To determine the range of possible values for the third side [tex]\( x \)[/tex] of a triangle with sides measuring 8 inches and 12 inches, we employ 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. There are three conditions to consider:
1. The sum of the first side and the second side must be greater than the third side.
2. The sum of the second side and the third side must be greater than the first side.
3. The sum of the first side and the third side must be greater than the second side.
Given sides [tex]\( a = 8 \)[/tex] inches and [tex]\( b = 12 \)[/tex] inches, we need to find the range for the third side [tex]\( x \)[/tex]:
[tex]\[
1. \quad 8 + 12 > x \implies 20 > x \implies x < 20
\][/tex]
[tex]\[
2. \quad 8 + x > 12 \implies x > 4
\][/tex]
[tex]\[
3. \quad 12 + x > 8 \implies x > -4 \quad (This condition is always true since \( x \) must be positive)
\][/tex]
The combination of these inequalities gives us the range for [tex]\( x \)[/tex]:
[tex]\[
4 < x < 20
\][/tex]
Thus, the range of possible values for [tex]\( x \)[/tex] is [tex]\( 4 < x < 20 \)[/tex].
Therefore, the correct inequality, according to the range found, is:
D. [tex]\( 4 < x < 20 \)[/tex]
