The inequality in the image is:

$-5 \leq \frac{6x - 5}{2} \leq 4$

Let's solve this step by step.

Step 1: Break the inequality into two parts:

$-5 \leq \frac{6x - 5}{2}$ and $\frac{6x - 5}{2} \leq 4$

Step 2: Solve each part of the inequality.

For $-5 \leq \frac{6x - 5}{2}$:

1. Multiply both sides by 2 to eliminate the denominator: $-5 \times 2 \leq 6x - 5 \quad \Rightarrow \quad -10 \leq 6x - 5$
2. Add 5 to both sides: $-10 + 5 \leq 6x \quad \Rightarrow \quad -5 \leq 6x$
3. Divide both sides by 6: $\frac{-5}{6} \leq x$ So, $x \geq \frac{-5}{6}$.

For $\frac{6x - 5}{2} \leq 4$:

1. Multiply both sides by 2: $6x - 5 \leq 4 \times 2 \quad \Rightarrow \quad 6x - 5 \leq 8$
2. Add 5 to both sides: $6x \leq 8 + 5 \quad \Rightarrow \quad 6x \leq 13$
3. Divide by 6: $x \leq \frac{13}{6}$

Step 3: Combine the two inequalities.

From both parts, we have: $\frac{-5}{6} \leq x \leq \frac{13}{6}$

Step 4: Write the solution in interval notation.

The solution set is: $\left[ \frac{-5}{6}, \frac{13}{6} \right]$

Thus, the correct choice is A, with the solution set $\left[ \frac{-5}{6}, \frac{13}{6} \right]$.

Would you like further details or assistance with any step? Here are five related questions to consider:

1. How do you graph inequalities on a number line?
2. Can you solve compound inequalities with different types of inequalities (e.g., strict vs. non-strict)?
3. What happens when you multiply or divide inequalities by negative numbers?
4. How can inequalities be used to represent real-world problems?
5. What is the difference between set notation and interval notation?

Tip: When solving inequalities, always be careful when multiplying or dividing by a negative number—this reverses the inequality sign!