To determine the range of possible lengths [tex]\( x \)[/tex] for the third side of a triangle when the other two sides are known, we need to apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Given two side lengths are 20 meters and 30 meters, let's denote these sides as [tex]\( \text{side1} \)[/tex] and [tex]\( \text{side2} \)[/tex] respectively.
1. Determine the lower bound: Calculate the absolute difference between the two known side lengths.
[tex]\[
\text{lower bound} = |30 - 20| = 10
\][/tex]
2. Determine the upper bound: Calculate the sum of the two known side lengths.
[tex]\[
\text{upper bound} = 20 + 30 = 50
\][/tex]
Therefore, the range of possible lengths [tex]\( x \)[/tex] for the third side of the triangle is:
[tex]\[
10 < x < 50
\][/tex]
So, the completed inequality is:
[tex]\[
10 < x < 50
\][/tex]
