https://file.aiquickdraw.com/mathbot/file/1726539217434rfmr6qwj.jpg

Let's solve the problems from the image one by one:

### 3. Given \( x \in [5, 10] \), find the minimum value \( M \) and the maximum value \( m \) such that \( m \leq \frac{12x - 1}{3x + 2} < M \).

We need to evaluate the expression \( \frac{12x - 1}{3x + 2} \) over the interval \( [5, 10] \). This involves calculating the extreme values of the function at the boundaries of the interval and determining its behavior within the interval.

### 4. If \( -3 < 2x - 1 < 5 \), then \( (5x + 2) \in (a, b) \). Find \( a \) and \( b \).

The inequality needs to be solved for \( x \) and then used to find the bounds for \( 5x + 2 \). We start by solving the inequality step by step.

### 5. Solve the inequality in \( \mathbb{R} \):
\[
\frac{3 - 5x}{2} < \frac{2x - 8}{4} < \frac{1}{4}x - 3
\]
This is a compound inequality that requires solving both parts individually and then combining the solutions.

### 6. For which values of \( k \) does the inequality 
\[
(4 - k^2)x^2 + (4k - 4)x - 4 < 0
\]
hold:
- (a) For all \( x \in \mathbb{R} \),
- (b) For \( x \in \left(-\frac{4}{5}, \frac{3}{4}\right) \).

We analyze the discriminant of the quadratic expression for the given conditions.

### 7. Solve the following inequalities:
- (a) \( x^2 + 2x + 4 \geq 0 \)
- (b) \( -x^2 - 5x + 6 < 0 \)
- (c) \( (2x - 3)(x + 2)(x - 1) \leq 0 \)
- (d) \( 5(x - 3)(3x - 2)(2x + 1)(2 - x^2) \leq 0 \)

Each of these inequalities needs to be solved using methods like factoring, quadratic formula, and analyzing the signs of the factors.

Would you like me to solve one of these problems in detail first or provide detailed solutions for all of them? Let me know!

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Here are 5 related questions based on these problems:
1. How do you find the maximum and minimum of a rational function on a closed interval?
2. What is the process for solving compound inequalities involving rational functions?
3. How can we determine the discriminant's role in solving quadratic inequalities?
4. What steps are used to analyze the signs of factors in polynomial inequalities?
5. How do boundary points affect the solution of polynomial inequalities?

**Tip**: When solving quadratic inequalities, always check if the quadratic opens upwards or downwards by examining the coefficient of \( x^2 \).

Si x ∈ [5,10], hallar el menor valor M y el mayor valor m, tal que, m ≤ (12x−1)/(3x+2) < M. Otros problemas involucran resolver inecuaciones cuadráticas y polinómicas, además de encontrar intervalos y restricciones.

Learn to solve inequalities involving quadratic equations, rational functions, and polynomials. This exercise covers solving quadratic inequalities, determining intervals, and evaluating rational functions.

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