Find the range of values for the third side of the triangle, x, based on the diagram. sides: x, 7

To find the range of values for the side \( x \) of a triangle given one side of length 7, we can use the triangle inequality theorem. The triangle inequality theorem states that for any triangle with sides of lengths \( a, b, \) and \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this case, we have two sides known: \( x \) and \( 7 \). Let's denote the third side as \( y \).

For the two sides we have, we can use the triangle inequalities as follows:

1. \( x + 7 > y \)
2. \( x + y > 7 \)
3. \( 7 + y > x \)

The inequalities will help us determine the valid range for \( x \).

However, since \( y \) is unknown, we can consider the simplest case where \( y \) is also related to \( x \). For the inequalities, we can combine them:

1. From \( x + 7 > y \), assuming \( y \) is at its minimum (0 when \( x \) is > 7).
2. From \( x + y > 7 \), and setting \( y = 0 \), gives \( x > 7 \).
3. The last inequality \( 7 + y > x \), gives \( x < 7 + y \).

Using the third inequality, we replace \( y \) with \( x \) since \( x + 0 > 7 \) can be applicable, thus:

The critical inequalities are:
- \( x + 0 > 7 \) implies \( x > 7 - 7\) or \( x > 0 \) is always valid.
- Set \( x + 0 > 7\), gives \( x > 7 \)
- Set \( y = x \) in the last, leads to \( 7+x > x\), which is always valid.

Combining these insights, we see that:

1. Considering the value needs not exceed is bounded by other existing side lengths.
2. Since \( x\) can be influenced by its relationship with \( 7 \) being stable.

After you consolidate this, you see:

**Final Conclusions**:

Thus, to validate triplet results and maintain triangle formation, the valid range of \( x \) can be denoted as:

\[
|x - 7| < y < x + y
\]
where both sides can vary; hence, realistically,
\[x > 0 \text{ and numerical handling leads to } x < 14 ,\text{beyond that results simply discredited.}\]

So, \(x\) lies inbetween \( (0,14) \) concretely observable.

Thus, the answer sets to:
- Valid range values must satisfy \( \: 0 < x < 14\) which holds true for maintaining triangle legitimacy.