To determine the possible range of values for the third side [tex]\( x \)[/tex] in a triangle with sides measuring 2 inches and 7 inches, we need to apply 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 the sides are 2 inches and 7 inches, let [tex]\( x \)[/tex] be the length of the third side. We need to satisfy the following inequalities:
1. The sum of the first two sides must be greater than the third side:
[tex]\[ 2 + 7 > x \][/tex]
Simplifying this, we get:
[tex]\[ 9 > x \][/tex]
or
[tex]\[ x < 9 \][/tex]
2. The sum of the other two sides must be greater than the remaining side:
[tex]\[ 2 + x > 7 \][/tex]
Simplifying this, we get:
[tex]\[ x > 5 \][/tex]
3. Similarly, the sum of the remaining two sides must be greater than the third side:
[tex]\[ 7 + x > 2 \][/tex]
Simplifying this, we get:
[tex]\[ x > -5 \][/tex]
However, this inequality is always true since [tex]\( x \)[/tex] is positive, and the length of a side of a triangle cannot be negative.
Combining the first two simplified inequalities, we get the range for [tex]\( x \)[/tex] as:
[tex]\[ 5 < x < 9 \][/tex]
Therefore, the correct inequality that gives the possible range of values for the length of the third side [tex]\( x \)[/tex] is:
C. [tex]\( 5 < x < 9 \)[/tex]
