To determine the possible lengths of the third side [tex]\( x \)[/tex] of a triangle where the other two sides measure 4 and 6, we can use the triangle inequality theorem, which states the following conditions must be satisfied for any triangle:
1. The sum of the lengths of any two sides must be greater than the length of the third side.
For sides [tex]\( a = 4 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = x \)[/tex], this gives us the following three inequalities:
1. [tex]\( 4 + 6 > x \)[/tex]
2. [tex]\( 4 + x > 6 \)[/tex]
3. [tex]\( 6 + x > 4 \)[/tex]
Let's analyze each of these in detail:
1. [tex]\( 4 + 6 > x \)[/tex] simplifies to [tex]\( 10 > x \)[/tex] or [tex]\( x < 10 \)[/tex]
2. [tex]\( 4 + x > 6 \)[/tex] simplifies to [tex]\( x > 2 \)[/tex]
3. [tex]\( 6 + x > 4 \)[/tex] simplifies to [tex]\( x > -2 \)[/tex], but this condition is always true if [tex]\( x > 2 \)[/tex], so we don't need to consider it separately.
Combining the two relevant inequalities, we have:
[tex]\[ 2 < x < 10 \][/tex]
So, the possible lengths for the third side [tex]\( x \)[/tex] are those that satisfy the inequality [tex]\( 2 < x < 10 \)[/tex].
Hence, the correct choice is:
C) [tex]\( 2 < x < 10 \)[/tex]
