https://file.aiquickdraw.com/mathbot/file/17240826100748kxspo7p.jpg

The image you've provided shows a rhombus \( PQRS \) with the following information given:

- Side \( PS = x \)
- Side \( PR = x \)
- Diagonal \( SQ = 3x - 7 \)
- Diagonal \( OR = 2x - 3 \)

In a rhombus, the diagonals bisect each other at right angles and are equal in length, but they are not equal to each other. However, all four sides of a rhombus are equal in length.

Since \( PS = PR \), and both of these are equal to the sides \( PQ \) and \( QR \), we can set up an equation using the property that the diagonals bisect each other.

Given that the diagonals of a rhombus are equal, the equation to find \( x \) will be:

\[ 3x - 7 = 2x - 3 \]

Let's solve for \( x \):

\[
3x - 7 = 2x - 3
\]

Subtract \( 2x \) from both sides:

\[
x - 7 = -3
\]

Add 7 to both sides:

\[
x = 4
\]

So, the value of \( x \) is \( \boxed{4} \).

Would you like more details, or do you have any questions? 

Here are five related questions:
1. How do we verify that all sides of a rhombus are equal after finding \( x \)?
2. What is the relationship between the diagonals in a rhombus?
3. How would the problem change if the figure was a square instead of a rhombus?
4. Can you find the lengths of the diagonals after determining \( x \)?
5. What would be the area of the rhombus using the given diagonal lengths?

**Tip:** Always check if the diagonals bisect each other at right angles when solving problems involving rhombuses.

Discover how to find the value of x in a rhombus using side and diagonal lengths. Solve the equation involving diagonals bisecting each other and explore properties of rhombuses. Suitable for high school students learning geometry.

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