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To solve the inequality \( |x - 4| \leq 2 \):

1. **Understand the inequality**: The absolute value inequality \( |x - 4| \leq 2 \) means that the expression \( x - 4 \) is within 2 units of 0. 

2. **Remove the absolute value**: This can be rewritten as two separate inequalities:
   \[
   -2 \leq x - 4 \leq 2
   \]

3. **Solve the compound inequality**:
   - First, solve \( -2 \leq x - 4 \):
     \[
     -2 \leq x - 4
     \]
     Add 4 to both sides:
     \[
     2 \leq x
     \]
   - Next, solve \( x - 4 \leq 2 \):
     \[
     x - 4 \leq 2
     \]
     Add 4 to both sides:
     \[
     x \leq 6
     \]

4. **Combine the solutions**: The solution to the inequality \( |x - 4| \leq 2 \) is:
   \[
   2 \leq x \leq 6
   \]

So, the solution to the inequality \( |x - 4| \leq 2 \) is \( 2 \leq x \leq 6 \).

Would you like any details or have any questions?

Here are some related questions to expand your understanding:

1. How do you solve an absolute value inequality where the inequality symbol is \(>\) instead of \( \leq \)?
2. What is the geometric interpretation of the solution to \( |x - 4| \leq 2 \)?
3. Can you solve the inequality \( |2x + 3| < 5 \) using a similar method?
4. How does the solution change if the inequality is \( |x - 4| \geq 2 \)?
5. What is the effect of adding a constant to the variable inside the absolute value on the solution set?

**Tip:** When solving absolute value inequalities, remember that the absolute value \( |A| \leq B \) implies \( -B \leq A \leq B \), and \( |A| \geq B \) implies \( A \leq -B \) or \( A \geq B \).

Learn how to solve the absolute value inequality |x - 4| ≤ 2 with step-by-step instructions. This problem covers key algebraic concepts, including absolute value inequalities and how to express them as compound inequalities. Suitable for students in Grades 7-9.

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