Certainly! To determine the possible range for the length of the third side of a triangle when the other two sides have lengths of 11 and 18, we can 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 remaining side.
Let's denote the length of the third side as [tex]\( x \)[/tex]. According to the Triangle Inequality Theorem, we will establish three inequalities:
1. The sum of the lengths of the first two sides must be greater than the third side:
[tex]\[ 11 + 18 > x. \][/tex]
2. The sum of the lengths of the first and the third sides must be greater than the second side:
[tex]\[ 11 + x > 18. \][/tex]
3. The sum of the lengths of the second and the third sides must be greater than the first side:
[tex]\[ 18 + x > 11. \][/tex]
Let's solve these inequalities one by one.
1. From [tex]\( 11 + 18 > x \)[/tex]:
[tex]\[ 29 > x. \][/tex]
or
[tex]\[ x < 29. \][/tex]
2. From [tex]\( 11 + x > 18 \)[/tex]:
[tex]\[ x > 7. \][/tex]
3. From [tex]\( 18 + x > 11 \)[/tex]:
[tex]\[ x > -7. \][/tex]
This inequality is already satisfied since any side lengths will be positive, hence [tex]\( x > -7 \)[/tex] does not provide additional useful information. Therefore, we can ignore this condition.
To combine these useful conditions, we have:
[tex]\[ 7 < x < 29. \][/tex]
Thus, the length of the third side must be:
[tex]\[ x \text{ must be greater than 7 but less than 29.} \][/tex]
Given the options provided, the correct statement about the length of the third side is:
[tex]\[ \text{less than 29.} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\text{less than 29}}. \][/tex]
