To determine the range of possible lengths [tex]\( x \)[/tex] for the third side of the triangle, we need to use the triangle inequality theorem. This theorem states that in any triangle, the length of any side must be less than the sum of the other two sides and more than the difference of the other two sides.
Given the sides of the triangle as [tex]\( 20 \)[/tex] meters and [tex]\( 30 \)[/tex] meters, let's apply the theorem:
1. Calculate the lower bound:
- The difference between the two given side lengths: [tex]\( |20 - 30| = 10 \)[/tex].
2. Calculate the upper bound:
- The sum of the two given side lengths: [tex]\( 20 + 30 = 50 \)[/tex].
Therefore, the third side [tex]\( x \)[/tex] must satisfy the inequality:
[tex]\[ 10 < x < 50. \][/tex]
So, the correct completed inequality is:
[tex]\[ \boxed{10} < x < \boxed{50} \][/tex]
