To determine the range of possible values for [tex]\( x \)[/tex], the length of the third side of a triangle with sides measuring 8 inches and 12 inches, we need to apply the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
In this problem, we have:
- [tex]\( a = 8 \)[/tex] inches
- [tex]\( b = 12 \)[/tex] inches
- [tex]\( c = x \)[/tex] inches (the unknown side)
Let's apply the inequalities one by one.
### 1. [tex]\( a + b > c \)[/tex]
[tex]\[ 8 + 12 > x \][/tex]
[tex]\[ 20 > x \][/tex]
This simplifies to:
[tex]\[ x < 20 \][/tex]
### 2. [tex]\( a + c > b \)[/tex]
[tex]\[ 8 + x > 12 \][/tex]
[tex]\[ x > 4 \][/tex]
### 3. [tex]\( b + c > a \)[/tex]
[tex]\[ 12 + x > 8 \][/tex]
This inequality simplifies to:
[tex]\[ x > -4 \][/tex]
However, since side lengths cannot be negative:
[tex]\[ x > 4 \][/tex]
Combining the results from the inequalities:
[tex]\[ 4 < x < 20 \][/tex]
So the correct range for the third side [tex]\( x \)[/tex] must satisfy the strict inequality:
[tex]\[ 4 < x < 20 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{4 < x < 20} \][/tex]
Options to compare:
- A. [tex]\( 4 \leq x \leq 20 \)[/tex] (This includes 4 and 20, which is not strict)
- B. [tex]\( 8 \leq x \leq 12 \)[/tex] (This range is too narrow)
- C. [tex]\( 8 < x < 12 \)[/tex] (This range is also too narrow)
- D. [tex]\( 4 < x < 20 \)[/tex] (This is the correct range)
So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
