Certainly! Let's solve the inequality step by step.
We need to solve the inequality:
[tex]\[ |x + 8| - 3 \geq 6 \][/tex]
### Step 1: Isolate the Absolute Value
First, isolate the absolute value expression on one side of the inequality:
[tex]\[ |x + 8| - 3 \geq 6 \][/tex]
Add 3 to both sides:
[tex]\[ |x + 8| \geq 9 \][/tex]
### Step 2: Split into Two Cases
The absolute value inequality [tex]\( |x + 8| \geq 9 \)[/tex] can be split into two cases because the absolute value of a number is either positive or negative:
1. [tex]\( x + 8 \geq 9 \)[/tex]
2. [tex]\( x + 8 \leq -9 \)[/tex]
Let's solve these two inequalities separately.
#### Case 1: [tex]\( x + 8 \geq 9 \)[/tex]
Subtract 8 from both sides:
[tex]\[ x \geq 1 \][/tex]
#### Case 2: [tex]\( x + 8 \leq -9 \)[/tex]
Subtract 8 from both sides:
[tex]\[ x \leq -17 \][/tex]
### Step 3: Combine the Solutions
Our solutions are from Case 1 and Case 2:
[tex]\[ x \geq 1 \quad \text{or} \quad x \leq -17 \][/tex]
### Final Answer
The solution to the inequality [tex]\(|x + 8| - 3 \geq 6 \)[/tex] is:
[tex]\[ x \leq -17 \quad \text{or} \quad x \geq 1 \][/tex]
Therefore, [tex]\( x \leq [?] \text{ or } x \geq \)[/tex] translates to:
[tex]\[ x \leq -17 \quad \text{or} \quad x \geq 1 \][/tex]
