To determine the possible values of [tex]\( h \)[/tex] in terms of [tex]\( x \)[/tex], given the side lengths of a triangle as [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 the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. We need to consider these three inequalities:
1. [tex]\( 3x + 7x > h \)[/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
3. [tex]\( 7x + h > 3x \)[/tex]
Let's break these down one by one:
### Step 1: Simplify the inequalities
1. First Inequality:
[tex]\[
3x + 7x > h
\][/tex]
[tex]\[
10x > h
\][/tex]
This simplifies to:
[tex]\[
h < 10x
\][/tex]
2. Second Inequality:
[tex]\[
3x + h > 7x
\][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[
h > 4x
\][/tex]
3. Third Inequality:
[tex]\[
7x + h > 3x
\][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[
7x + h - 3x > 0
\][/tex]
[tex]\[
4x + h > 0
\][/tex]
This simplifies to:
[tex]\[
h > -4x
\][/tex]
However, since [tex]\( h \)[/tex] represents a side length of a triangle and must be positive, this inequality is always true and doesn't provide any new information.
### Step 2: Combine the valid inequalities
From the simplified inequalities, we have:
[tex]\[
h < 10x
\][/tex]
and
[tex]\[
h > 4x
\][/tex]
Combining these two, we get:
[tex]\[
4x < h < 10x
\][/tex]
### Conclusion
The expression that describes the possible values of [tex]\( h \)[/tex] is [tex]\( 4x < h < 10x \)[/tex]. Therefore, the correct answer is:
[tex]\[
4x < h < 10x
\][/tex]
