To solve the inequality:

\[
\frac{5}{3t-1} < 2
\]

### Step 1: Eliminate the denominator by multiplying both sides by \( 3t - 1 \).
Since the sign of \( 3t - 1 \) can change depending on the value of \( t \), we need to consider two cases.

### Case 1: \( 3t - 1 > 0 \) (i.e., \( t > \frac{1}{3} \))

In this case, we can multiply both sides by \( 3t - 1 \) without changing the inequality sign:

\[
5 < 2(3t - 1)
\]

Expanding the right-hand side:

\[
5 < 6t - 2
\]

Now, solve for \( t \):

\[
5 + 2 < 6t \implies 7 < 6t
\]

\[
t > \frac{7}{6}
\]

So, for \( t > \frac{1}{3} \), the solution is \( t > \frac{7}{6} \).

### Case 2: \( 3t - 1 < 0 \) (i.e., \( t < \frac{1}{3} \))

In this case, we need to reverse the inequality sign when multiplying by \( 3t - 1 \):

\[
5 > 2(3t - 1)
\]

Expanding the right-hand side:

\[
5 > 6t - 2
\]

Now, solve for \( t \):

\[
5 + 2 > 6t \implies 7 > 6t
\]

\[
t < \frac{7}{6}
\]

Thus, for \( t < \frac{1}{3} \), the solution is \( t < \frac{7}{6} \).

### Final Solution:
Combining the two cases, the solution is:

\[
t < \frac{7}{6}
\]

Thus, the inequality holds for \( t < \frac{7}{6} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve similar rational inequalities?
2. What are the rules for multiplying or dividing inequalities by negative numbers?
3. What happens when the denominator of a fraction includes variables?
4. How can you check solutions of inequalities on a number line?
5. How do you solve compound inequalities?

**Tip:** Always check if the expression has restrictions, like dividing by zero, when dealing with rational inequalities!

This guide explains how to solve the rational inequality 5/(3t-1) < 2 step-by-step. Explore key algebraic concepts such as rational inequalities and sign change when multiplying by negative values. Suitable for Grades 9-11 students.

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