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Let's solve the given rational inequality:

\[
\frac{4x + 5}{3x - 1} > 4
\]

### Step-by-step Solution:

1. **Move all terms to one side**:
   To simplify the inequality, let's subtract 4 from both sides:

   \[
   \frac{4x + 5}{3x - 1} - 4 > 0
   \]

2. **Express the left side with a common denominator**:
   The expression can be written with a common denominator:

   \[
   \frac{4x + 5}{3x - 1} - \frac{4(3x - 1)}{3x - 1} > 0
   \]

   Simplifying the second fraction:

   \[
   \frac{4x + 5 - 12x + 4}{3x - 1} > 0
   \]

   This simplifies to:

   \[
   \frac{-8x + 9}{3x - 1} > 0
   \]

3. **Find critical points**:
   The critical points occur where the numerator or denominator is zero. Let's find these:

   - Numerator: \( -8x + 9 = 0 \)  
     Solving for \(x\):  
     \[
     x = \frac{9}{8}
     \]

   - Denominator: \( 3x - 1 = 0 \)  
     Solving for \(x\):  
     \[
     x = \frac{1}{3}
     \]

4. **Determine sign changes**:
   We now test the sign of the expression \( \frac{-8x + 9}{3x - 1} \) in the intervals determined by the critical points \( x = \frac{1}{3} \) and \( x = \frac{9}{8} \).

   The intervals are:
   - \( (-\infty, \frac{1}{3}) \)
   - \( (\frac{1}{3}, \frac{9}{8}) \)
   - \( (\frac{9}{8}, \infty) \)

   We can test points in each interval to determine where the expression is positive.

   - For \( x = 0 \) (in \( (-\infty, \frac{1}{3}) \)):
     \[
     \frac{-8(0) + 9}{3(0) - 1} = \frac{9}{-1} = -9 \quad \text{(negative)}
     \]

   - For \( x = \frac{1}{2} \) (in \( (\frac{1}{3}, \frac{9}{8}) \)):
     \[
     \frac{-8\left(\frac{1}{2}\right) + 9}{3\left(\frac{1}{2}\right) - 1} = \frac{-4 + 9}{1.5 - 1} = \frac{5}{0.5} = 10 \quad \text{(positive)}
     \]

   - For \( x = 2 \) (in \( (\frac{9}{8}, \infty) \)):
     \[
     \frac{-8(2) + 9}{3(2) - 1} = \frac{-16 + 9}{6 - 1} = \frac{-7}{5} \quad \text{(negative)}
     \]

5. **Solution in interval notation**:
   The solution is where the expression is positive, which occurs in the interval \( \left( \frac{1}{3}, \frac{9}{8} \right) \).

Thus, the solution is:

\[
\left( \frac{1}{3}, \frac{9}{8} \right)
\]

Would you like a detailed explanation of any specific step? 

Here are 5 related questions to explore further:

1. How do we find common denominators in rational expressions?
2. Why do we test values in different intervals to determine signs?
3. What does the critical point of a rational inequality signify?
4. Can this method be applied to solve other types of inequalities?
5. What is the importance of excluding points where the denominator is zero?

**Tip**: Always check for points that make the denominator zero since they are not part of the solution set!

Solve the following rational inequality: (4x + 5) / (3x - 1) > 4

Learn how to solve the rational inequality (4x + 5) / (3x - 1) > 4 using algebraic methods. This step-by-step solution includes common denominator techniques and critical point analysis. Suitable for high school students in Grades 10-12.

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