The set of all values of x satisfying the inequations `|x-1| le 5 " and " |x|ge 2,` is

Options

A. `[-4, 6]`
B. `[-4, -2]`
C. `[-4, -2] cup [2, 6]`
D. `[2, 6]`

Explanation

To solve the given inequalities \( |x - 1| \leq 5 \) and \( |x| \geq 2 \), we will break them down step by step.

### Step 1: Solve the first inequality \( |x - 1| \leq 5 \)

The absolute value inequality \( |x - 1| \leq 5 \) can be rewritten as:
\[ -5 \leq x - 1 \leq 5 \]

Now, we will solve this compound inequality:

1. Add 1 to all parts of the inequality:
\[ -5 + 1 \leq x - 1 + 1 \leq 5 + 1 \]
This simplifies to:
\[ -4 \leq x \leq 6 \]

So, the solution for the first inequality is:
\[ x \in [-4, 6] \]

### Step 2: Solve the second inequality \( |x| \geq 2 \)

The absolute value inequality \( |x| \geq 2 \) can be rewritten as:
\[ x \leq -2 \quad \text{or} \quad x \geq 2 \]

This means the solution for the second inequality is:
\[ x \in (-\infty, -2] \cup [2, \infty) \]

### Step 3: Find the intersection of the two solutions

Now we need to find the intersection of the two sets:
1. From the first inequality, we have \( x \in [-4, 6] \).
2. From the second inequality, we have \( x \in (-\infty, -2] \cup [2, \infty) \).

Now, we will find the common values between these two sets:

- For the interval \( (-\infty, -2] \):
- The intersection with \( [-4, 6] \) gives us \( [-4, -2] \).

- For the interval \( [2, \infty) \):
- The intersection with \( [-4, 6] \) gives us \( [2, 6] \).

### Final Solution

Combining both intervals, we have:
\[ x \in [-4, -2] \cup [2, 6] \]