To solve the equation \( \frac{2}{x} + 3 = \frac{3}{4x} + \frac{25}{8} \), we'll follow these steps:

1. **Isolate the terms with \( x \)** on one side.
2. **Eliminate the fractions** by finding a common denominator.
3. **Solve the resulting equation** for \( x \).

### Step 1: Isolate the terms with \( x \)
We start by subtracting \( \frac{3}{4x} \) from both sides:

\[
\frac{2}{x} - \frac{3}{4x} = \frac{25}{8} - 3
\]

### Step 2: Simplify the equation

To subtract the fractions on the left side, we need a common denominator. The common denominator of \( x \) and \( 4x \) is \( 4x \). So, we rewrite the fractions:

\[
\frac{8}{4x} - \frac{3}{4x} = \frac{25}{8} - \frac{24}{8}
\]

This simplifies to:

\[
\frac{5}{4x} = \frac{1}{8}
\]

### Step 3: Cross-multiply and solve for \( x \)

Cross-multiply to get rid of the fractions:

\[
5 \times 8 = 4x \times 1
\]

Simplifying:

\[
40 = 4x
\]

Divide both sides by 4:

\[
x = 10
\]

### Final Answer
The solution is \( x = 10 \).

---

Would you like more details or have any questions?

Here are five related problems you can try:

1. Solve \( \frac{4}{x} + 2 = \frac{5}{2x} + 7 \).
2. Solve \( \frac{3}{x} - 1 = \frac{2}{5x} + \frac{6}{7} \).
3. Solve \( \frac{5}{2x} + 4 = \frac{7}{3x} + \frac{9}{2} \).
4. Solve \( \frac{8}{x} + 1 = \frac{6}{3x} + \frac{15}{4} \).
5. Solve \( \frac{1}{x} + 2 = \frac{2}{4x} + \frac{3}{5} \).

**Tip:** Always check for extraneous solutions when dealing with rational equations by substituting back into the original equation.

Learn how to solve the rational equation 2/x + 3 = 3/4x + 25/8 step-by-step with a detailed solution. This problem involves equations with fractions and requires cross-multiplication to eliminate fractions and solve for x. Suitable for high school students and advanced learners.

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