To determine how many unique triangles can be formed with side lengths of 9 inches, 2 inches, and a third side length (let's call it \( x \)), we need to apply the triangle inequality theorem. The triangle inequality states that for three sides \( a, b, \) and \( c \) of a triangle, the following conditions must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
In this scenario, we have two known sides: 9 inches and 2 inches. We will denote the unknown side as \( x \). Thus, the triangle inequality conditions will be applied as follows:
- \( 9 + 2 > x \) ⇒ \( 11 > x \) or \( x < 11 \)
- \( 9 + x > 2 \) ⇒ \( x > 2 - 9 \) ⇒ \( x > -7 \) (This condition is always satisfied since \( x \) must be positive)
- \( 2 + x > 9 \) ⇒ \( x > 9 - 2 \) ⇒ \( x > 7 \)
From these inequalities, we can derive that:
- \( x < 11 \)
- \( x > 7 \)
Combining these two inequalities gives us:
\[ 7 < x < 11 \]
Since \( x \) must be a whole number, the only possible whole numbers that satisfy this inequality are:
- \( x = 8 \)
- \( x = 9 \)
- \( x = 10 \)
Thus, Sam can create triangles with the following side lengths:
- \( 2, 9, 8 \)
- \( 2, 9, 9 \)
- \( 2, 9, 10 \)
These are three unique triangles that can be formed with the given side lengths.
Therefore, the answer is:
3