Solve `|1-x|gt3`.

Options

A. `xgt4` or `xlt-1`
B. `xgt2` or `xlt-2`
C. `xgt5` or `xlt-2`
D. `xgt4` or `xlt-2`

Answer

To solve the inequality \( |1 - x| > 3 \), we can break it down into two separate cases based on the definition of absolute value.

### Step 1: Set up the cases
The expression \( |1 - x| > 3 \) implies two scenarios:
1. \( 1 - x > 3 \)
2. \( 1 - x < -3 \)

### Step 2: Solve the first case
For the first case \( 1 - x > 3 \):
- Subtract 1 from both sides:
\[ -x > 3 - 1 \]
\[ -x > 2 \]
- Multiply both sides by -1 (remember to flip the inequality sign):
\[ x < -2 \]

### Step 3: Solve the second case
For the second case \( 1 - x < -3 \):
- Subtract 1 from both sides:
\[ -x < -3 - 1 \]
\[ -x < -4 \]
- Multiply both sides by -1 (again, flip the inequality sign):
\[ x > 4 \]

### Step 4: Combine the results
From the two cases, we have:
1. \( x < -2 \)
2. \( x > 4 \)

Thus, the solution to the inequality \( |1 - x| > 3 \) is:
\[ x < -2 \quad \text{or} \quad x > 4 \]

### Final Answer
The solution set is \( (-\infty, -2) \cup (4, \infty) \).

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