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The inequality given is:

\[
|x + 8| \leq 2
\]

To solve this absolute value inequality, we follow these steps:

### Step 1: Break down the absolute value inequality
For any expression of the form \( |A| \leq B \), the inequality can be rewritten as:

\[
-B \leq A \leq B
\]

In our case, \( A = x + 8 \) and \( B = 2 \). So, we rewrite the inequality as:

\[
-2 \leq x + 8 \leq 2
\]

### Step 2: Solve the compound inequality
Now, we solve for \( x \) by isolating it in the middle:

1. Subtract 8 from all parts of the inequality:

\[
-2 - 8 \leq x + 8 - 8 \leq 2 - 8
\]

This simplifies to:

\[
-10 \leq x \leq -6
\]

### Step 3: Graph the solution
The solution is all \( x \)-values between \( -10 \) and \( -6 \), inclusive. On a number line, this is represented by a solid line from \( -10 \) to \( -6 \), with filled circles at \( -10 \) and \( -6 \) to indicate that these values are included (since the inequality is \( \leq \) and not strict \( < \)).

Would you like further details or have any questions?

### Related Questions:
1. How do you solve absolute value inequalities with \( > \) instead of \( \leq \)?
2. Can you solve a similar inequality like \( |x - 5| \geq 4 \)?
3. What happens when the absolute value inequality includes a negative number on the right-hand side, such as \( |x + 3| \leq -1 \)?
4. Can this technique be applied to quadratic inequalities?
5. How do you graph inequalities when the solutions are not within a single interval?

### Tip:
When solving absolute value inequalities, remember to check if the inequality is possible. For example, \( |x| \leq -3 \) would have no solution because absolute values are always non-negative.

Graph the solution to the inequality |x + 8| ≤ 2 on the number line.

Learn how to graph the absolute value inequality |x + 8| ≤ 2 on a number line. This problem involves breaking down the inequality into a compound inequality and solving it to find the solution set. Suitable for students in Grades 7-9.

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