1. **Problem Statement:** Solve the quadratic inequality $x^2 - 5x + 6 < 0$. 2. **Formula and Rules:** To solve a quadratic inequality, first find the roots of the corresponding quadratic equation $x^2 - 5x + 6 = 0$ by factoring or using the quadratic formula. 3. **Find the roots:** Factor the quadratic: $$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$$ The roots are $x = 2$ and $x = 3$. 4. **Analyze intervals:** The roots divide the number line into three intervals: $(-\infty, 2)$, $(2, 3)$, and $(3, \infty)$. 5. **Test each interval:** - For $x < 2$, pick $x=1$: $(1-2)(1-3) = (-1)(-2) = 2 > 0$ (not less than 0). - For $2 < x < 3$, pick $x=2.5$: $(2.5-2)(2.5-3) = (0.5)(-0.5) = -0.25 < 0$ (satisfies inequality). - For $x > 3$, pick $x=4$: $(4-2)(4-3) = (2)(1) = 2 > 0$ (not less than 0). 6. **Solution:** The inequality $x^2 - 5x + 6 < 0$ holds for $x$ in the interval $(2, 3)$. **Final answer:** $$\boxed{(2, 3)}$$