To solve the inequality [tex]\( |3X + 6| \geq 9 \)[/tex], we need to address it by considering the nature of absolute values. The absolute value [tex]\( |A| \geq B \)[/tex] for any expression [tex]\( A \)[/tex] and positive number [tex]\( B \)[/tex] translates to two scenarios:
1. [tex]\( A \geq B \)[/tex]
2. [tex]\( A \leq -B \)[/tex]
Let's apply these scenarios to our inequality [tex]\( |3X + 6| \geq 9 \)[/tex].
### Scenario 1: [tex]\( 3X + 6 \geq 9 \)[/tex]
1. Subtract 6 from both sides:
[tex]\[
3X \geq 3
\][/tex]
2. Divide both sides by 3:
[tex]\[
X \geq 1
\][/tex]
### Scenario 2: [tex]\( 3X + 6 \leq -9 \)[/tex]
1. Subtract 6 from both sides:
[tex]\[
3X \leq -15
\][/tex]
2. Divide both sides by 3:
[tex]\[
X \leq -5
\][/tex]
### Combining Both Scenarios
The solutions from both scenarios are:
1. [tex]\( X \geq 1 \)[/tex]
2. [tex]\( X \leq -5 \)[/tex]
So, the complete solution to the inequality [tex]\( |3X + 6| \geq 9 \)[/tex] is:
[tex]\[
X \leq -5 \quad \text{or} \quad X \geq 1
\][/tex]
In interval notation, the solution is:
[tex]\[
(-\infty, -5] \cup [1, \infty)
\][/tex]
