To find the value of \( x \) given the lengths \( DO = 4x + 8 \) and \( CO = 15x - 3 \), we can equate the two expressions since both represent the length of the same segment \( O \) to points \( D \) and \( C \) in rectangle \( ABCD \).
Set the two expressions equal to one another:
\[ 4x + 8 = 15x - 3 \]
Now, let's solve for \( x \).
- First, subtract \( 4x \) from both sides:
\[ 8 = 15x - 4x - 3 \]
This simplifies to:
\[ 8 = 11x - 3 \]
- Next, add 3 to both sides:
\[ 8 + 3 = 11x \]
This simplifies to:
\[ 11 = 11x \]
- Finally, divide both sides by 11:
\[ x = 1 \]
Thus, the value of \( x \) is:
\[ \boxed{1} \]
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