To determine the range of possible values for the third side [tex]\( x \)[/tex] of a triangle when the other two sides are 8 inches and 12 inches, we can use the triangle inequality theorem. The triangle inequality theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. The difference between the lengths of any two sides must be less 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
- [tex]\( x \)[/tex] inches (the third side)
According to the triangle inequality theorem, we need to satisfy the following conditions:
1. [tex]\( a + b > x \)[/tex]
2. [tex]\( a + x > b \)[/tex]
3. [tex]\( b + x > a \)[/tex]
Substituting the given values, we get:
1. [tex]\( 8 + 12 > x \)[/tex] simplifies to [tex]\( 20 > x \)[/tex]
2. [tex]\( 8 + x > 12 \)[/tex] simplifies to [tex]\( x > 4 \)[/tex]
3. [tex]\( 12 + x > 8 \)[/tex] simplifies to [tex]\( x > -4 \)[/tex] which is always true since [tex]\( x \)[/tex] is a positive length
Combining the inequalities from the above conditions, we get:
[tex]\[ 4 < x < 20 \][/tex]
Therefore, the range of possible values for the third side [tex]\( x \)[/tex] is such that [tex]\( x \)[/tex] is greater than 4 inches and less than 20 inches. This matches option D:
[tex]\[ \boxed{4 < x < 20} \][/tex]
