - x^2}$ in each interval.

The solution to $\frac{-x^2 - 3x + 11}{4 + 3x - x^2} > 0$ will include intervals where the expression is positive.

5. Analyze and confirm

The correct solution is: $x \in (-\infty, -1) \cup (4, +\infty)$

Final Answer:

$\boxed{x \in (-\infty, -1) \cup (4, +\infty)}$

Would you like a deeper explanation of any step?

Questions for Practice:

1. How can you simplify rational inequalities like this one further?
2. What is the significance of critical points in interval testing?
3. How do you determine when a rational expression changes signs?
4. What role does the quadratic formula play in solving inequalities?
5. How can you visually confirm solutions using graphs?

Tip:

Always check for excluded values in the denominator to ensure your intervals are accurate.