To solve this problem, we need to apply the concept of inverse variation. If [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], it means [tex]\( y \cdot x = k \)[/tex] for some constant [tex]\( k \)[/tex].
Given:
[tex]\[ y = 8 \text{ when } x = 3 \][/tex]
First, we find the constant [tex]\( k \)[/tex]:
[tex]\[ y \cdot x = k \][/tex]
[tex]\[ 8 \cdot 3 = k \][/tex]
[tex]\[ k = 24 \][/tex]
Next, we use this constant to find the value of [tex]\( x \)[/tex] when [tex]\( y = 12 \)[/tex]:
[tex]\[ y \cdot x = k \][/tex]
[tex]\[ 12 \cdot x = 24 \][/tex]
To isolate [tex]\( x \)[/tex], we divide both sides of the equation by 12:
[tex]\[ x = \frac{24}{12} \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, the value of [tex]\( x \)[/tex] when [tex]\( y = 12 \)[/tex] is [tex]\( 2 \)[/tex].
The correct answer is:
C. 2
