To solve the absolute value inequality [tex]\(3|z + 3| \geq 9\)[/tex], let's go through the problem step-by-step.
### Step 1: Isolate the absolute value expression
First, we will isolate the absolute value term on one side of the inequality:
[tex]\[ 3 |z + 3| \geq 9 \][/tex]
Divide both sides by 3 to simplify:
[tex]\[ |z + 3| \geq 3 \][/tex]
### Step 2: Break down the absolute value inequality
We need to consider the definition of absolute value to break this into two separate inequalities. By definition, [tex]\(|A| \geq B\)[/tex] implies:
[tex]\[ A \geq B \quad \text{or} \quad A \leq -B \][/tex]
Here, [tex]\(A\)[/tex] is [tex]\(z + 3\)[/tex] and [tex]\(B\)[/tex] is 3. Therefore, the inequality [tex]\(|z + 3| \geq 3\)[/tex] translates to:
[tex]\[ z + 3 \geq 3 \quad \text{or} \quad z + 3 \leq -3 \][/tex]
### Step 3: Solve each inequality separately
#### For [tex]\(z + 3 \geq 3\)[/tex]:
Subtract 3 from both sides:
[tex]\[ z \geq 0 \][/tex]
#### For [tex]\(z + 3 \leq -3\)[/tex]:
Subtract 3 from both sides:
[tex]\[ z \leq -6 \][/tex]
### Step 4: Combine the solutions
Combining both parts of our solution, we have:
[tex]\[ z \geq 0 \quad \text{or} \quad z \leq -6 \][/tex]
### Step 5: Express the solution in interval notation
To write the solution in interval notation, we combine the inequalities into intervals:
[tex]\[ z \in (-\infty, -6] \cup [0, \infty) \][/tex]
So, the solution to the inequality [tex]\(3|z + 3| \geq 9\)[/tex] expressed in interval notation is:
[tex]\[
(-\infty, -6] \cup [0, \infty)
\][/tex]
This represents all values of [tex]\(z\)[/tex] that satisfy the given inequality.
