To solve for the range of possible values for [tex]\( h \)[/tex] using the triangle inequality theorem, we need to analyze the side lengths given:
1. Side 1: [tex]\( 3x \)[/tex] cm
2. Side 2: [tex]\( 7x \)[/tex] cm
3. Side 3: [tex]\( h \)[/tex] cm
The triangle inequality 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]
Let’s apply these inequalities to our triangle:
1. [tex]\( 3x + 7x > h \)[/tex]
[tex]\[ 10x > h \][/tex]
[tex]\[ h < 10x \][/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
[tex]\[ h > 7x - 3x \][/tex]
[tex]\[ h > 4x \][/tex]
3. [tex]\( 7x + h > 3x \)[/tex]
[tex]\[ h > 3x - 7x \][/tex]
[tex]\[ h > -4x \][/tex]
Since [tex]\( h > -4x \)[/tex] will always be true as long as [tex]\( h > 4x \)[/tex], it isn't a restrictive condition in this context.
Combining these inequalities, we get:
[tex]\[ 4x < h < 10x \][/tex]
Therefore, the expression that describes the possible values of [tex]\( h \)[/tex] in cm is:
[tex]\[ 4x < h < 10x \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{4x < h < 10x} \][/tex]
