To determine the range of possible values for the third side [tex]\( x \)[/tex] of a triangle with sides measuring 2 inches and 7 inches, we need to use the triangle inequality theorem. This theorem states:
1. The length of any side of a triangle must be less than the sum of the lengths of the other two sides.
2. The length of any side of a triangle must be greater than the absolute difference of the lengths of the other two sides.
Let’s denote the given sides as [tex]\(a = 2\)[/tex] inches and [tex]\(b = 7\)[/tex] inches. The third side length is denoted as [tex]\(x\)[/tex].
According to the triangle inequality theorem:
1. [tex]\( x \)[/tex] must be greater than the difference of the given sides:
[tex]\[
x > |a - b|
\][/tex]
Calculating the absolute difference:
[tex]\[
|2 - 7| = 5
\][/tex]
Therefore:
[tex]\[
x > 5
\][/tex]
2. [tex]\( x \)[/tex] must be less than the sum of the given sides:
[tex]\[
x < a + b
\][/tex]
Summing the lengths:
[tex]\[
2 + 7 = 9
\][/tex]
Therefore:
[tex]\[
x < 9
\][/tex]
Combining these two inequalities, we get:
[tex]\[
5 < x < 9
\][/tex]
So, the range of possible values for the third side [tex]\( x \)[/tex] is:
[tex]\[
5 < x < 9
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C. \ 5 < x < 9}
\][/tex]
