The problem is to solve the equation $5|3-x| - 51 = 4$ for $x$. This involves isolating the absolute value term and then considering both positive and negative cases.

1. Problem Description

The problem is to solve the equation $5|3-x| - 51 = 4$ for $x$. This involves isolating the absolute value term and then considering both positive and negative cases.

2. Solution Steps

First, add 51 to both sides of the equation:

$5|3-x| - 51 + 51 = 4 + 51$

$5|3-x| = 55$

Next, divide both sides by 5:

$\frac{5|3-x|}{5} = \frac{55}{5}$

$|3-x| = 11$

Now, we consider two cases:

Case 1: $3-x = 11$

Subtract 3 from both sides:

$3 - x - 3 = 11 - 3$

$-x = 8$

$x = -8$

Case 2: $3-x = -11$

Subtract 3 from both sides:

$3 - x - 3 = -11 - 3$

$-x = -14$

$x = 14$

Therefore, the two solutions for $x$ are $-8$ and $14$.

3. Final Answer

$x = -8, 14$