To determine the range of possible values for [tex]\( x \)[/tex], the length of the third side of the triangle, we 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 third side.
Given sides are 2 inches and 7 inches. Let [tex]\( x \)[/tex] be the third side. We need to satisfy the following three conditions based on the triangle inequality theorem:
1. [tex]\( x + 2 > 7 \)[/tex] (sum of [tex]\( x \)[/tex] and 2 greater than 7)
2. [tex]\( x + 7 > 2 \)[/tex] (sum of [tex]\( x \)[/tex] and 7 greater than 2)
3. [tex]\( 2 + 7 > x \)[/tex] (sum of 2 and 7 greater than [tex]\( x \)[/tex])
Let's solve each inequality:
1. [tex]\( x + 2 > 7 \)[/tex]
[tex]\[
x > 5
\][/tex]
2. [tex]\( x + 7 > 2 \)[/tex]
[tex]\[
x > -5
\][/tex]
Since [tex]\( x \)[/tex] is a length, it must be positive, and [tex]\( x > -5 \)[/tex] is always true for any positive value of [tex]\( x \)[/tex].
3. [tex]\( 2 + 7 > x \)[/tex]
[tex]\[
9 > x
\][/tex]
[tex]\[
x < 9
\][/tex]
Combining these inequalities, we get:
[tex]\[
5 < x < 9
\][/tex]
Therefore, the range of possible values for [tex]\( x \)[/tex] is given by the inequality [tex]\( 5 < x < 9 \)[/tex].
The correct answer is [tex]\( \boxed{A} \)[/tex].
