To determine the range of possible values for the third side [tex]\(x\)[/tex] 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, the sum of the lengths of any two sides must be greater than the length of the third side.
Let’s denote the three sides of the triangle as [tex]\(a = 8\)[/tex], [tex]\(b = 12\)[/tex], and [tex]\(x\)[/tex] (the unknown side). According to the triangle inequality theorem, we have the following three inequalities:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex] with their given lengths:
1. [tex]\(8 + 12 > x\)[/tex]
[tex]\[20 > x\][/tex]
[tex]\[x < 20\][/tex]
2. [tex]\(8 + x > 12\)[/tex]
[tex]\[x > 12 - 8\][/tex]
[tex]\[x > 4\][/tex]
3. [tex]\(12 + x > 8\)[/tex]
[tex]\[x > 8 - 12\][/tex]
Since [tex]\(x\)[/tex] must be positive, this inequality will always hold true. Thus, it doesn’t affect the range we are trying to find.
Combining the valid inequalities from the above steps, we get:
[tex]\[4 < x < 20\][/tex]
Therefore, the correct inequality that gives the range of possible values for [tex]\(x\)[/tex] is:
C. [tex]\(4 < x < 20\)[/tex]
