To determine the range of possible values for [tex]\( x \)[/tex], the length of the third side of a triangle, given that the other two sides measure 2 inches and 7 inches, we 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.
We'll denote the sides of the triangle as follows:
- Side [tex]\( a \)[/tex] = 2 inches
- Side [tex]\( b \)[/tex] = 7 inches
- Side [tex]\( x \)[/tex] = third side (the value we need to find the range for)
According to the triangle inequality theorem, we have three conditions to satisfy:
1. [tex]\( a + b > x \)[/tex]
2. [tex]\( a + x > b \)[/tex]
3. [tex]\( b + x > a \)[/tex]
Let's apply these conditions step-by-step:
1. [tex]\( a + b > x \)[/tex]
[tex]\[
2 + 7 > x \Rightarrow 9 > x \Rightarrow x < 9
\][/tex]
2. [tex]\( a + x > b \)[/tex]
[tex]\[
2 + x > 7 \Rightarrow x > 7 - 2 \Rightarrow x > 5
\][/tex]
3. [tex]\( b + x > a \)[/tex]
[tex]\[
7 + x > 2 \Rightarrow this condition is always true for any positive \( x \) since 7 + x will always be greater than 2.
\][/tex]
To combine these, we need to find the intersection of the inequalities derived from the valid conditions:
From condition 1:
[tex]\[
x < 9
\][/tex]
From condition 2:
[tex]\[
x > 5
\][/tex]
There is no contradiction between these inequalities. Therefore, the range of possible values for [tex]\( x \)[/tex] satisfying both conditions is:
[tex]\[
5 < x < 9
\][/tex]
Thus, the inequality that represents the range of possible values for [tex]\( x \)[/tex] is:
[tex]\[
C. \, 5 < x < 9
\][/tex]
So, the correct answer is:
C. [tex]\( 5 < x < 9 \)[/tex]
