Which of the following values x do not satisfy the inequality `(x^(2) - 3x + 2 lt 0)` at all ?

Options

A. `1 le x le 2`
B. `2 lt x lt 4`
C. `-5 le x le -1`
D. `5 lt x le 10`

Answer

To solve the inequality \( x^2 - 3x + 2 < 0 \) and determine which values of \( x \) do not satisfy this inequality at all, we will follow these steps:

### Step 1: Factor the quadratic expression
We start with the quadratic expression \( x^2 - 3x + 2 \). We need to factor it.

To factor, we look for two numbers that multiply to \( 2 \) (the constant term) and add up to \( -3 \) (the coefficient of \( x \)). The numbers \( -1 \) and \( -2 \) fit this requirement.

Thus, we can factor the expression as:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 2: Set the factored expression less than zero
Now we rewrite the inequality using the factored form:
\[ (x - 1)(x - 2) < 0 \]

### Step 3: Determine the critical points
The critical points occur where the expression equals zero:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
So, the critical points are \( x = 1 \) and \( x = 2 \).

### Step 4: Test intervals around the critical points
We will test the sign of the expression \( (x - 1)(x - 2) \) in the intervals determined by the critical points:
1. \( (-\infty, 1) \)
2. \( (1, 2) \)
3. \( (2, \infty) \)

- **Interval \( (-\infty, 1) \)**: Choose \( x = 0 \):
\[ (0 - 1)(0 - 2) = (-1)(-2) = 2 \quad (\text{positive}) \]

- **Interval \( (1, 2) \)**: Choose \( x = 1.5 \):
\[ (1.5 - 1)(1.5 - 2) = (0.5)(-0.5) = -0.25 \quad (\text{negative}) \]

- **Interval \( (2, \infty) \)**: Choose \( x = 3 \):
\[ (3 - 1)(3 - 2) = (2)(1) = 2 \quad (\text{positive}) \]

### Step 5: Determine where the inequality holds
From our tests, we find:
- The expression \( (x - 1)(x - 2) < 0 \) holds true in the interval \( (1, 2) \).

### Step 6: Identify values of \( x \) that do not satisfy the inequality
The inequality \( (x - 1)(x - 2) < 0 \) does not hold for:
- \( x < 1 \)
- \( x > 2 \)
- \( x = 1 \) or \( x = 2 \) (where the expression equals zero)

### Conclusion
The values of \( x \) that do not satisfy the inequality \( x^2 - 3x + 2 < 0 \) at all are those in the intervals \( (-\infty, 1] \) and \( [2, \infty) \).