The inequality you're working with is:

$|4x - 5| \leq 9$

Steps to solve:

1. Understand the absolute value inequality: The inequality $|A| \leq B$ means that the expression $A$ is between $-B$ and $B$. So, $-B \leq A \leq B$.

   Applying this to your inequality:

   $-9 \leq 4x - 5 \leq 9$

2. Solve the compound inequality:

   To solve this, we'll break it into two parts:

   $-9 \leq 4x - 5 \quad \text{and} \quad 4x - 5 \leq 9$

   Solving the first part:

   $-9 \leq 4x - 5$ Add 5 to both sides:

   $-9 + 5 \leq 4x \quad \Rightarrow \quad -4 \leq 4x$

   Now divide by 4:

   $-1 \leq x$

   Solving the second part:

   $4x - 5 \leq 9$ Add 5 to both sides:

   $4x \leq 14$

   Now divide by 4:

   $x \leq \frac{14}{4} = 3.5$

3. Combine the results:

   The combined solution is:

   $-1 \leq x \leq 3.5$

Final Answer:

• The solution to the inequality is $-1 \leq x \leq 3.5$.

So, the correct choice is B: "The solution is $-1 \leq x \leq 3.5$."

Would you like further clarification or have any questions on this? Here's a set of related questions to help reinforce your understanding:

1. How would the solution change if the inequality was $|4x - 5| > 9$?
2. What if the absolute value was on the right side with a negative number, such as $|4x - 5| \leq -9$?
3. How would you graph the solution $-1 \leq x \leq 3.5$?
4. What are the steps to solve an absolute value inequality where the right side is not zero?
5. How do you check the solution to an absolute value inequality?

Tip: Remember that when solving absolute value inequalities, always break them into two separate inequalities based on the form $|A| \leq B$ or $|A| \geq B$.