If the sides of a triangle are 7, 12 and x, and x is an integer, then find the number of possible values of x.

To find the number of possible integer values for the third side of the triangle, we can apply the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that for any triangle with sides of lengths a, b, and c, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side. This can be expressed as three inequalities:
1. $a+b>c$
2. $a+c>b$
3. $b+c>a$

A convenient way to combine these inequalities to find the range of the unknown third side x is:
$|a-b|<x<a+b$
In other words, the length of the third side must be strictly greater than the absolute difference of the other two sides, and strictly less than the sum of the other two sides.

Given the two known side lengths are 7 and 12, we can set:
$a=12$
$b=7$

Substituting these values into the inequality, we get:
$|12-7|<x<12+7$
Simplifying both sides:
$5<x<19$

Since x must be an integer, the possible integer values for x are:
$x\in \{6,7,8,9,10,11,12,13,14,15,16,17,18\}$

To find the total number of integer values in this range, we can use the formula for the number of integers strictly between two integers A and B (where A < B):
$\text{Number of values}=B-A-1$
Applying this to our range:
$19-5-1=13$

Thus, there are 13 possible integer values for x.