To determine the range of possible lengths [tex]\( x \)[/tex] for the third side of the A-frame restaurant shaped as a triangle with given side lengths of 20 meters and 30 meters, we can utilize the triangle inequality theorem. This theorem states that the length of one side of a triangle must be less than the sum of the other two sides and greater than the difference of the other two sides.
Given side lengths [tex]\( a = 20 \)[/tex] meters and [tex]\( b = 30 \)[/tex] meters:
1. Calculate the lower bound:
[tex]\[
\text{Lower bound} = \left| 20 - 30 \right| = 10 \, \text{meters}
\][/tex]
2. Calculate the upper bound:
[tex]\[
\text{Upper bound} = 20 + 30 = 50 \, \text{meters}
\][/tex]
Thus, the length of the third side [tex]\( x \)[/tex] must satisfy the inequality:
[tex]\[
10 < x < 50
\][/tex]
So, the completed inequality is:
[tex]\[
\boxed{10} < x < \boxed{50}
\][/tex]
