To solve the equation [tex]\( |3x + 12| = 6 \)[/tex], we need to consider two separate cases because the absolute value can be either positive or negative.
### Case 1: [tex]\( 3x + 12 = 6 \)[/tex]
1. Start with the equation:
[tex]\[ 3x + 12 = 6 \][/tex]
2. Isolate the term involving [tex]\( x \)[/tex] by subtracting 12 from both sides:
[tex]\[ 3x = 6 - 12 \][/tex]
[tex]\[ 3x = -6 \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 3:
[tex]\[ x = \frac{-6}{3} \][/tex]
[tex]\[ x = -2 \][/tex]
### Case 2: [tex]\( 3x + 12 = -6 \)[/tex]
1. Start with the equation:
[tex]\[ 3x + 12 = -6 \][/tex]
2. Isolate the term involving [tex]\( x \)[/tex] by subtracting 12 from both sides:
[tex]\[ 3x = -6 - 12 \][/tex]
[tex]\[ 3x = -18 \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 3:
[tex]\[ x = \frac{-18}{3} \][/tex]
[tex]\[ x = -6 \][/tex]
So the solutions to the equation [tex]\( |3x + 12| = 6 \)[/tex] are:
[tex]\[ x = -2 \text{ or } x = -6 \][/tex]
### Solutions
[tex]\[
x = -2 \quad \text{or} \quad x = -6
\][/tex]
### Graphing the solutions on the number line
To graph the solutions, we will plot points at [tex]\( x = -2 \)[/tex] and [tex]\( x = -6 \)[/tex] on the number line.
1. Plot a point at [tex]\( x = -2 \)[/tex].
2. Plot a point at [tex]\( x = -6 \)[/tex].
The final graph looks like this:
```
<-3 -2 -1 0 1 2 3 4 5 6 7 8->
X X
-6
```
