Solve `x:|x-1|=4`.

Answer

To solve the equation \( x : |x - 1| = 4 \), we can rewrite it as:

\[ |x - 1| = 4 \]

Now, we will consider two cases based on the definition of absolute value.

### Case 1: \( x - 1 \geq 0 \)

In this case, the absolute value function can be simplified as:

\[ |x - 1| = x - 1 \]

Substituting this into the equation gives:

\[ x - 1 = 4 \]

Now, we solve for \( x \):

\[ x - 1 + 1 = 4 + 1 \]
\[ x = 5 \]

### Case 2: \( x - 1 < 0 \)

In this case, the absolute value function can be simplified as:

\[ |x - 1| = -(x - 1) = -x + 1 \]

Substituting this into the equation gives:

\[ -x + 1 = 4 \]

Now, we solve for \( x \):

\[ -x + 1 - 1 = 4 - 1 \]
\[ -x = 3 \]
\[ x = -3 \]

### Final Solution

From both cases, we have two solutions:

\[ x = 5 \quad \text{and} \quad x = -3 \]

### Summary of Solutions
The solutions to the equation \( x : |x - 1| = 4 \) are:

\[ x = 5 \quad \text{and} \quad x = -3 \]

---