31+4•|4-x| = 47

To solve the equation \( 31 + 4 \cdot |4 - x| = 47 \), follow these steps:

1. Isolate the absolute value term: \[ 4 \cdot |4 - x| = 47 - 31 \] \[ 4 \cdot |4 - x| = 16 \]

2. Divide both sides by 4: \[ |4 - x| = \frac{16}{4} \] \[ |4 - x| = 4 \]

3. Set up the two cases for the absolute value: Case 1: \( 4 - x = 4 \)
   Case 2: \( 4 - x = -4 \)

4. Solve each case:

   • For Case 1: \[ 4 - x = 4 \implies -x = 0 \implies x = 0 \]

   • For Case 2: \[ 4 - x = -4 \implies -x = -8 \implies x = 8 \]

5. Final solutions: The solutions to the equation are \( x = 0 \) and \( x = 8 \).