To solve the inequality $|3x - 5| \leq 4$, we need to break it down into two cases because the absolute value expression $|A| \leq B$ can be rewritten as two simultaneous inequalities:

$-4 \leq 3x - 5 \leq 4$

Step 1: Solve the two inequalities

We will now split the inequality into two parts and solve for $x$.

First part: $3x - 5 \geq -4$

1. Add 5 to both sides: $3x \geq -4 + 5$ $3x \geq 1$
2. Divide both sides by 3: $x \geq \frac{1}{3}$

Second part: $3x - 5 \leq 4$

1. Add 5 to both sides: $3x \leq 4 + 5$ $3x \leq 9$
2. Divide both sides by 3: $x \leq 3$

Step 2: Combine the two parts

Now, we combine the two results: $\frac{1}{3} \leq x \leq 3$

Step 3: Express the solution in interval notation

The solution is the set of $x$-values that satisfy both inequalities. In interval notation, this is: $\left[ \frac{1}{3}, 3 \right]$

Thus, the solution to the inequality $|3x - 5| \leq 4$ is $\left[ \frac{1}{3}, 3 \right]$.

Would you like any additional details or have further questions?

Here are five related questions:

1. How do you solve an inequality involving a strict inequality $|A| < B$?
2. What changes when solving inequalities with $|A| \geq B$?
3. Can you solve $|3x - 5| \geq 4$ and express the answer in interval notation?
4. How do you solve absolute value equations, e.g., $|3x - 5| = 4$?
5. What happens when an absolute value inequality has no solution?

Tip: When solving absolute value inequalities, always consider breaking it into two parts, one with a positive and one with a negative bound!