The sides of a triangle are equal to integral numbers of units. Two sides are 4 and 6 units long, respectively; what is the MINIMUM value for the triangle's perimeter?

To find the minimum value for the triangle's perimeter, we need to consider the triangle inequality theorem, which 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 that two sides of the triangle are 4 units and 6 units long, we need to find the minimum value for the third side. Let's denote the length of the third side as "x." According to the triangle inequality theorem, we have the following inequalities:

4 + 6 > x
10 > x

x + 4 > 6
x > 2

x + 6 > 4
x > -2

Since the sides of the triangle are integral numbers of units, the length of the third side must be an integer greater than 2. Therefore, the minimum value for the third side is 3.
To calculate the minimum perimeter, we add up the lengths of all three sides:
Perimeter = 4 + 6 + 3 = 13
Hence, the minimum value for the triangle's perimeter is 13 units.