What is the solution set for the inequality: x^2 - 5x + 6 < 0?

What is the solution set for the inequality: x^2 - 5x + 6 < 0?

Step 1: Factor the quadratic expression x^2 - 5x + 6. This can be factored into (x - 2)(x - 3).
Step 2: Identify the critical points where the expression equals zero. Set (x - 2)(x - 3) = 0. The solutions are x = 2 and x = 3.
Step 3: Determine the intervals to test. The critical points divide the number line into three intervals: (-∞, 2), (2, 3), and (3, ∞).
Step 4: Test a point from each interval in the inequality (x - 2)(x - 3) < 0.
Step 5: For the interval (-∞, 2), test x = 0: (0 - 2)(0 - 3) = 6 (not less than 0).
Step 6: For the interval (2, 3), test x = 2.5: (2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25 (less than 0).
Step 7: For the interval (3, ∞), test x = 4: (4 - 2)(4 - 3) = 2 (not less than 0).
Step 8: The solution set where the inequality (x - 2)(x - 3) < 0 holds true is the interval (2, 3).

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