To determine the range of possible values for the third side of a triangle when two sides are given as 8 inches and 12 inches, we use the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ a + b > c \][/tex]
[tex]\[ a + c > b \][/tex]
[tex]\[ b + c > a \][/tex]
Here [tex]\( a = 8 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = x \)[/tex] (the third side we are looking for). Let’s apply the theorem step by step:
1. Adding sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ 8 + 12 > x \][/tex]
[tex]\[ 20 > x \][/tex]
or
[tex]\[ x < 20 \][/tex]
2. Adding sides [tex]\( a \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[ 8 + x > 12 \][/tex]
[tex]\[ x > 4 \][/tex]
3. Adding sides [tex]\( b \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[ 12 + x > 8 \][/tex]
This inequality is always true for positive [tex]\( x \)[/tex] because [tex]\( x > 4 \)[/tex], which is implicitly included in the previous inequality ([tex]\(x > 4\)[/tex]).
So, combining the resulting inequalities, we get:
[tex]\[ 4 < x < 20 \][/tex]
Therefore, the range of possible values for [tex]\( x \)[/tex] is:
[tex]\[ 4 < x < 20 \][/tex]
This matches the inequality given in option D.
Thus, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
