To solve the inequality [tex]\(3x < -9\)[/tex], let's go through the steps systematically.
### Step 1: Isolate the variable [tex]\(x\)[/tex]
We start with the inequality:
[tex]\[ 3x < -9 \][/tex]
To isolate [tex]\(x\)[/tex], we need to divide both sides of the inequality by 3:
[tex]\[ \frac{3x}{3} < \frac{-9}{3} \][/tex]
Simplifying this, we get:
[tex]\[ x < -3 \][/tex]
### Step 2: Represent the solution set on a number line
The inequality [tex]\(x < -3\)[/tex] means that [tex]\(x\)[/tex] can be any number less than [tex]\(-3\)[/tex].
### Step 3: Number Line Representation
To represent this on a number line:
1. Draw a horizontal line, which represents all possible values of [tex]\(x\)[/tex].
2. Place a point or an open circle on [tex]\(-3\)[/tex]. An open circle is used to show that [tex]\(-3\)[/tex] is not included in the solution set (since the inequality is strictly [tex]\(<\)[/tex], not [tex]\(\le\)[/tex]).
3. Shade the line to the left of [tex]\(-3\)[/tex] to indicate all the numbers that are less than [tex]\(-3\)[/tex].
Here is the graphical representation:
```
<---(O)----->
-3
```
- The open circle at [tex]\(-3\)[/tex] indicates that [tex]\(-3\)[/tex] is not included in the solution ([tex]\(x \neq -3\)[/tex]).
- The shading to the left indicates all values less than [tex]\(-3\)[/tex].
### Summary
The number line representing the solution set for the inequality [tex]\(3x < -9\)[/tex] includes all numbers less than [tex]\(-3\)[/tex], but not [tex]\(-3\)[/tex] itself.
