If `|x-1| gt 5`, then

Options

A. `x in (-4,6)`
B. `x in [-4,6]`
C. `x in (-infty,-4) cup (6, infty)`
D. `x in [-infty, -4) cup [6,infty)`

Explanation

To solve the inequality \( |x - 1| > 5 \), we will break it down into two separate cases based on the definition of absolute value.

### Step 1: Set up the two cases
The inequality \( |x - 1| > 5 \) means that the expression inside the absolute value, \( x - 1 \), is either greater than 5 or less than -5. Therefore, we can write:

1. \( x - 1 > 5 \)
2. \( x - 1 < -5 \)

### Step 2: Solve the first case
For the first case \( x - 1 > 5 \):

\[ x - 1 > 5 \]

Adding 1 to both sides:

\[ x > 6 \]

### Step 3: Solve the second case
For the second case \( x - 1 < -5 \):

\[ x - 1 < -5 \]

Adding 1 to both sides:

\[ x < -4 \]

### Step 4: Combine the solutions
From the two cases, we have:

1. \( x < -4 \)
2. \( x > 6 \)

This means the solution to the inequality \( |x - 1| > 5 \) is:

\[ x < -4 \quad \text{or} \quad x > 6 \]

### Step 5: Write the solution in interval notation
In interval notation, the solution can be expressed as:

\[ (-\infty, -4) \cup (6, \infty) \]

### Final Answer
Thus, the solution to the inequality \( |x - 1| > 5 \) is:

\[ (-\infty, -4) \cup (6, \infty) \]

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