Of course! Let's solve the inequality [tex]\( |3x| \geq 18 \)[/tex] step by step.
### Step 1: Understanding Absolute Value
The absolute value notation [tex]\( |3x| \)[/tex] means that [tex]\( 3x \)[/tex] can either be positive or negative. Specifically:
[tex]\[ |3x| \geq 18 \][/tex]
means:
[tex]\[ 3x \geq 18 \quad \text{or} \quad 3x \leq -18 \][/tex]
We need to solve these two separate inequalities.
### Step 2: Solve the First Inequality
Let's start with the first inequality:
[tex]\[ 3x \geq 18 \][/tex]
To isolate [tex]\( x \)[/tex], divide both sides of the inequality by 3:
[tex]\[ x \geq \frac{18}{3} \][/tex]
[tex]\[ x \geq 6 \][/tex]
### Step 3: Solve the Second Inequality
Now, let's solve the second inequality:
[tex]\[ 3x \leq -18 \][/tex]
Similarly, divide both sides by 3:
[tex]\[ x \leq \frac{-18}{3} \][/tex]
[tex]\[ x \leq -6 \][/tex]
### Step 4: Combine the Solutions
The solutions for the inequality [tex]\( |3x| \geq 18 \)[/tex] are:
[tex]\[ x \geq 6 \][/tex]
or
[tex]\[ x \leq -6 \][/tex]
### Conclusion
We have two distinct intervals as solutions:
[tex]\[ x \geq 6 \][/tex]
and
[tex]\[ x \leq -6 \][/tex]
So the final solution for the inequality [tex]\( |3x| \geq 18 \)[/tex] is [tex]\( x \in (-\infty, -6] \cup [6, \infty) \)[/tex]. Therefore, the numerical critical points in the solution are:
[tex]\[ x = -6 \][/tex]
and
[tex]\[ x = 6 \][/tex]
These represent the boundary points where the inequality changes its truth value.
