To determine the range of possible values for the third side [tex]\( x \)[/tex] in a triangle with sides measuring 2 inches and 7 inches, we need to use the triangle inequality theorem. The triangle inequality 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:
- Side1 = 2 inches
- Side2 = 7 inches
- Side3 = [tex]\( x \)[/tex] inches
We need to ensure the following inequalities hold true:
1. [tex]\( \text{Side1} + \text{Side2} > \text{Side3} \)[/tex]
[tex]\[ 2 + 7 > x \][/tex]
[tex]\[ 9 > x \][/tex]
[tex]\[ x < 9 \][/tex]
2. [tex]\( \text{Side1} + \text{Side3} > \text{Side2} \)[/tex]
[tex]\[ 2 + x > 7 \][/tex]
[tex]\[ x > 5 \][/tex]
3. [tex]\( \text{Side2} + \text{Side3} > \text{Side1} \)[/tex]
[tex]\[ 7 + x > 2 \][/tex]
[tex]\[ x > -5 \][/tex]
Since [tex]\( x \)[/tex] represents a length, this condition [tex]\( x > -5 \)[/tex] is always true and does not provide new information for positive values of [tex]\( x \)[/tex].
Combining the two meaningful 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]
So, the correct answer is:
A. [tex]\( 5 < x < 9 \)[/tex]
