Let's solve the inequality \( \frac{5x + 2}{6x - 1} > 3 \).

### Step 1: Eliminate the denominator by multiplying both sides by \(6x - 1\)
Assuming \(6x - 1 \neq 0\), we multiply both sides by \(6x - 1\):

\[
\frac{5x + 2}{6x - 1} \cdot (6x - 1) > 3 \cdot (6x - 1)
\]

This simplifies to:

\[
5x + 2 > 3(6x - 1)
\]

### Step 2: Expand and simplify
Now expand the right-hand side:

\[
5x + 2 > 18x - 3
\]

### Step 3: Rearrange the terms
Subtract \(5x\) from both sides:

\[
2 > 13x - 3
\]

Add \(3\) to both sides:

\[
5 > 13x
\]

### Step 4: Solve for \(x\)
Divide both sides by 13:

\[
x < \frac{5}{13}
\]

### Step 5: Consider the domain restrictions
The denominator \(6x - 1\) cannot be zero, so:

\[
6x - 1 = 0 \implies x = \frac{1}{6}
\]

Thus, \(x \neq \frac{1}{6}\).

### Final Answer:
The solution is \( x < \frac{5}{13} \) and \( x \neq \frac{1}{6} \).

Would you like further details or clarification?

### Related Questions:
1. How would the solution change if the inequality were \( \frac{5x + 2}{6x - 1} < 3 \)?
2. What happens if the expression had absolute values, i.e. \( \left| \frac{5x + 2}{6x - 1} \right| > 3 \)?
3. Can you solve a similar inequality with quadratic terms in the denominator?
4. How does multiplying both sides of an inequality by a negative number affect the solution?
5. What is the graphical interpretation of the solution set for this inequality?

### Tip:
Always check for restrictions in the domain when solving rational inequalities, such as values that make the denominator zero.

Learn how to solve the rational inequality 5x + 2 / 6x - 1 > 3 with a clear, step-by-step solution. This guide covers key concepts in inequalities and rational expressions, including domain restrictions and solving inequalities involving fractions. Suitable for students in Grades 9-11.

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