If `x in [0,5]`, then what is the probability that `x^2 - 3x + 2 > 0`

Options

A. `4//5`
B. `1//5`
C. `2//5`
D. none of these

Answer

To solve the problem, we need to determine the probability that the expression \( x^2 - 3x + 2 > 0 \) holds true for \( x \) in the interval \([0, 5]\).

### Step-by-Step Solution:

1. **Identify the quadratic expression**:
The expression we are working with is:
\[ x^2 - 3x + 2 \]

2. **Factor the quadratic expression**:
We can factor the quadratic expression:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

3. **Set the expression greater than zero**:
We need to solve the inequality:
\[ (x - 1)(x - 2) > 0 \]

4. **Find the critical points**:
The critical points occur where the expression equals zero:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
Thus, the critical points are \( x = 1 \) and \( x = 2 \).

5. **Test intervals**:
We will test the intervals determined by the critical points:
- Interval 1: \( (-\infty, 1) \)
- Interval 2: \( (1, 2) \)
- Interval 3: \( (2, \infty) \)

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

- For \( 1 < x < 2 \) (e.g., \( x = 1.5 \)):
\[ (1.5 - 1)(1.5 - 2) = (0.5)(-0.5) = -0.25 < 0 \quad \text{(False)} \]

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

6. **Determine the solution set**:
From the tests, we find that the expression is greater than zero in the intervals:
\[ (-\infty, 1) \quad \text{and} \quad (2, \infty) \]

7. **Consider the given range \( [0, 5] \)**:
We need to find where these intervals intersect with \( [0, 5] \):
- The interval \( (-\infty, 1) \) intersects with \( [0, 5] \) at \( [0, 1] \).
- The interval \( (2, \infty) \) intersects with \( [0, 5] \) at \( (2, 5] \).

8. **Calculate the lengths of the intervals**:
- Length of \( [0, 1] \) is \( 1 - 0 = 1 \).
- Length of \( (2, 5] \) is \( 5 - 2 = 3 \).

9. **Total favorable outcomes**:
The total length of the intervals where the inequality holds true is:
\[ 1 + 3 = 4 \]

10. **Total outcomes in the interval [0, 5]**:
The total length of the interval \( [0, 5] \) is:
\[ 5 - 0 = 5 \]

11. **Calculate the probability**:
The probability \( P \) that \( x^2 - 3x + 2 > 0 \) is given by:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{4}{5} \]

### Final Answer:
The probability that \( x^2 - 3x + 2 > 0 \) for \( x \in [0, 5] \) is \( \frac{4}{5} \).