To determine the possible values of [tex]\( h \)[/tex] that would make a valid triangle with the given side lengths [tex]\( 3x \)[/tex] cm, [tex]\( 7x \)[/tex] cm, and [tex]\( h \)[/tex] cm, we need to apply the triangle inequality theorem. 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]
Here, the sides of the triangle are [tex]\( 3x \)[/tex], [tex]\( 7x \)[/tex], and [tex]\( h \)[/tex]. Let's apply the triangle inequality theorem to these sides:
1. [tex]\( 3x + 7x > h \)[/tex]
[tex]\[
10x > h
\][/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
[tex]\[
h > 4x
\][/tex]
3. [tex]\( 7x + h > 3x \)[/tex]
[tex]\[
h > -4x \quad (\text{This inequality is not relevant as it will always be satisfied for positive \( x \)})
\][/tex]
Combining the valid inequalities from above, we get:
[tex]\[
4x < h < 10x
\][/tex]
So, the expression that describes the possible values of [tex]\( h \)[/tex] in cm is:
[tex]\[
4x < h < 10x
\][/tex]
