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.

Understand how to solve the absolute value inequality -3 + |x - 4| <= -1 step by step. This guide explains key algebraic concepts, including isolating the absolute value term, solving compound inequalities, and interpreting the solution. Suitable for Grades 8-10.

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