Let's solve each equation one by one.

1. $\frac{3x}{2} = \frac{2}{5}$

We can solve this proportion by cross-multiplying: $3x \times 5 = 2 \times 2$ $15x = 4$ $x = \frac{4}{15}$

2. $\frac{4}{7} = \frac{2x}{3}$

Again, solve by cross-multiplying: $4 \times 3 = 7 \times 2x$ $12 = 14x$ $x = \frac{12}{14} = \frac{6}{7}$

3. $\frac{4x + 2}{18} = \frac{6x + 3}{17}$

Cross-multiplying: $(4x + 2) \times 17 = (6x + 3) \times 18$ Expanding both sides: $68x + 34 = 108x + 54$ Simplifying: $68x - 108x = 54 - 34$ $-40x = 20$ $x = \frac{20}{-40} = -\frac{1}{2}$

6. $\frac{x - 4}{6} = \frac{5}{1}$

Cross-multiplying: $(x - 4) \times 1 = 5 \times 6$ $x - 4 = 30$ $x = 30 + 4 = 34$

7. $\frac{x}{3} = \frac{2x - 1}{5}$

Cross-multiplying: $x \times 5 = (2x - 1) \times 3$ Expanding both sides: $5x = 6x - 3$ Simplifying: $5x - 6x = -3$ $-x = -3$ $x = 3$

8. $\frac{2x + 1}{18} = \frac{4x - 3}{21}$

Cross-multiplying: $(2x + 1) \times 21 = (4x - 3) \times 18$ Expanding both sides: $42x + 21 = 72x - 54$ Simplifying: $42x - 72x = -54 - 21$ $-30x = -75$ $x = \frac{75}{30} = \frac{5}{2}$

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Final solutions:

1. $x = \frac{4}{15}$
2. $x = \frac{6}{7}$
3. $x = -\frac{1}{2}$
4. $x = 34$
5. $x = 3$
6. $x = \frac{5}{2}$

Would you like to go into more detail on any of these solutions?

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Related questions:

1. How do you solve equations with ratios and proportions?
2. What is cross-multiplication and when should you use it?
3. Can you explain more about simplifying equations?
4. How do negative and positive solutions change the interpretation of a problem?
5. What are common mistakes to avoid in solving proportional equations?

Tip: Always check your solutions by substituting them back into the original equation!