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The image contains three inequalities to solve. Here's a detailed step-by-step solution for each inequality:

### 1. Inequality: \(\frac{x}{x+1} \geq 2\)

**Solution:**

1. Rewrite the inequality:
   \[
   \frac{x}{x+1} - 2 \geq 0
   \]

2. Combine into a single fraction:
   \[
   \frac{x - 2(x + 1)}{x + 1} \geq 0
   \]

3. Simplify the numerator:
   \[
   \frac{x - 2x - 2}{x + 1} \geq 0 \implies \frac{-x - 2}{x + 1} \geq 0
   \]

4. Find the critical points by setting the numerator and denominator to zero:
   - Numerator: \(-x - 2 = 0 \implies x = -2\)
   - Denominator: \(x + 1 = 0 \implies x = -1\)

5. Determine the sign intervals:
   - Test intervals: \(x < -2\), \(-2 < x < -1\), \(x > -1\).
   - Check signs in each interval.

6. Final answer: \(x \in (-\infty, -2] \cup (-1, \infty)\).

### 2. Inequality: \(\frac{3}{2x + 3} < \frac{1}{x - 2}\)

**Solution:**

1. Cross-multiply, considering the restrictions \(2x + 3 \neq 0\) and \(x - 2 \neq 0\).
   
   \[
   3(x - 2) < (2x + 3) \cdot 1
   \]

2. Simplify the inequality:
   
   \[
   3x - 6 < 2x + 3
   \]

3. Rearrange terms:
   
   \[
   x < 9
   \]

4. Check the valid range:
   - Restrictions: \(x \neq -\frac{3}{2}\) and \(x \neq 2\).
   
5. Final answer: \(x \in (-\frac{3}{2}, 2) \cup (2, 9)\).

### 3. Inequality: \(\frac{(5 - x)(x^2 + 2)}{(x - 3)(x + 3)} \geq 0\)

**Solution:**

1. Identify critical points:
   - Numerator: \(5 - x = 0 \implies x = 5\).
   - Denominator: \(x - 3 = 0 \implies x = 3\) and \(x + 3 = 0 \implies x = -3\).

2. Test sign changes around the critical points.

3. Final answer:
   - Check intervals and signs considering restrictions.

### Do you want a detailed explanation for any specific part of these solutions?

### Relative Questions:
1. What are the common methods to solve rational inequalities?
2. How do you determine sign changes around critical points in inequalities?
3. What are the steps to check for restrictions in rational inequalities?
4. Can you explain the importance of testing intervals for inequality solutions?
5. How do cross-multiplication affect inequalities involving fractions?

**Tip:** Always check for restrictions on the domain when solving inequalities with fractions!

Solve the inequalities: 
1) x/(x+1) ≥ 2 
2) 3/(2x+3) < 1/(x-2) 
3) ((5-x)(x²+2))/((x-3)(x+3)) ≥ 0

Learn how to solve complex rational inequalities such as x/(x+1) ≥ 2, 3/(2x+3) < 1/(x-2), and ((5-x)(x²+2))/((x-3)(x+3)) ≥ 0. This guide explains the process, including combining fractions, testing intervals, and identifying critical points. Suitable for students in Grades 10-12.

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