`x^2-4x+3ge0`

Options

A. [1,3]
B. `(oo,-3]uu]1,oo)`
C. `(-oo,1)uu(3oo)`
D. `R-(1,3)`

Answer

To solve the inequality \( x^2 - 4x + 3 \geq 0 \), we will follow these steps:

### Step 1: Factor the quadratic expression
We start with the quadratic expression \( x^2 - 4x + 3 \). We need to factor it.

\[ x^2 - 4x + 3 = (x - 1)(x - 3) \]

### Step 2: Set the factors to zero
Next, we find the roots of the equation by setting each factor equal to zero.

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Step 3: Determine the intervals
The roots divide the number line into intervals. The intervals are:
- \( (-\infty, 1) \)
- \( (1, 3) \)
- \( (3, \infty) \)

### Step 4: Test the sign of the expression in each interval
We will test a point from each interval to determine where the expression \( (x - 1)(x - 3) \) is greater than or equal to zero.

1. **Interval \( (-\infty, 1) \)**: Choose \( x = 0 \)
\[ (0 - 1)(0 - 3) = (-1)(-3) = 3 \quad (\text{positive}) \]

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

3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \)
\[ (4 - 1)(4 - 3) = (3)(1) = 3 \quad (\text{positive}) \]

### Step 5: Include the points where the expression equals zero
Since we have the inequality \( \geq 0 \), we include the points where the expression equals zero, which are \( x = 1 \) and \( x = 3 \).

### Step 6: Write the solution
From our tests, we find that the expression is non-negative in the intervals \( (-\infty, 1] \) and \( [3, \infty) \).

Thus, the solution to the inequality \( x^2 - 4x + 3 \geq 0 \) is:

\[ (-\infty, 1] \cup [3, \infty) \]

### Final Answer
The solution set is \( (-\infty, 1] \cup [3, \infty) \).

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