To determine the possible values for the third side [tex]\( x \)[/tex] of a triangle with given sides measuring 2 inches and 7 inches, we will use 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 remaining side.
Given sides:
1. [tex]\( \text{side}_1 = 2 \)[/tex] inches
2. [tex]\( \text{side}_2 = 7 \)[/tex] inches
We'll let [tex]\( x \)[/tex] be the length of the third side. The triangle inequality theorem gives us three conditions:
1. [tex]\( \text{side}_1 + \text{side}_2 > x \)[/tex]:
[tex]\[
2 + 7 > x
\][/tex]
[tex]\[
9 > x \quad \text{(or equivalently, } x < 9)
\][/tex]
2. [tex]\( \text{side}_1 + x > \text{side}_2 \)[/tex]:
[tex]\[
2 + x > 7
\][/tex]
[tex]\[
x > 5
\][/tex]
3. [tex]\( \text{side}_2 + x > \text{side}_1 \)[/tex]:
[tex]\[
7 + x > 2
\][/tex]
[tex]\[
x > -5 \quad \text{(This is always true since \( x \) must be positive)}
\][/tex]
The more stringent conditions derived from the triangle inequality theorem are:
[tex]\[
5 < x < 9
\][/tex]
Therefore, the correct inequality that describes the range of possible values for [tex]\( x \)[/tex] is:
[tex]\[
5 < x < 9
\][/tex]
So, the correct answer is:
C. [tex]\( 5 < x < 9 \)[/tex]
