https://r2.erweima.ai/mathbot/image/1723734643146b06dsj05.jpg

INECUACIONES RESOLVER CON EL METODO DE PUNTOS CRITICOS

To solve the inequality provided in the image using the method of critical points, let's follow the step-by-step process.

### Problem:
We need to solve the inequality:
\[
\frac{2(x - 9)(x + 3)}{4} < \frac{3(x - 1)(x + 5)}{3}
\]

### Step 1: Simplify the inequality
First, let's simplify both sides of the inequality.

1. Divide both sides by the constants 4 and 3:
   \[
   \frac{2(x - 9)(x + 3)}{4} \quad \text{and} \quad \frac{3(x - 1)(x + 5)}{3}
   \]
   simplify to:
   \[
   \frac{(x - 9)(x + 3)}{2} < (x - 1)(x + 5)
   \]

2. Multiply both sides by 2 to eliminate the fraction on the left:
   \[
   (x - 9)(x + 3) < 2(x - 1)(x + 5)
   \]

### Step 2: Expand both sides
Expand both quadratic expressions:
1. Left side:
   \[
   x^2 + 3x - 9x - 27 = x^2 - 6x - 27
   \]
2. Right side:
   \[
   2(x^2 + 5x - x - 5) = 2(x^2 + 4x - 5) = 2x^2 + 8x - 10
   \]

So, the inequality now is:
\[
x^2 - 6x - 27 < 2x^2 + 8x - 10
\]

### Step 3: Rearrange the inequality
Bring all terms to one side:
\[
x^2 - 6x - 27 - 2x^2 - 8x + 10 < 0
\]
Simplify:
\[
-x^2 - 14x - 17 < 0
\]

We can multiply by -1 to make it more straightforward:
\[
x^2 + 14x + 17 > 0
\]

### Step 4: Find the critical points
To solve this inequality, let's find the roots of the quadratic equation:
\[
x^2 + 14x + 17 = 0
\]

Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = 14\), and \(c = 17\):
\[
x = \frac{-14 \pm \sqrt{14^2 - 4(1)(17)}}{2(1)}
\]
\[
x = \frac{-14 \pm \sqrt{196 - 68}}{2}
\]
\[
x = \frac{-14 \pm \sqrt{128}}{2}
\]
\[
x = \frac{-14 \pm 8\sqrt{2}}{2}
\]
\[
x = -7 \pm 4\sqrt{2}
\]

So, the roots are \(x_1 = -7 + 4\sqrt{2}\) and \(x_2 = -7 - 4\sqrt{2}\).

### Step 5: Test the intervals
The critical points split the real line into intervals:
- \( (-\infty, -7 - 4\sqrt{2}) \)
- \( (-7 - 4\sqrt{2}, -7 + 4\sqrt{2}) \)
- \( (-7 + 4\sqrt{2}, \infty) \)

Since the inequality is \( x^2 + 14x + 17 > 0 \), the quadratic expression is positive outside the roots, so the solution is:
\[
x \in (-\infty, -7 - 4\sqrt{2}) \cup (-7 + 4\sqrt{2}, \infty)
\]

### Conclusion:
Matching with the given options:
\[
X \in (-\infty, -7\sqrt{2} - 9) \cup (\sqrt{5} - 7, \infty)
\]

This doesn't exactly match any of the given choices, so it suggests that the correct answer is "Ninguna de las anteriores."

Do you need more details or have any questions about this solution?

### Related Questions:
1. How do you solve inequalities involving quadratic expressions?
2. What are the steps for applying the quadratic formula?
3. How do you determine the intervals where a quadratic inequality holds true?
4. What is the significance of the critical points in solving inequalities?
5. Can you explain how to test intervals for inequalities?
6. How do you handle inequalities that result from a quadratic equation?
7. What happens if the discriminant in the quadratic formula is negative?
8. How do you interpret the solution set of an inequality graphically?

### Tip:
When solving inequalities involving quadratic expressions, carefully analyze the sign of the expression in each interval defined by the critical points to determine where the inequality is satisfied.

Learn how to solve inequalities using the method of critical points with a step-by-step example problem. This tutorial covers key algebraic concepts including inequalities, quadratic equations, and the quadratic formula. Suitable for high school students and learners.

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