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

Options

A. `(4)/(5)`
B. `(1)/(5)`
C. `(2)/(5)`
D. `(3)/(5)`

Answer

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

### Step-by-Step Solution:

1. **Identify the inequality**: We start with the inequality:
\[ x^2 - 3x + 2 \geq 0 \]

2. **Factor the quadratic expression**: We can factor the quadratic:
\[ (x - 1)(x - 2) \geq 0 \]

3. **Determine the critical points**: The critical points from the factors are \( x = 1 \) and \( x = 2 \). These points divide the number line into intervals.

4. **Test the intervals**: We need to test the sign of the expression in the intervals determined by the critical points:
- Interval 1: \( (-\infty, 1) \)
- Interval 2: \( (1, 2) \)
- Interval 3: \( (2, \infty) \)

- For \( x < 1 \) (e.g., \( x = 0 \)):
\[ (0 - 1)(0 - 2) = 1 \cdot 2 = 2 \quad (\text{positive}) \]
- For \( 1 < x < 2 \) (e.g., \( x = 1.5 \)):
\[ (1.5 - 1)(1.5 - 2) = 0.5 \cdot (-0.5) = -0.25 \quad (\text{negative}) \]
- For \( x > 2 \) (e.g., \( x = 3 \)):
\[ (3 - 1)(3 - 2) = 2 \cdot 1 = 2 \quad (\text{positive}) \]

5. **Determine where the inequality holds**: The expression \( (x - 1)(x - 2) \geq 0 \) holds for:
- \( x \leq 1 \) or \( x \geq 2 \)

6. **Find the valid range within [0, 5]**: We need to find the values of \( x \) in the interval \([0, 5]\) that satisfy the inequality:
- From \( 0 \) to \( 1 \): \( [0, 1] \)
- From \( 2 \) to \( 5 \): \( [2, 5] \)

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

8. **Total length of intervals satisfying the inequality**:
\[ 1 + 3 = 4 \]

9. **Calculate the total length of the interval [0, 5]**:
\[ 5 - 0 = 5 \]

10. **Calculate the probability**:
\[ \text{Probability} = \frac{\text{Length of intervals satisfying the inequality}}{\text{Total length of the interval}} = \frac{4}{5} \]

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