If 

*EF* = 3*x* + 8

, 

*FG* = 11*x*

, and 

*EG* = 12*x* + 10

, what is *EF*?

Given:

- \( EF = 3x + 8 \)
- \( FG = 11x \)
- \( EG = 12x + 10 \)

We need to determine the value of \( EF \). To do this, we can apply the triangle inequality to ensure the sides form a valid triangle.

### Triangle Inequality Conditions:

1. \( EF + FG > EG \)
2. \( EF + EG > FG \)
3. \( FG + EG > EF \)

### 1. \( EF + FG > EG \)
\[
(3x + 8) + 11x > 12x + 10
\]
\[
14x + 8 > 12x + 10
\]
\[
14x - 12x > 10 - 8
\]
\[
2x > 2
\]
\[
x > 1
\]

### 2. \( EF + EG > FG \)
\[
(3x + 8) + (12x + 10) > 11x
\]
\[
15x + 18 > 11x
\]
\[
15x - 11x > -18
\]
\[
4x > -18
\]
This inequality is always true for any positive \( x \).

### 3. \( FG + EG > EF \)
\[
11x + (12x + 10) > 3x + 8
\]
\[
23x + 10 > 3x + 8
\]
\[
23x - 3x > 8 - 10
\]
\[
20x > -2
\]
This inequality is also always true for any positive \( x \).

### Conclusion:
The only significant restriction is \( x > 1 \). So, \( EF = 3x + 8 \) for any \( x > 1 \).

Thus, \( EF \) is determined by the value of \( x \). Since the problem does not specify \( x \), \( EF \) is a function of \( x \), where \( x \) must be greater than 1. 

### Example Calculation:
If \( x = 2 \) (since \( x > 1 \)):
\[
EF = 3(2) + 8 = 6 + 8 = 14
\]
Thus, for \( x = 2 \), \( EF = 14 \).

Would you like to explore how \( EF \) changes with different values of \( x \), or need any other details?

### Related Questions:
1. What happens if \( x = 1 \) or \( x < 1 \) in this problem?
2. How can we graph the relationship between \( x \) and \( EF \)?
3. How do the inequalities ensure the sides form a valid triangle?
4. Can \( x \) take non-integer values, and how does that affect \( EF \)?
5. How can we verify if a given set of sides forms a triangle using their lengths?

**Tip:** When dealing with triangle problems, always check if the values satisfy the triangle inequality theorem, as it ensures the sides can indeed form a triangle.

Explore how to solve a triangle inequality problem involving sides EF, FG, and EG. Understand the application of the Triangle Inequality Theorem to determine the value of EF based on given relationships. Suitable for advanced high school students.

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