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Let's solve the problem step-by-step. The question asks us to find the set of possible values of \( x \) for which:

1. \( 4x^2 - 25 < 0 \)
2. \( 12 - 5x - 3x^2 > 0 \)

We need to solve both inequalities and then find the intersection of the solutions. I'll begin with each inequality individually.

### Solving \( 4x^2 - 25 < 0 \)

This is a quadratic inequality. We can rewrite it as:

\[
4x^2 - 25 = 0 \quad \Rightarrow \quad 4x^2 = 25 \quad \Rightarrow \quad x^2 = \frac{25}{4}
\]

Taking square roots:

\[
x = \pm \frac{5}{2}
\]

This means the quadratic factorizes as:

\[
(2x - 5)(2x + 5) < 0
\]

To solve this, we test the sign changes around the critical points \( x = \frac{5}{2} \) and \( x = -\frac{5}{2} \). The inequality \( (2x - 5)(2x + 5) < 0 \) is satisfied when \( -\frac{5}{2} < x < \frac{5}{2} \).

### Solving \( 12 - 5x - 3x^2 > 0 \)

Rearrange the inequality:

\[
-3x^2 - 5x + 12 > 0
\]

Multiply the entire inequality by \( -1 \) (note that this reverses the inequality sign):

\[
3x^2 + 5x - 12 < 0
\]

Now, solve the quadratic equation \( 3x^2 + 5x - 12 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3 \), \( b = 5 \), and \( c = -12 \).

Let's calculate the discriminant and roots.The discriminant is \( 169 \), which is positive, meaning the quadratic has two real roots:

\[
x_1 = 1.33 \quad \text{(or } \frac{4}{3}\text{)}, \quad x_2 = -3
\]

Thus, the quadratic \( 3x^2 + 5x - 12 < 0 \) can be factored as:

\[
3(x - 1.33)(x + 3) < 0
\]

To solve this inequality, test the sign of \( 3(x - 1.33)(x + 3) \) in the intervals determined by the roots \( x = -3 \) and \( x = 1.33 \). The inequality holds when:

\[
-3 < x < 1.33
\]

### Combining the Two Solutions

Now, we combine the results of both inequalities:

1. From \( 4x^2 - 25 < 0 \), we have \( -\frac{5}{2} < x < \frac{5}{2} \) or approximately \( -2.5 < x < 2.5 \).
2. From \( 3x^2 + 5x - 12 < 0 \), we have \( -3 < x < 1.33 \).

The intersection of these two intervals is:

\[
-2.5 < x < 1.33
\]

Thus, the solution set is \( x \in (-2.5, 1.33) \).

Would you like more details or have any questions?

### Follow-up Questions:
1. What are the general steps to solve quadratic inequalities?
2. How do you find the critical points of a quadratic inequality?
3. What happens when you multiply or divide both sides of an inequality by a negative number?
4. How do you interpret the solution intervals when solving compound inequalities?
5. Why is the intersection of the solution sets important?

### Tip:
Always graph the quadratic functions to visualize the regions where the inequalities hold. This can help in understanding the solution set better.

Find the set of possible values of x for which 4x^2 - 25 < 0 and 12 - 5x - 3x^2 > 0.

Learn how to solve the quadratic inequalities 4x^2 - 25 < 0 and 12 - 5x - 3x^2 > 0 using factorization, quadratic formulas, and interval testing. Suitable for students in Grades 9-11, this step-by-step guide covers key concepts in algebraic manipulation and quadratic inequalities.

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