The inequality to solve is:

$|5t - 6| \leq 4$

Step 1: Break the absolute value inequality into two cases.

By definition of absolute value, the inequality $|A| \leq B$ translates into:

$-B \leq A \leq B$

In this case, the expression inside the absolute value is $5t - 6$, and $B = 4$. So, the inequality becomes:

$-4 \leq 5t - 6 \leq 4$

Step 2: Solve the compound inequality.

First, solve the left side:

$-4 \leq 5t - 6$

Add 6 to both sides:

$2 \leq 5t$

Now, divide by 5:

$\frac{2}{5} \leq t$

Next, solve the right side:

$5t - 6 \leq 4$

Add 6 to both sides:

$5t \leq 10$

Now, divide by 5:

$t \leq 2$

Step 3: Combine the solutions.

The solution set is:

$\frac{2}{5} \leq t \leq 2$

Step 4: Write the solution in interval notation.

In interval notation, the solution is:

$\left[ \frac{2}{5}, 2 \right]$

Therefore, the correct choice is A, with the solution set being $\left[ \frac{2}{5}, 2 \right]$.

Would you like more details or have any questions?

Here are 5 related questions to further explore:

1. How would the solution change if the inequality was $|5t - 6| > 4$?
2. How can you graph this solution set on a number line?
3. What is the meaning of "absolute value" geometrically?
4. How would you solve $|5t - 6| \geq 4$ as a compound inequality?
5. Can you generalize the process for any inequality involving absolute values?

Tip: Always double-check your compound inequality by solving both sides separately before combining the results.