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The image contains a set of inequalities involving absolute values. I'll go through solving a few of them step by step. Let's start with problems 8 and 9.

### Problem 8: \(|2 - x| + |3 - 2x| \leq 3\)

To solve this, we need to consider different cases based on the expressions inside the absolute values.

1. **Case 1**: \(x \geq 2\)
   - \(2 - x \leq 0\), so \(|2 - x| = x - 2\)
   - \(3 - 2x \leq 0\), so \(|3 - 2x| = 2x - 3\)
   - The inequality becomes: \((x - 2) + (2x - 3) \leq 3\)
   - Simplifying: \(3x - 5 \leq 3\)
   - \(3x \leq 8\) → \(x \leq \frac{8}{3}\)

   Since we assumed \(x \geq 2\), the solution in this case is \(2 \leq x \leq \frac{8}{3}\).

2. **Case 2**: \( \frac{3}{2} \leq x < 2\)
   - \(2 - x > 0\), so \(|2 - x| = 2 - x\)
   - \(3 - 2x \leq 0\), so \(|3 - 2x| = 2x - 3\)
   - The inequality becomes: \((2 - x) + (2x - 3) \leq 3\)
   - Simplifying: \(x - 1 \leq 3\)
   - \(x \leq 4\)

   Since we assumed \( \frac{3}{2} \leq x < 2\), the solution in this case is \( \frac{3}{2} \leq x < 2\).

3. **Case 3**: \(x < \frac{3}{2}\)
   - \(2 - x > 0\), so \(|2 - x| = 2 - x\)
   - \(3 - 2x > 0\), so \(|3 - 2x| = 3 - 2x\)
   - The inequality becomes: \((2 - x) + (3 - 2x) \leq 3\)
   - Simplifying: \(-3x + 5 \leq 3\)
   - \(-3x \leq -2\) → \(x \geq \frac{2}{3}\)

   Since we assumed \(x < \frac{3}{2}\), the solution in this case is \(\frac{2}{3} \leq x < \frac{3}{2}\).

Combining all cases, the solution is \(\frac{2}{3} \leq x \leq \frac{8}{3}\).

### Problem 9: \(2x + 3 \geq 4x + 5\)

To solve this linear inequality:

1. Subtract \(2x\) from both sides:
   \[
   3 \geq 2x + 5
   \]

2. Subtract 5 from both sides:
   \[
   -2 \geq 2x
   \]

3. Divide both sides by 2:
   \[
   x \leq -1
   \]

So the solution for Problem 9 is \(x \leq -1\).

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Would you like me to continue solving the other inequalities or provide further details on these? Here are some related questions:

1. How do we solve absolute value inequalities with more complex expressions?
2. What are the general steps to solve linear inequalities?
3. How can we graph the solutions to these inequalities on a number line?
4. What methods can be used to check the solutions of these inequalities?
5. How would these problems change if they were equations instead of inequalities?

**Tip:** When solving absolute value inequalities, always consider the different cases based on the critical points where the expressions inside the absolute values equal zero.

Solve the given set of inequalities involving absolute values, fractions, and rational expressions.

Learn how to solve complex inequalities involving absolute values, rational expressions, and linear terms. This tutorial covers multiple examples, including steps for solving each inequality and simplifying rational expressions.

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