Let's analyze the problem using the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's denote the sides of the triangle as [tex]\( a = 8 \)[/tex] inches, [tex]\( b = 12 \)[/tex] inches, and [tex]\( c = x \)[/tex] inches (where [tex]\( x \)[/tex] is the unknown side length).
We need to satisfy the following three inequalities based on the triangle inequality theorem:
1. The sum of the first two sides must be greater than the third side:
[tex]\[ a + b > x \][/tex]
[tex]\[ 8 + 12 > x \][/tex]
[tex]\[ 20 > x \][/tex]
which simplifies to:
[tex]\[ x < 20 \][/tex]
2. The sum of [tex]\( a \)[/tex] and [tex]\( c \)[/tex] must be greater than [tex]\( b \)[/tex]:
[tex]\[ a + x > b \][/tex]
[tex]\[ 8 + x > 12 \][/tex]
[tex]\[ x > 4 \][/tex]
3. The sum of [tex]\( b \)[/tex] and [tex]\( c \)[/tex] must be greater than [tex]\( a \)[/tex]:
[tex]\[ b + x > a \][/tex]
[tex]\[ 12 + x > 8 \][/tex]
This simplifies to:
[tex]\[ x > -4 \][/tex]
However, since side length must always be positive, this condition is inherently satisfied for any positive [tex]\( x \)[/tex].
Combining the valid inequalities from the above analysis, we get:
[tex]\[ 4 < x < 20 \][/tex]
Thus, the range of possible values for [tex]\( x \)[/tex] is:
[tex]\[ 4 < x < 20 \][/tex]
This corresponds to the answer choice:
D. [tex]\( 4 < x < 20 \)[/tex]
