To solve the inequality $-3 + |x - 4| \leq -1$, let's go through it step by step.

Step 1: Isolate the absolute value expression

Add 3 to both sides of the inequality:

$|x - 4| \leq -1 + 3$

$|x - 4| \leq 2$

Step 2: Solve the absolute value inequality

For the absolute value inequality $|x - 4| \leq 2$, we split it into two cases:

$-2 \leq x - 4 \leq 2$

Step 3: Solve the compound inequality

Now, solve for $x$ by adding 4 to each part of the inequality:

$-2 + 4 \leq x \leq 2 + 4$

$2 \leq x \leq 6$

Final Answer:

The solution to the inequality is:

$2 \leq x \leq 6$

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you solve inequalities involving absolute values greater than a constant?
2. What are the key steps for solving compound inequalities?
3. How would the solution change if the inequality sign were $\geq$ instead of $\leq$?
4. Can you graph the solution to $2 \leq x \leq 6$?
5. How would you solve $|x + 1| - 2 \geq 5$?

Tip: Always remember to isolate the absolute value term before solving.