X2-3x+2<0

To solve the inequality \(x^2 - 3x + 2 < 0\), we first need to factor the quadratic expression.

1. Factor the expression: The quadratic \(x^2 - 3x + 2\) can be factored as: \[ (x - 1)(x - 2) \]

2. Set the factors to zero: Next, we find the points where the expression equals zero: \[ (x - 1)(x - 2) = 0 \] This gives us: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

3. Determine intervals: The critical points divide the number line into intervals. The intervals to test are:

   • \( (-\infty, 1) \)
   • \( (1, 2) \)
   • \( (2, \infty) \)

4. Test a point in each interval:

   • For \(x < 1\) (e.g., \(x = 0\)): \((0 - 1)(0 - 2) = ( -1)( -2) = 2 > 0\) (not in solution)

   • For \(1 < x < 2\) (e.g., \(x = 1.5\)): \((1.5 - 1)(1.5 - 2) = (0.5)( -0.5) = -0.25 < 0\) (in solution)

   • For \(x > 2\) (e.g., \(x = 3\)): \((3 - 1)(3 - 2) = (2)(1) = 2 > 0\) (not in solution)

5. Combine the results: The expression \(x^2 - 3x + 2 < 0\) holds true in the interval: \[ (1, 2) \]

6. Final solution: The solution to the inequality \(x^2 - 3x + 2 < 0\) is: \[ \boxed{(1, 2)} \]