If x is real and `x^(2) - 3x + 2 gt 0, x^(2)- 3x - 4 le 0,` then which one of the following is correct?

Options

A. `|x| le 2`
B. ` 2 le x le 4`
C. ` -1 le x le 1`
D. ` 2 lt x le 4`

Answer

To solve the inequalities given in the question step by step, we will break down each part and find the solution.

### Step 1: Solve the first inequality \( x^2 - 3x + 2 > 0 \)

1. **Factor the quadratic expression**:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]
We set this greater than zero:
\[ (x - 1)(x - 2) > 0 \]

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

3. **Test intervals**:
We will test the sign of the product in the intervals determined by the roots:
- For \( x < 1 \) (e.g., \( x = 0 \)): \( (0 - 1)(0 - 2) = 2 > 0 \) (positive)
- For \( 1 < x < 2 \) (e.g., \( x = 1.5 \)): \( (1.5 - 1)(1.5 - 2) = -0.25 < 0 \) (negative)
- For \( x > 2 \) (e.g., \( x = 3 \)): \( (3 - 1)(3 - 2) = 2 > 0 \) (positive)

4. **Conclusion for the first inequality**:
The solution for \( (x - 1)(x - 2) > 0 \) is:
\[ x \in (-\infty, 1) \cup (2, \infty) \]

### Step 2: Solve the second inequality \( x^2 - 3x - 4 \leq 0 \)

1. **Factor the quadratic expression**:
\[ x^2 - 3x - 4 = (x + 1)(x - 4) \]
We set this less than or equal to zero:
\[ (x + 1)(x - 4) \leq 0 \]

2. **Find the roots**:
The roots of the equation are \( x = -1 \) and \( x = 4 \).

3. **Test intervals**:
We will test the sign of the product in the intervals determined by the roots:
- For \( x < -1 \) (e.g., \( x = -2 \)): \( (-2 + 1)(-2 - 4) = 6 > 0 \) (positive)
- For \( -1 < x < 4 \) (e.g., \( x = 0 \)): \( (0 + 1)(0 - 4) = -4 < 0 \) (negative)
- For \( x > 4 \) (e.g., \( x = 5 \)): \( (5 + 1)(5 - 4) = 6 > 0 \) (positive)

4. **Conclusion for the second inequality**:
The solution for \( (x + 1)(x - 4) \leq 0 \) is:
\[ x \in [-1, 4] \]

### Step 3: Find the common solution

1. **Combine the solutions**:
From the first inequality, we have \( x \in (-\infty, 1) \cup (2, \infty) \).
From the second inequality, we have \( x \in [-1, 4] \).

2. **Identify the intersection**:
- The interval \( (-\infty, 1) \) intersects with \( [-1, 4] \) at \( [-1, 1) \).
- The interval \( (2, \infty) \) intersects with \( [-1, 4] \) at \( (2, 4] \).

3. **Final solution**:
The combined solution is:
\[ x \in [-1, 1) \cup (2, 4] \]