To determine the possible values for [tex]\(x\)[/tex], the length of the third side of the triangle, we need to use the triangle inequality theorem. The 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 the sides 2 inches and 7 inches, we set up the following inequalities:
1. [tex]\(2 + 7 > x\)[/tex]
2. [tex]\(2 + x > 7\)[/tex]
3. [tex]\(7 + x > 2\)[/tex]
Let's analyze and simplify each inequality one by one:
1. [tex]\(2 + 7 > x\)[/tex]:
[tex]\[
9 > x \quad \Rightarrow \quad x < 9
\][/tex]
2. [tex]\(2 + x > 7\)[/tex]:
[tex]\[
2 + x > 7 \quad \Rightarrow \quad x > 5
\][/tex]
3. [tex]\(7 + x > 2\)[/tex]:
[tex]\[
7 + x > 2 \quad \Rightarrow \quad x > -5 \quad \text{(which is always true for positive \(x\))}
\][/tex]
The critical inequalities are [tex]\(x < 9\)[/tex] and [tex]\(x > 5\)[/tex], as the third inequality [tex]\(x > -5\)[/tex] doesn't restrict the positive values of [tex]\(x\)[/tex]. Combining these results, we get:
[tex]\[
5 < x < 9
\][/tex]
Hence, the range of possible values for [tex]\(x\)[/tex] is [tex]\(5 < x < 9\)[/tex], which corresponds to choice C.
So the correct answer is:
C. [tex]\(5 < x < 9\)[/tex]
