The values of x which satisfy both the inequations `x^(2)-1 le 0 and x^(2)-x-2 ge0` lie in

Options

A. (-1, 2)
B. (-1,-1)
C. (1,2)
D. `{-1}`

Answer

To solve the inequalities \( x^2 - 1 \leq 0 \) and \( x^2 - x - 2 \geq 0 \), we will approach each inequality step by step.

### Step 1: Solve the first inequality \( x^2 - 1 \leq 0 \)

1. **Factor the inequality**:
\[ x^2 - 1 = (x - 1)(x + 1) \leq 0 \]

2. **Find the roots**:
The roots of the equation \( x^2 - 1 = 0 \) are \( x = -1 \) and \( x = 1 \).

3. **Test intervals**:
We will test the intervals created by the roots:
- Interval 1: \( (-\infty, -1) \)
- Interval 2: \( (-1, 1) \)
- Interval 3: \( (1, \infty) \)

- For \( x < -1 \) (e.g., \( x = -2 \)):
\[ (-2 - 1)(-2 + 1) = (-3)(-1) = 3 \quad (\text{positive}) \]
- For \( -1 < x < 1 \) (e.g., \( x = 0 \)):
\[ (0 - 1)(0 + 1) = (-1)(1) = -1 \quad (\text{negative}) \]
- For \( x > 1 \) (e.g., \( x = 2 \)):
\[ (2 - 1)(2 + 1) = (1)(3) = 3 \quad (\text{positive}) \]

4. **Determine the solution**:
The inequality \( (x - 1)(x + 1) \leq 0 \) is satisfied in the interval:
\[ [-1, 1] \]

### Step 2: Solve the second inequality \( x^2 - x - 2 \geq 0 \)

1. **Factor the inequality**:
\[ x^2 - x - 2 = (x - 2)(x + 1) \geq 0 \]

2. **Find the roots**:
The roots of the equation \( x^2 - x - 2 = 0 \) are \( x = -1 \) and \( x = 2 \).

3. **Test intervals**:
We will test the intervals created by the roots:
- Interval 1: \( (-\infty, -1) \)
- Interval 2: \( (-1, 2) \)
- Interval 3: \( (2, \infty) \)

- For \( x < -1 \) (e.g., \( x = -2 \)):
\[ (-2 - 2)(-2 + 1) = (-4)(-1) = 4 \quad (\text{positive}) \]
- For \( -1 < x < 2 \) (e.g., \( x = 0 \)):
\[ (0 - 2)(0 + 1) = (-2)(1) = -2 \quad (\text{negative}) \]
- For \( x > 2 \) (e.g., \( x = 3 \)):
\[ (3 - 2)(3 + 1) = (1)(4) = 4 \quad (\text{positive}) \]

4. **Determine the solution**:
The inequality \( (x - 2)(x + 1) \geq 0 \) is satisfied in the intervals:
\[ (-\infty, -1] \cup [2, \infty) \]

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

1. **Solutions from the first inequality**:
\[ [-1, 1] \]

2. **Solutions from the second inequality**:
\[ (-\infty, -1] \cup [2, \infty) \]

3. **Intersection**:
The intersection of \( [-1, 1] \) and \( (-\infty, -1] \cup [2, \infty) \) is:
\[ \{-1\} \]

### Final Result
The values of \( x \) that satisfy both inequalities lie in:
\[ \{-1\} \]