To determine the range of possible lengths for the third side [tex]\( x \)[/tex] of the triangle-shaped restaurant, we need to apply the triangle inequality theorem.
According to the triangle inequality theorem, the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than the absolute difference of the lengths of the other two sides. When applied to our scenario with sides of lengths 20 meters and 30 meters, we get:
1. The lower bound for [tex]\( x \)[/tex]:
[tex]\( x \)[/tex] must be greater than the absolute difference of the two given sides.
[tex]\[
x > |20 - 30| \implies x > 10
\][/tex]
2. The upper bound for [tex]\( x \)[/tex]:
[tex]\( x \)[/tex] must be less than the sum of the two given sides.
[tex]\[
x < 20 + 30 \implies x < 50
\][/tex]
Putting these together, the length [tex]\( x \)[/tex] must satisfy the following inequality:
[tex]\[
10 < x < 50
\][/tex]
So, the complete inequality that describes the range of possible lengths [tex]\( x \)[/tex] of the third side of the restaurant is:
[tex]\[
10 < x < 50
\][/tex]
