To find the range of possible lengths [tex]\( x \)[/tex] of the third side of the triangle, we need to apply the triangle inequality theorem. According to the theorem, the sum of any two sides of a triangle must be greater than the length of the third side.
Given the side lengths 20 meters and 30 meters:
1. [tex]\( x \)[/tex] must be greater than the absolute difference between the two given sides:
[tex]\[
x > |20 - 30|
\][/tex]
Simplifying this, we get:
[tex]\[
x > 10
\][/tex]
2. [tex]\( x \)[/tex] must be less than the sum of the two given sides:
[tex]\[
x < 20 + 30
\][/tex]
Simplifying this, we get:
[tex]\[
x < 50
\][/tex]
Therefore, the range of possible lengths [tex]\( x \)[/tex] of the third side of the restaurant is:
[tex]\[
10 < x < 50
\][/tex]
