To solve the problem of determining the possible length [tex]\( x \)[/tex] for the third side of a triangle with given sides [tex]\( 20 \)[/tex] meters and [tex]\( 30 \)[/tex] meters, we need to use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given the sides [tex]\( 20 \)[/tex] meters (side1) and [tex]\( 30 \)[/tex] meters (side2), we can write the following inequalities:
1. [tex]\( 20 + 30 > x \)[/tex]
2. [tex]\( 20 + x > 30 \)[/tex]
3. [tex]\( 30 + x > 20 \)[/tex]
Let’s solve these inequalities one by one:
1. [tex]\( 20 + 30 > x \)[/tex]
[tex]\[
50 > x \quad \text{or} \quad x < 50
\][/tex]
2. [tex]\( 20 + x > 30 \)[/tex]
[tex]\[
x > 30 - 20 \quad \text{or} \quad x > 10
\][/tex]
3. [tex]\( 30 + x > 20 \)[/tex]
[tex]\[
x > 20 - 30 \quad \text{or} \quad x > -10
\][/tex]
Since lengths cannot be negative, the third inequality [tex]\( x > -10 \)[/tex] does not impose any additional restriction beyond [tex]\( x > 10 \)[/tex].
Combining the inequalities from steps 1 and 2, the length [tex]\( x \)[/tex] of the third side must satisfy:
[tex]\[
10 < x < 50
\][/tex]
Thus, the correct inequality to describe the range of possible lengths [tex]\( x \)[/tex] for the third side is:
[tex]\[
10 < x < 50
\][/tex]
