\( 3.2 ; 2 x+1 \) and \( 3 x-3 \) are the first three terms of a linear pattern. Calculate the value of \( x \).

We have an arithmetic sequence with the first three terms given by: \[ 3.2,\quad 2x+1,\quad 3x-3. \] In an arithmetic sequence, the difference between consecutive terms is constant. Therefore, we set the difference between the second and first term equal to the difference between the third and second term: \[ (2x+1) - 3.2 = (3x-3) - (2x+1). \] **Step 1: Simplify both sides** Left side: \[ (2x+1) - 3.2 = 2x + 1 - 3.2 = 2x - 2.2. \] Right side: \[ (3x-3) - (2x+1) = 3x - 3 - 2x - 1 = x - 4. \] **Step 2: Set up the equation** \[ 2x - 2.2 = x - 4. \] **Step 3: Solve for \( x \)** Subtract \( x \) from both sides: \[ 2x - x - 2.2 = -4, \] \[ x - 2.2 = -4. \] Add \( 2.2 \) to both sides: \[ x = -4 + 2.2, \] \[ x = -1.8. \] Thus, the value of \( x \) is: \[ \boxed{-1.8}. \]