If 6,18, x are sides of a triangle and x is an integer, then the number of possible values of x is:

The correct answer is 11 (Option 3).

To find the number of possible integer values for the side length $x$ of a triangle with known side lengths 6 and 18, we apply the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side. Additionally, the difference between any two side lengths must be strictly less than the third side length.

For a triangle with sides $a$, $b$, and $c$:

$|a-b|<c<a+b$

Given the two side lengths $a=6$ and $b=18$:

1. Lower bound for $x$:
$x>18-6$
$x>12$

2. Upper bound for $x$:
$x<18+6$
$x<24$

Combining these two inequalities gives the range of possible values for $x$:

$12<x<24$

Since $x$ is an integer, the possible values for $x$ are:

$13,14,15,16,17,18,19,20,21,22,23$

To count the total number of integer values in the strictly open interval $(12,24)$:

$\text{Number of values}=23-13+1=11$

Thus, there are 11 possible integer values for $x$.