To find the value of \( x \) given the segments \( BE \) and \( EC \) in rectangle \( BCDA \), we know that:
- The rectangle has two sides \( BC \) and \( CD \), and point \( E \) lies on line \( BC \).
- The segments \( BE \) and \( EC \) must add up to the total length of \( BC \).
From the problem, we have:
\[ BE = 2x + 9 \] \[ EC = 12x - 11 \]
Since point \( E \) is between points \( B \) and \( C \), we can write:
\[ BE + EC = BC \]
However, we don't have the total length \( BC \). But since \( E \) lies on line \( BC \), for the sake of solving for \( x \), we'll equate the two segments, assuming they represent segments of the same length to some extent (implying it's just a common equation).
Thus, we can set up the equation:
\[ BE + EC = BC \]
Assuming \( BE + EC = BC \) as a single value (and thus both expressions equating),
We can express the lengths as:
\[ 2x + 9 + 12x - 11 = BC \]
Since we don't have enough information to find BC directly, a possible equation would be to set the two expressions equal to each other to find when they equalize their respective points. Thus:
\[ 2x + 9 = 12x - 11 \]
Now, we'll solve for \( x \):
- Subtract \( 2x \) from both sides:
\[ 9 = 10x - 11 \]
- Add \( 11 \) to both sides:
\[ 20 = 10x \]
- Divide by \( 10 \):
\[ x = 2 \]
Thus, the value of \( x \) is \( \boxed{2} \).