The inequality you're dealing with is:

\[
\frac{1}{3x - 2} \leq 4
\]

Let's solve it step by step.

### Step 1: Isolate the fraction
Start by multiplying both sides of the inequality by \(3x - 2\), but you must be careful: if \(3x - 2\) is positive, the inequality stays the same; if \(3x - 2\) is negative, the inequality flips direction.

We need to solve two cases: 
1. \( 3x - 2 > 0 \)
2. \( 3x - 2 < 0 \)

#### Case 1: \( 3x - 2 > 0 \) 
For \( 3x - 2 \) to be positive:
\[
3x - 2 > 0 \implies x > \frac{2}{3}
\]

Now, solving the inequality for \(x > \frac{2}{3}\):

\[
\frac{1}{3x - 2} \leq 4
\]

Multiply both sides by \(3x - 2\):

\[
1 \leq 4(3x - 2)
\]

Simplify:

\[
1 \leq 12x - 8
\]

Add 8 to both sides:

\[
9 \leq 12x
\]

Divide by 12:

\[
x \geq \frac{9}{12} = \frac{3}{4}
\]

For this case, the solution is \( x \geq \frac{3}{4} \) and \( x > \frac{2}{3} \). Therefore, the solution for this case is \( x \geq \frac{3}{4} \).

#### Case 2: \( 3x - 2 < 0 \)
For \( 3x - 2 \) to be negative:

\[
3x - 2 < 0 \implies x < \frac{2}{3}
\]

In this case, we need to solve the inequality:

\[
\frac{1}{3x - 2} \leq 4
\]

Multiplying both sides by \(3x - 2\) (note that since \(3x - 2\) is negative, we flip the inequality):

\[
1 \geq 4(3x - 2)
\]

Simplify:

\[
1 \geq 12x - 8
\]

Add 8 to both sides:

\[
9 \geq 12x
\]

Divide by 12:

\[
x \leq \frac{9}{12} = \frac{3}{4}
\]

For this case, \( x \leq \frac{3}{4} \) and \( x < \frac{2}{3} \). Therefore, the solution for this case is \( x < \frac{2}{3} \).

### Final Solution:
The solution to the inequality is:
\[
x \in (-\infty, \frac{2}{3}) \cup \left[ \frac{3}{4}, \infty \right)
\]

Do you have any questions or want further details?

### Related questions:
1. How do you solve rational inequalities in general?
2. Can this method be applied to quadratic inequalities?
3. What are the key differences between strict and non-strict inequalities?
4. How do you find the domain of rational functions?
5. What happens if the fraction involves more than one variable?

**Tip:** Always check for restrictions in inequalities involving fractions to avoid division by zero!

Learn how to solve the rational inequality 1/(3x - 2) ≤ 4 with detailed steps. This solution covers key concepts in algebra, rational functions, and inequalities, including the handling of rational expressions and inequality flipping. Suitable for students in Grades 9-11.

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