The problem asks us to identify which sets of side lengths can form a triangle. We need to apply the triangle inequality theorem to each set of side lengths. The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side.

1. Problem Description

The problem asks us to identify which sets of side lengths can form a triangle. We need to apply the triangle inequality theorem to each set of side lengths. The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side.

2. Solution Steps

The triangle inequality theorem states that for any triangle with side lengths $a$, $b$, and $c$, the following inequalities must hold:

$a + b > c$

$a + c > b$

$b + c > a$

If all three inequalities hold, then a triangle can be formed.

Let's check each set of side lengths:

* 3 inches, 4 inches, 8 inches:

$3 + 4 > 8$ => $7 > 8$ (False)

$3 + 8 > 4$ => $11 > 4$ (True)

$4 + 8 > 3$ => $12 > 3$ (True)

Since one of the inequalities is false, this set of side lengths cannot form a triangle.

* 3 inches, 5 inches, 6 inches:

$3 + 5 > 6$ => $8 > 6$ (True)

$3 + 6 > 5$ => $9 > 5$ (True)

$5 + 6 > 3$ => $11 > 3$ (True)

Since all three inequalities are true, this set of side lengths can form a triangle.

* 7 inches, 8 inches, 16 inches:

$7 + 8 > 16$ => $15 > 16$ (False)

$7 + 16 > 8$ => $23 > 8$ (True)

$8 + 16 > 7$ => $24 > 7$ (True)

Since one of the inequalities is false, this set of side lengths cannot form a triangle.

* 2 inches, 4 inches, 7 inches:

$2 + 4 > 7$ => $6 > 7$ (False)

$2 + 7 > 4$ => $9 > 4$ (True)

$4 + 7 > 2$ => $11 > 2$ (True)

Since one of the inequalities is false, this set of side lengths cannot form a triangle.

3. Final Answer

Only the set of side lengths 3 inches, 5 inches, and 6 inches can form a triangle.