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.