To solve the linear equation \( 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1 \), we can start by simplifying both sides.
First, simplify the left side:
\[ 2.25 - 7.75 = -5.5, \] so the left side simplifies to:
\[ -5.5 - 11j + 1.5j = -5.5 - 9.5j. \]
The equation now looks like this:
\[ -5.5 - 9.5j = 0.5j - 1. \]
Next, let's rearrange the equation to combine like terms. Start by moving all \( j \) terms to one side and constant terms to the other side.
Add \( 9.5j \) to both sides:
\[ -5.5 = 0.5j + 9.5j - 1, \]
which simplifies to:
\[ -5.5 = 10j - 1. \]
Now, add 1 to both sides:
\[ -5.5 + 1 = 10j, \]
which simplifies to:
\[ -4.5 = 10j. \]
To solve for \( j \), divide both sides by 10:
\[ j = \frac{-4.5}{10} = -0.45. \]
Thus, the solution to the equation is
\[ \boxed{-0.45}. \]