Let's solve the compound inequality:

\[
3x + \frac{1}{2} \leq 2x + \frac{5}{3} < \frac{x}{4} + 25
\]

We will solve it by breaking it into two parts:

### 1. Solve \(3x + \frac{1}{2} \leq 2x + \frac{5}{3}\)

Subtract \(2x\) from both sides:

\[
x + \frac{1}{2} \leq \frac{5}{3}
\]

Next, subtract \(\frac{1}{2}\) from both sides:

\[
x \leq \frac{5}{3} - \frac{1}{2}
\]

To subtract these fractions, find a common denominator:

\[
\frac{5}{3} = \frac{10}{6}, \quad \frac{1}{2} = \frac{3}{6}
\]

Now, subtract:

\[
x \leq \frac{10}{6} - \frac{3}{6} = \frac{7}{6}
\]

So, the solution to the first part is:

\[
x \leq \frac{7}{6}
\]

### 2. Solve \(2x + \frac{5}{3} < \frac{x}{4} + 25\)

Subtract \(2x\) from both sides:

\[
\frac{5}{3} < \frac{x}{4} - 2x + 25
\]

Combine the \(x\) terms on the right-hand side:

\[
\frac{5}{3} < \frac{-7x}{4} + 25
\]

Next, subtract 25 from both sides:

\[
\frac{5}{3} - 25 < \frac{-7x}{4}
\]

Convert 25 to a fraction with a denominator of 3:

\[
\frac{5}{3} - \frac{75}{3} < \frac{-7x}{4}
\]

Simplify:

\[
\frac{-70}{3} < \frac{-7x}{4}
\]

Multiply both sides by \(-4/7\) to solve for \(x\) (remember to reverse the inequality sign):

\[
x < \frac{4}{7} \times \frac{70}{3} = \frac{280}{21} = \frac{40}{3} \approx 13.33
\]

### Combined Solution

The solution is the intersection of \(x \leq \frac{7}{6}\) and \(x < \frac{40}{3}\). Since \(x \leq \frac{7}{6}\) is more restrictive:

\[
\boxed{x \leq \frac{7}{6}}
\]

This is the solution to the compound inequality.

Would you like more details or have any questions?

Here are some related questions to further explore:
1. How do you handle inequalities with different denominators?
2. What steps should you take to simplify compound inequalities?
3. How do you interpret the intersection of inequalities?
4. Can you provide an example where inequalities have no intersection?
5. What methods can you use to check your solutions to inequalities?

**Tip:** Always be cautious when subtracting fractions with different denominators. Ensuring the correct common denominator is crucial for accuracy.

Learn how to solve the compound inequality 3x + 1/2 ≤ 2x + 5/3 < x/4 + 25 with a detailed step-by-step solution. This problem involves inequalities and fraction arithmetic, focusing on solving and interpreting compound inequalities. Suitable for high school students (Grades 10-12).

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